Spurt and Instability in a Two-Layer Johnson-Segalman Liquid1
|
|
- Malcolm Wilcox
- 5 years ago
- Views:
Transcription
1 Theoret. Comput. Fluid Dynamics (1995) 7: Theoretical and Computational Fluid Dynamics Springer-Verlag 1995 Spurt and Instability in a Two-Layer Johnson-Segalman Liquid1 Yuriko Yamamuro Renardy Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA , U.S.A. Communicated by Kang Ping Chen and Thomas B. Gatski Received 6 December 1993 and accepted 20 July 1994 Abstract. The Johnson-Segalman model is an example of a model that exhibits a nonmonotone curve for the shear stress in terms of shear rate. There are many works based on such models for an explanation of the spurt phenomenon but they have concerned the one-dimensional problem. This paper concerns a model problem, taking a one-dimensionally stable "spurted solution," viewed in two dimensions. A two-layer arrangement between walls in parallel shear, with a thin layer in the higher shear rate and the bulk of the fluid in the lower shear rate, is examined for linear stability in two dimensions. The spectrum is computed numerically for normal mode solutions. Instabilities with dominant growth rates for short waves are found. Thus, the one-dimensionally stable solutions of this model are actually two-dimensionally unstable. This work is dedicated to Professor Daniel D. Joseph on his 65th birthday in appreciation of his help through the years. There once was an engineer Whose ideas were fun and clear. He showed us flows, amazing us. Why, he's our Pythagoras! There once was a pipe flow. Two viscous fluids on the go. Both would like us to bestow Easy rules for them to know. "Past results we can dispute." Their energy, we do compute. Dan studies how flows bifurcate. Many theorems he would state Describing flows that rotate. All such flows can find their fate Let they Dan investigate. This research was sponsored by ONR Grant No. N J-1664 and NSF Grant CTS
2 464 Y.Y. Renardy 1. Introduction Some constitutive models, including the Johnson-Segalman model and the Giesekus model (Kolkka et al., 1988; Kolkka and Ierley, 1989) have a nonmonotone shear-stress-shear-rate curve. We focus on the Johnson-Segalman fluid. Johnson and Seglman (1977) begin with a constitutive equation that is derived systematically from both the molecular theory of Gaussian networks, and the molecular bead-spring model with Hookean springs. In order to allow the model to describe such viscoelastic properties as a shearthinning viscosity, they generalize the model. They do this by introducing a scalar parameter and "by developing a continuum theory that allows two histories of deformation which may be nonaffine. One history is made to give rise to the current state of stress according to the constitutive equation from molecular theory. The second is made the observed smooth continuum deformation. The two motions are connected by an appropriate constitutive equation." They apply their theory to shear flows. A critical discussion of its application to elongational flows is given by Adewale and Leonov (1993). The model contains one relaxation time, which is thought of as the dominant relaxation time of the polymeric melt, one "polymer viscosity" which is identified as the contribution to the total viscosity from the component with the dominant relaxation time, and a second "solvent viscosity" which is identified with the viscosity arising from the shorter relaxation times present in the melt; the apparent viscosity is shear-dependent. Figure l shows the steady shear stress w( ) against shear rate ~ for the Johnson-Segalman fluid: w( ) = (1 - a 2) ~ e{, (1) where ~ denotes the retardation parameter and a is a model constant with ½ < a < 1. The range of e < ~, a < 1, leads to the nonomonotone behavior, and popular values to model polymer melts have been picked for this figure. It has been proposed (Hunter and Slemrod, 1983; Malkus et al., 1990, 1991) that such differential constitutive laws may explain the spurt phenomenon. Melt fracture is an instability that is observed in the extrusion of molten polymers. The spurt phenomenon is a stage of melt fracture that occurs in some types of polymer melts. The stages involved are described for linear low-density polyethylene by Denn (1990), Kalika and Denn (1987), and Moynihan et al. (1990). Below a certain critical shear stress, the surface of the extrudate is smooth. At a first critical shear stress, very short wavelength disturbances called sharkskin appear on the extrudate surface. At a second critical shear stress, the surface becomes alternately sharkskin and smooth and the flow is known as stick-slip or spurt. The flow rate increases in this regime. At a third critical shear stress, there is onset of a more severe extrudate distortion, called wavy fracture. Larson (1992) gives a review of experiments and theories, and in particular describes the dependence of the stages of melt fracture on the materials and flow conditions. There are a number of suggested causes for the onset of melt fracture: partial slip at the polymer/die interface, and the role of prestressing conditions upstream of the exit (Moynihan et al., 1990). The slip hypothesis is that for low flow rates, the material adheres to the wall, while at higher flow rates, it goes through alternate sticking and slipping, and at the highest flow rates, the material slips. Models for the wall-stress-slip-velocity dependence have been developed by Leonov (1984, 1990) and Leonov and Srinivasan (1993). Another possible and totally different explanation is based on the role of bulk properties, and has attracted much interest from analysts. This is the nonmonotone constitutive model of the type in (1). Given the shear stress, there are two possible values for the shear rate once a critical value is reached, ignoring the decreasing part of the curve in Figure 1. The sudden increase of the flow rate in the spurt regime Figure 1. Graph of the shear stress w( ) versus shear rate ~ for the Johnson Segalman fluid with e = 0.05, a = 0.8. The increasing parts of the curve along < ~ _< and _< ~ < yield two valnes of ~ for a given w(~). The decreasing part of the curve is unstable (Kolkka et al., 1988; Malkus et al., 1990). The maximum and minimum values of the shear stress in the nonmonotone part are and 0.725, respectively.
3 Spurt and Instability in a Two-Layer Johnson-Segalman Liquid 465 is modeled by the two shear rates coexisting in a flow, with the higher shear rate close to the wall to mimic slip at observable scales. K01kka et al. (1988) refer to the work of Yerushalmi et al. (1970) on the linear stability of plane Couette flow for nonmonotone shear-thinning fluids, and extend their work to show that the one-dimensional solutions along the increasing portions of the curve in Figure I are linearly stable but along the decreasing portion of the curve they are linearly unstable. Malkus et al. (1990) show that the problem is well posed when the retardation parameter e is strictly positive, unlike the generalized Newtonian model which suffers from the Hadamard instability when the steady shear stress decreases with shear rate. Malkus et al. (1991) and Nohel and coworkers (1990, 1993) analyze systems of ordinary differential equations that approximate the dynamics of one-dimensional shear flow through a slit die. In particular, they treat highly elastic very viscous flows to model the spurt experiments. The velocity profile through the slit die is calculated as a function of the distance from the wall and time. In a cross section of the die the shear stress has its maximum value at the wall and is zero at the centerline. They determine the global dynamics using a phase.plane analysis. One example is the case of quasi-static loading, in which the pressure gradient is increased in small steps so that a steady state is achieved at each step. Under loading, the shear stress at the wall increases until the first hump in Figure 1 is reached. Past this point, the ve][ocity profile develops a kink at the wall, as the fluid at the wall jumps to the higher shear rate, and the bulk of the fluid remains at the lower shear rate. This jump in the shear rate at the top of the curve is called "top jumping." The layered solution is a "spurted solution." As the driving pressure gradient is increased further, the kink moves into the interior of the flow. Other properties such as latency and hysteresis under loading and unloading are predicted and they suggest experiments that may be performed to verify their model. Kolkka et al. (1988) and the works of Malkus et al. fit the data at the onset of the spurt regime in the experiments of Vinogradov et al. (1972). The separate case of one-dimensional piston-driven shear flow for prescribed volumetric flow rate is treated by Malkus et al. (1994). They predict oscillations in the extrudate due to a Hopf bifurcation to periodic solutions, as the volumetric flow rate is increased beyond a critical value. These solutions have a layered structure except at very low flow rates. They draw the analogy between this periodicity and the stick-slip regime, where periodic pressure pulsations have been observed by Lim (1988) and Lim and Schowalter (1989). In this paper we pursue the idea of the one-dimensional spurted solution. We take the following simple model problem (as opposed to a physical model). A two-dimensional spurted Couette flow is driven by the motion of the upper wall. The lower layer of fluid occupies the bulk of the flow and has a shear rate which is just below the critical value. They upper layer has the higher shear rate and occupies a thin region. The two layers are separated by a free surface, but are composed of the same liquids. Admittedly, spurt experiments are pressure-driven or piston-driven flows and would be better modeled by a Poiseuille flow, with a lower shear stress at the center of the velocity profile, and the highest shear attained close to the wall. However, since stick-slip apparently occurs first at the wall and then moves inward, the aim of our Couette-ftow analysis is to model the region close to the wall at the onset of an instability. Linear stability to perturbations which are periodic in the direction of the flow is examined by computing the eigenvalues numerically. The results are presented in Section 3. It is!round that there are always short-wave instabilities. The most unstable mode is found to be the interracial mode. Depending on the relative depths of the fluids, other modes which may be termed internal modes may also cause instabilities but with lesser growth rates than the interracial mode. In the case of quasi-static loading, the onedimensional results of Malkus et al. (1990) suggest that a likely situation is top-jumping at the hump of the curve in Figure 1. In such an arrangement, long and order-one waves are found to be stable, and short waves are unstable, the growth rate increases with wave number until it levels offfor the shortest waves. A number of internal or bulk modes may also be unstable for a range of wave numbers. As the kink in the velocity profile moves inward from the wall, it is found that the interracial mode is unstable down to longer waves, with comparable growth rates for order-one waves and shorter. Any instability from the internal modes also shift from short to order-one wave numbers. Near the trough of the curve in Figure 1, spurted solutions are unstable for all wavelengths of perturbation, with dominant growth rates occurring for order-one and short waves. Since the one-dimensionally stable layered solutions are unstable to two-dimensional perturbations, how would this affect the dynamic evolution predicted previously with the one-dimensional models, and how
4 466 Y.Y. Renardy can these steady spurted solutions be achieved during an actual experiment? It is possible that even if a one-dimensional steady solution is unstable, it may be describing what is happening in an average sense. The short-wave interfacial instability is an intriguing feature. The eigenfunction for this instability decays exponentially fast away from the interface position. Its effect is localized in a boundary layer near the interface, not disturbing the bulk of the flow, which in this case is steady. As the disturbance wavelength becomes longer, the corresponding eigenfunction begins to affect the bulk of the flow. Thus, the presence of a short-wave interfacial instability for a just-spurted arrangement is reminiscent of the loss of gloss of the extrudate surface at the first instability of the onset of melt fracture, described by the formation of the spurted solution with the layer at the wall too thin to be observable, together with the short-wave instability which leaves the bulk of the flow steady. In any case, there is uncertainty about spurt models, and the results of this paper raise further questions. Chen and Joseph (1992) suggest a different "apparent slip" explanation, which motivated them to do a stability analysis of two upper-convected Maxwell liquids in layers. The idea is that at some critical Weissenberg number, high molecular-weight polymers are detached from the wall and migrate to the lower shear-rate region, creating a lubricating layer near the wall. Hence, the layer near the wall is a polymer-depleted layer as opposed to a layer of a different material, and therefore surace tension between the two layers is negligible. In this way, one can start with a single layer and end up with two layers without resorting to the use of nonmonotone constitutive equations. They treat core-annular flow in which the pipe radius is slightly larger than the nominal radius separating the high and low molecular-weight polymers. The stratification in density, viscosity, and relaxation times affect the stability of this two-layer flow. The viscosity of the polymer core is much larger than the average viscosity of the low molecular-weight annulus, and the core is more elastic. Both instability and stability are possible depending on the conditions. The instability may be suggestive of sharkskin, with the physical mechanism being the leaching of polymers from the solution at the wall. The stress-induced diffusion of polymer chains has been experimentally observed by a number of research groups for polymer solutions (Tirrell and Malone, 1977). On the other hand, the situation is different for polymer melts. In the case of polymer melts, the distribution of molecular weights is narrow. It is not clear how a mechanism like leaching would happen in a polymer melt. If the separation were present, there would be a strong dependence of spurt on the molecular-weight distribution, and this has not been reported. The observed critical stress (Vinogradov et al., 1972) is independent of molecular weight. In the absence of a separation mechanism, one choice is the adoption of the nonmonotone model which describes the initial flow of one fluid and subsequent separation into two layers of the identical fluid, and provides an explanation of where the second layer came from with a single constitutive equation across the entire flow. 2. Governing Equations We model a freshly spurted flow with the following two-layer Couette flow of a Johnson-Segalman liquid. Two layers of a fluid of density p, polymer or shear viscosity/~, Newtonian or solvent viscosity t/, and relaxation time 2 lie in the (x*, z*) plane between infinite parallel plates located at z* = 0, l*. Asterisks are used for dimensional variables. The upper plate moves with velocity (Uv*, 0) and the bottom plate is at rest. In the basic flow, layer 1 occupies 0 _< z* _< 11" and layer 2 occupies l* _< z* _< l*. The velocity, distance, time, and pressure are made dimensionless with respect to l*/2,1", A, and ]~/2. The dimensionless upper plate speed is defined to be the Weissenberg number, representing the average shear rate multiplied by the relaxation time. The Reynolds number, retardation parameter, and Weissenberg number, respectively, are defined by 1"212 rl U * 2 R = (~2,)' ~ ~t' W = Up = I* (2) The equation of motion is \ & + (u.v)n) = V-T - Vp + 8Au. (3)
5 ,(l-a) Spurt and Instability in a Two-Layer Johnson-Segalman Liquid 467 The total stress is T + e(vu + Vu T) - pi. The constitutive law for the Johnson-Segalman fluid (Johnson and Segalman, 1977) is where D/Dt is the Oldroyd derivative DT T + ~- = Vu + (Vu) r, (4) DT 0T (1 +a)(vu)t ~ ~ (1 +a)t(vu)t. Dt - & + (u'v)t 2 + (Vu)TT + T(Vu) 2 (5) These equations reduce to those of the upper-convected Maxwell liquid when a = 1. For the fluid to be rod-climbing, ½ < a < 1. For the nonmonotone behavior of Figure 1, a < 1 and ~ < -~. At the kink in the velocity profile, the extra stress has a jump corresponding to the jump in the velocity gradient. This discontinuity is transported along with the fluid particles, as evident from the material derivative terms in (5) of the constitutive law. Hence the kink in the base flow lies on a material surface. At this interface, the velocity and total tangential stress are continuous, the jump in the normal stress is zero, and the kinematic free surface condition holds. The dimensionless basic velocity (U(z), 0) satisfying no-slip at the walls is U", ~klz, O<_z<_ll, ~z) ~- ~k2( z _ 1) + Up, l 1 <_ z <_ 1, (6) where kl is chosen so that it lies in the interval of shear rate where the variation with the shear stress is nonmonotone. At the interface z = l 1, continuity of shear stress yields the value of k2: k2 kt 1 + k](1 - a 2) ~- ekx k~(1 - a 2) +- ek2" (7) Continuity of velocity yields the Weissenberg number or the upper plate speed: W = Up = kll 1 + k212, = 1. (8) The normal stress condition gives the difference in the constant basic pressures. The basic extra stress tensor in fluid i (i = 1, 2) is C2 C3/ (9) ki C 1 =(1 +a)cak,, C2-1 + k2(1-az) ' C 3 = -(1-a)C2k v Solutions that are perturbations of the above basic flow are sought in normal modes proportional to exp(i~x + at). The perturbations to the velocity, pressure, and interface position are denoted by (u, v), p, and h, respectively, with u = 34`/~z = 4`', v -= - ic@, where 4` is the streamfunction. We use a prime to denote differentiation with respect to z. The perturbation to the extra stress tensor is denoted by rl 2 The equations of motion (3) in each fluid yield R(~ + i~u(z))(4`"- c~24`) = e(4`iv _ 2c~24`, + ~40) + i~t,11 + T 2 + ~2Tl 2 _ i~t22. (11) The constitutive equations (4)-(5) yield (a+ic~u(z)+ 1)Ttl--(1 +a)u't12+(1 -a)c2ct24`-2i~(ac1 + 1)0'- (1 + a)c24`" = 0, (12) r _,(1 +a) (a i~u(z))r~2 - t; '~i2 + tj ~ ~ + 4'"((1 - a2)k~c2-1)- ~24`(k~C2(1 + a 2) + 1) --- 0, (13) (a iag(z))t22 + (1 -- a)u't `c~2(1 + a)c 2 + 2ic@'(aC 3 + 1) + 4`"(1 - a)c 2 = 0. (14)
6 468 Y.Y. Renardy The boundary conditions of no slip are ~ = ~' = 0 at z = 0, 1. The conditions at the interface are posed at the unknown position z = 11 + h(x, t) where h(x, t) is small, and the conditions are linearized at z = 11. Continuity of velocity yields and h[u'~ + [0'~ = 0 (15) [0~ = 0, (16) where ~x~ denotes x(fluid 1) - x(fluid 2). The total stress is Ttota 1 = T + e(vu + V(u) r) - pi, where T here denotes the sum of the basic stress (9) and the perturbation (10), u denotes the sum of the basic velocity (6) and the perturbation, and p denotes the basic plus the perturbation pressure. Continuity of shear stress is then It "Ttota I" n~ , where the tangent to the interface is t=(1,h,)/x/1 +h~, n = (-h~, 1)/x/i- + h 2. Thus, and the normal to the interface is ~T12 ~ + ieh[c 3 - C1~ + e[~," + e2~9~ = 0. (17) Here, ~C 2 + eki~ = 0 and ~P~ = 0 in the basic flow. The balance of normal stress is [n'ttota,-n~ = 0, where again T denotes the basic stress (9) plus the perturbation (10): In terms of the perturbations, this is ~[T22 + 2evz -- p~ = O. We express this in terms of the streamfunction: -- c~t22 ~ -1- ig(30~2{ '~ -- [~"~) + R(ia~[~k'~ -- ctu(i1)[o' ~ + c~tpu'~) + ~Tll ~ - i[t'12 ~ (18) The kinematic free surface condition yields ha + U(ll)icth + ieol(ll) = 0. (19) 3. Results on the Spectrum The strips of continuous spectra are found by reducing the governing equations to a single equation for the streamfunction and setting the coefficient of the highest derivative to zero. Results on compact operators (Dunford and Schwartz, 1958) are applied to the ordinary differential equation for the streamfunction defined on the bounded interval. It is known that if the coefficient of the highest derivative is not zero, then the eigenvalues are discrete. If the coefficient of the highest derivative vanishes, then the eigenfunctions may be singular and yield a continuous spectrum. In order to do the reduction, we define 21(Z ) = (7 "~- 1 "~ i~u(z), 0 < z < 1. (20) This is the term that multiplies the stress component Tll in the constitutive equation (12) which is re-expressed as rll,h = rlrl~ + r2(0), F 1 = (1 + a)u', F2(O) = -(1 - a)c2~2~ + 2ietk~(aC 1 + 1) + (1 + a)c2ozz. (21) (22) (23) The constitutive equation for the component T22 also involves ~,, 21, and T12; F3(O) = 21 T22 = r3(~,) + F4T12, C~2(1 q- a)c21 p -- 2i~(1 + ac3)o= -- (1 -- a)c2~b=, F 4 = --(1 -- a)u'. (24) (25) (26)
7 Spurt and Instability in a Two-Layer Johnson-Segalman Liquid 469 We substitute into the Navier-Stokes equation (11) and express the stress terms on the right-hand side of it in terms of T12. In order to do this, we require T'11 and T22 in terms of T12. From (21), 2~T'11 + 2'1T11 = rlti~ + rl, where the prime denotes d/dz, and substitution from (21) for T 11 yields We multiply this through by 21: From (24), and substitution from (24) for T22 yields 21Ti ~ _ -2'1 (rt T12 + r2 ) + rl Ti 2 + rl" 21 2~rql = -41(rlTlz + r2) + 21(F1Ti2 + rl). (27)! t! 21T22 + 4'1T22 = F 3 + F4T12, so that t 4t t! 41 T22 = --"~2!1 (F 3 + F4T12 ) -t- F 3 + F4T12,,tl 412 T22,..= _ 2,1(F3 q_ F4T12) -t- 21(F 3-1- F4 T'12 ). (28) We use (27)-(28) to re-express the stress terms in (11) in terms of 7"12: iet'tl + T12 -'}-e2tlz- ic~t'z2 = 2--~t (-- 24](-- C2~2@ -q- io@z(ac 1 + ac 3 + 2) + C2@zz ) + 221(- C2o~21[tz 71- ic~ozz(ac 1 + ac 3 + 2) + C2@zzz) ) T12 2 r2 2 ~2 Y' +--~-1 (2c~ U + e ~1) + ~ ie2u' + T~2. (29) We next express T12 in terms of ~. We re-express (13) as 41T12 = FsT22 + F6Tll + F7(O), (30) Fs= U,(1 +a) 2 ' F6 = _U,(1-a) 2 ' FT= -tp"((1-az)kicz-1)+c~zt~(kic2(1-t-a2)+l)" We substitute from (21) and (24) and find 22T12 = FsF 3 + F6F2 + 21F7, 22(z) = 22 + U,2( 1 _ az), (31) which shows that the highest derivative of ~k in T 1 z is ~,". Equations (30)-(31) show that we should multiply the Navier-Stokes equation (11) through by 21222, and find that the highest derivative of ~, Oiv, is multiplied by g4~ az((1 -- a2)u'c2(41 + 1) - 21) = 22( (1 - a2)u'c2(41 + 1) + 41). This yields The first strip is 21=0, e(22+u'2(1-a2))-(1-az)u'cz(21+l)+21=o. Reo- = -1, Im a in [0, -sup]. The second equation in (32) yields two constant values of 21 for each fluid given by A-t-(AZ-4eB(e-A))l/22e ' B=U'2(1-a2)' A-I + B' (32) (33) (34)
8 470 Y.Y. Renardy resulting in two strips per fluid at = ic~u(z). (35) This set of eigenvalues gives rise to instability on the decreasing portion of the curve in Figure 1 for the following reason. If the square root in (34) is taken over a negative term, then 21 is a complex conjugate with negative real part, and o- is stable. Thus, if 2 t is to cause an instability, it must be real and must be greater than one. If 2 t = 1, then (32) yields C2(ki) = 0 where C 2 is defined in (10). Thus when 21 is greater than one, this yields the condition that C'z(ki) < 0. This is consistent with the results of Malkus et al. (1990) where the phase plane analysis of the one-dimensional time-dependent problem showed that if any point on the velocity profile lands on the decreasing part of the shear-stress-shear-rate curve, then it is a saddle point and is not a steady-state solution. The numerical calculation of the eignvalues was done with the Chebyshev-tau method (Gottlieb and Orszag, 1983; Orszag, 1971). This approximates the eigenvalues for C~-eigenfunctions with infinite-order accuracy. The streamfunction, T t 1, T12, and T22 are discretized in the z-direction in terms of Chebyshev polynomials up to order N, N - 2, N - 2, and N, respectively, in each fluid. Together with h, the total number of unknowns is 8N - 1. The equation of motion is approximated to the (N - 4)th degree, the constitutive equations for Tll and T12 to the (N- 2)th degree, and for T2z up to the (N- 1)th degree. Together with the four boundary conditions and five interface conditions, this yields 8N - 1 equations. The eigenvalues are computed in complex quadruple precision with an NAG routine. The numerical results of this paper have been convergence tested. An additional test for the numerical results is the case c~ = 0, for which the solution to the linearized problem is known in closed form. In particular, for R = 0, (31) which comes from the constitutive equations and the Navier-Stokes equation are and T12((g + 1) 2 + U'2(1 - a2)) + 0"(U'(1 -- a2)c2 + (1 - a2)u'c2-1)(a + 1)) = 0 (36) e01v + T~ 2 = 0. (37) These imply that Oi~ = 0. The streamfunctions are therefore cubic polynomials in z: = al + a2z + a3z 2 -~ a4z 3 in layer 1, = bt + b2(1 - z) + b3(1 - z) 2 + b4(1 - z) 3 in layer 2. The boundary conditions at z = 0, 1 yield a t = 0, a z -- 0, b 1 = 0, and b 2 = 0. At the interface, the kinematic condition reduces to h = 0. The remaining conditions are the continuity of the two velocity components, the shear-stress balance and the normal-stress balance. This yields a four-by-four determinant equation for the eigenvalues. The eigenvalues computed with the Chebyshev-tau scheme have been checked to satisfy the determinant equation. The stability at ~ = 0 is inferred from the works of Malkus et al. (1990). An inspection of the governing equations shows that the growth rate of the interfacial eigenvalue a is proportional to cd as ~ 0. All other eigenvalues are real or complex conjugates and are found to be stable. We begin by reminding the reader of results available for the one-layer case. The one-layer flow is expected to be linearly stable, from previous results on related constitutive models (Renardy and Renardy, 1986). A computation at a = 0.8, ~ = 0.05, and kl = k 2 = Up = 0.8 confirms this for the increasing portions of the shear-stress-shear-rate curve. Due to the reasons following (34), the one-layer flow is unstable along the decreasing part of the curve. A number of works on the linear stability of two-layer flows of related constitutive models have appeared, many with applications to the production of multilayered materials (Joseph and Renardy, 1992). The upper-convected Maxwell (UCM) model is the case e = 0 and a = 1. This is a singular limit in the Johnson-Segalman model since e multiplies the highest derivative in the governing equations. The short-wave asymptotic analysis for the interfacial mode was done for the UCM model by Renardy (1988) for zero surface tension and equal densities. The eigenfunction decays exponentially fast away from the interface, so that the conditions at the walls do not enter the calculation. This is a boundary-layer-type singular perturbation problem. The growth rate of the interfacial eigenvalue is expanded in powers of l/e". At leading order, the governing equation for the streamfunction is a fourth-order ordinary differential equation with the complication being the variable coefficients. This is solved with a decoupling technique used by Gorodtsov and Leonov (1967). The growth rate is order one, and the competition between viscous
9 Spurt and Instability in a Two-Layer Johnson-Segalman Liquid 471 and elastic stratifications appears. In Newtonian flows the effect of viscosity stratification appears at growth rates of O(1/cd) (Hooper and Boyd, 1983), so the effect of elastic stratification is stronger than this. The order-one growth rate for short waves is seen also in the numerical results for the Johnson-Segalman model. Chen (1991b) derives the long-wave expansion (e small) for the interfacial mode for two-layer UCM and Oldroyd-B liquids in Couette flow. The Oldroyd-B model reduces to the UCM model when the characteristic retardation time is zero. He notes that the jump in the first normal stress difference across the interface is not balanced for the basic flow, and this induces an elastic instability. This instability occurs J when the more elastic fluid occupies less than half of the total volume inside the channel. He computes neutral stability curves for this and for core-annular flow (Chen, 1991a; Chen and Zhang, 1993). Chen and Joseph (1992) analyze the short-wave case for core-annular flow. S u and Khomami (1991, 1992a, 1992b) treat the stability of two-layer plane Poiseuille flow of power-law fluids, second-order fluids, and Oldroyd-B fluids. They calculate the long-wave formulas and numerically compute the neutral stability curves over wave numbers of order one. For the Oldroyd-B model, the purely elastic instability is again found (Su and Khomami, 1992b). In the absence of other effects such as surface tension, the interface is unstable for long and order one wave numbers if the more-elastic fluid occupies less than half of the channel. If the less-elastic fluid is in the thinner layer, then stability depends on the elasticities, depth ratio, and wave number. The maximum growth rates occur at wave numbers, made dimensionless with respect to the thickness of the more elastic layer, of order one. Growth rates are shown for situations where long waves are unstable and the highest growth rates occur at order-one wave numbers. The opposite situation of the stable long waves and unstable short waves arises in the numerical results below for the Johnson-Segalman model. A direct comparison between the numerical results for the Johnson-Segalman model and the Oldroyd-B model is hampered by the question of how to evaluate the effective elasticities in each layer for the Johnson-Segalman case, since both the viscosities and the elasticities compete to control stability. The Reynolds number for polymer melts is close to zero. Results are shown for R = 0, a = 0.8 and e = A comparison between R = 0 and R = 1 at k 1 = 1, 11 = 0.9 has shown that the modes which cause instability for low R are captured by looking at R = 0. The inclusion of nonzero R leads to a family of solutions which do not cause instability at low R. In an experiment which begins at low shear rates, the flow is initially in a one-fluid arrangement and, under quasi-static loading, follows the increasing part of the curve in Figure 1 until it reaches the crest at = At this point, the flow develops spurt with two regions of different shear rates. At the onset of spurt, the region of high shear is expected to be relatively thin. The phase plane analysis of Malkus et al. (1990, 1991) on their one-dimensional model predicts the flow will approach a steady spurt solution in which the jump in the strain rate occurs at the maximum stress ("top-jumping") with the kink in the velocity profile located as close as possible to the wall. This occurs for the case of quasi-static loading where the stress is increased in small steps. As an example of top-jumping, consider the lower fluid of depth occupying the bulk of the flow with shear rate k 1 = 1.85, and a thin layer of depth 12 = 0.01 at the wall with shear rate k 2 = The Weissenberg number is Up = Figure 2 illustrates the growth rate versus the wave number of the Re o- 0.2 Figure 2. Growth rate versus wave number for the unstable mode for a=0.8, e=0.05, 11=0.99, k 1=1.85, k 2= , w(k0= w(k2)=0.92, and Up= Other eigenvalues are stable '
10 472 Y.Y. Renardy perturbation. In this range of wave numbers only the interfacial mode was found to be unstable. There is stability for long and order-one waves. Instability sets in for wave numbers greater than about eight and the maximum growth rate is achieved for ~ ~ 42, with growth rates for shorter waves at about the same order of magnitude thereafter. Thus, the wavelength for instabilities is on the scale of the thickness of the thin spurted layer. The short-wave growth rate appears to be at most order one, just as in the UCM case discussed above (Chen and Joseph, 1992; Renardy, 1988). Instabilities may set in for the bulk modes at shorter wavelengths than the ones investigated here. The significant amount of short-wave instability is reminiscent of the loss of gloss observed in the sharkskin regime, with very fine extrudate distortions. Larson (1992) writes that "sharkskin is a surface roughness that usually modulates the extrudate diameter by no more than one percent or so." In the case of an abrupt loading where the change in shear stress in an experiment is not small, it may be possible for spurt to occur for a shear stress below that of top-jumping. As an example of this, consider the bulk of the fluid of depth l 1 = 0.99 at the shear rate k 1 = 1, with a thin layer of fluid at the wall at the shear rate k 2 = , w(k~) = w(k2) = The Weissenberg number is Up = Figure 3 shows the growth rate or Re o- versus the wave number ~ of the disturbance. The interfacial mode is stable for long waves and is unstable for wave numbers larger than 3.8. The maximum growth rate is attained at c~ ~ 40 and thereafter remains at about the same level. A second mode is unstable for short waves in the region 41 < c~ and appears to become stable for wavelengths shorter than the ones recorded here. At ~ ~ 67, a third mode appears to be nearing instability. The instability of the bulk modes is a feature that has not appeared in previous works on two-layer flows. For the Oldroyd-B model, it would be expected that the bulk modes be unstable for sufficiently high flow rates. In the Johnson-Segalman case we have the bulk modes being unstable at shear rates that are less than the ones where they were stable (see also Renardy, 1996). The effect of an abrupt loading rather than top-jumping is that instabilities have moved down to the longer waves and long waves are less stable. The short waves attain slightly larger growth rates. The predominantly short-wave instability of Figure 2 suggests that it more closely describes sharkskin than the situation in Figure 3. As an example of a situation where the region of high shear has traveled inward into the fluid, la is chosen to be 0.9. Figure 4 shows the growth rate for the unstable modes for 0 < c~ _< 70 for 11 = 0.9, k~ = 1, k 2 = , w(kl) = w(k2) = 0.79, and U v = The most unstable mode is the interracial mode which is unstable for the entire range of c~ < 70 which was examined, and at ~ = 70 the growth rate appears to be tapering to a constant. The maximum growth rate is reached at ~ ~ 4 and approximately this level is 1: 1![ Re ~ ' Figure 3. Growth rate versus wave number for the least stable mode for a=0.8, e=0.05, 11=0.99, kl=l, k2=10.584, w(kl)= w(k2)=0.79, and Up= Other eigenvalues are stable. 0.5 ~ e 2 Re cr 0 / Mode r Mode 1-1.! Figure 4. Growth rate versus wave number for the least stable modes for a=0.8, ~=0.05, 11=0.9, kl=l, k2=10.584, w(kl) = w(k2) = 0.79, and U v = The most unstable mode is the interfacial mode. The next unstable mode, labeled Mode 2, merges into the continuous spectrum at approximately e = 3.3. The picture in Figure 5(c) shows the location of the eigenvalues for e = 3.3.
11 Spurt and Instability in a Two-Layer Johnson-Segalman Liquid Imff 0 ~ '8 Re o" -'6 -'4 -'2 (a) Im o- -( o -1( -25-2o -io Re (b) I Im cr o -10 -s 6 5 Re cr (c) k Figure 5. The spectrum for a = 0.8, 8 = 0.05, l 1 = 0.9, k 1 = 1, k 2 = , w(kl) = w(k2) = 0.79, and U, = (a) ~ = 0, (b) e = 1.7, and (c) c~ = 3.3. attained for shorter waves. There is a second mode which is unstable over 4.1 < c~ < 11 and has a maximum growth rate at c~ ~ 5.6. A third mode is unstable over 7 < e < 11. The second and third modes are bulk modes, which pop out of the continuous spectrum and this is illustrated next. Figure 5(a) shows the spectrum at c~ -- 0, where the continuous spectra, by (32)-(35), have diminished to the points G , , , and i. Around the continuous spectra, there are a number of discrete modes. This is similar to the behavior found for the two-layer UCM liquid, in which the continuous spectrum consists of two strips. In between these strips, four discrete modes were captured by the short-wave asymptotic expansion for the interracial mode (Renardy, 1988). The next least-stable mode for c~ = 0 in Figure 5(a) is o- = _ /i and is denoted by Mode 1 in Figure 4. At e ~ 1.7, this mode merges into a continuous spectrum at Re o- = , < Im o- < 0. The computed spectrum for c~ = 1.7 is shown in Figure 5(b). There are four other strips of continuous spectra at: (i) Reo-= 1, <Imo-<0. (ii) Re o- = , < Im cr < 0. (iii) Re a = , < Im ~r < (iv) Re o- = , <Im a < There are a number of isolated discrete modes close to these spectra in Figure 5(b). The unstable mode labeled Mode 2 in Figure 4 emerges at e ~ 3.3 from the continuous spectrum located at Re ~ = - 1, < Im o- < 0. Figure 5(c) shows the computed spectrum at c~ = 3.3, where the other
12 474 Y.Y. Renardy Rea L- 0 1' ' Figure 6. Growth rate versus wave number for the least stable modes for a=0.8, e= 0.05, 11 =0.99, k t =0.87, k z = , w(kl) = w(k2)= 0.727, and Up = The interfacial mode is unstable over all the wave numbers shown. strips of continuous spectra are at: (i) Re o- = and Re a = with < Im a < 0. (ii) Re o- = with < Im a < and < Im cr < Once a flow has undergone spurt, quasi-static unloading, in which the shear stress is reduced in small steps until a smooth flow is recovered may be applied. At this point, the value of the shear stress is lower than is required for the onset of spurt. This hysteresis effect on unloading has been described in the onedimensional model equation of Malkus et al. (1991), where a smooth flow is achieved once the shear stress falls below the trough of the curve. This is called bottom-jumping. The trough in Figure 1 occurs at w(~) = 0.725, = 0.866, If the flow is in a two-layer state around the trough, then consider 11 = 0.99, k 1 = 0.87, k 2 = 7.392, and U~ Figure 6 shows the growth rates versus wave numbers. The interfacial mode is unstable over all wave numbers. The stability of long waves is influenced by both the elasticities and by the viscosities of each of the regions. A number of bulk modes are also unstable. Compared with the situations in Figures 2 and 3, the instabilities have moved down to the longer waves and the long-wave stability is lost. The unstable modes achieve higher growth rates than when the shear stress was higher. The spectrum for the top-jumping situation appears more closely related to sharkskin than the spectrum for bottom-jumping. The results show that the two-layer arrangement modeling spurt along the nonmonotone part of Figure 1 is unstable to short-wave disturbances. Thus the path described by the arrows in Figure 11 of Malkus et al. (1991) lands on steady states which are two-dimensionally unstable. At the onset of spurt for top-jumping, the largest growth rates are found for short wavelengths, consistent with the observation of the loss of gloss. As the fluid in high shear increases in depth, the largest growth rates move in to the order-one wavelength with the same order of magnitude for shorter waves. The spurted arrangement at the trough of the shear-stress-shear-strain curve is unstable for all wavelengths and the growth rates are larger than for higher shear stresses. The Johnson-Segalman model has been examined for the stability of steady states exhibiting spurt to two-dimensional perturbations. References Adewale, K.E.P., and Leonov, A.I. (1993). On modeling spurt flows of polymers, J. Non-Newtonian Fluid Mech., 49, Chen, K. (1991a). Interfacial instability due to elastic stratification in concentric coextrusion of two viscoelastic fluids, J. Non- Newtonian Fluid Mech., 40, Chen, K. (199 lb). Elastic instability of the interface in Couette flow of viscoelastic liquids, J. Non-Newtonian Fluid. Mech., 40, Chen, K., and Joseph, D.D. (1992). Elastic shortwave instability in extrusion flows of viscoelastic liquids, J. Non-Newtonian Fluid Mech., 42, Chen, K., and Zhang, Y. (1993). Stability of the interface in co-extrusion flow of two viscoelastic fluids through a pipe, J. Fluid Mech., 247, Denn, M.M. (1990). Issues in viscoelastic fluid mechanics, Ann. Rev. Fluid Mech., 22, Dunford, N., and Schwartz, J.T. (1958). Linear Operators, vol. 1, Wiley, New York.
13 Spurt and Instability in a Two-Layer Johnson Segalman Liquid 475 Gorodtsov, V.A., and Leonov, A.I. (1967). On a linear instability of a plane parallel Couette flow of viscoelastic fluid, J. Appl. Math. Mech., 31, Gottlieb, D., and Orszag, S. (1983). Numerical Analysis of Spectral Methods: Theory and Applications, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, PA. Hopper, A.P., and Boyd, W.G.C. (1983). Shear flow instability at the interface between two viscous fluids, J. Fluid Mech., 128, Hunter, J., and Slemrod, M. (1983). Viscoelastic fluid flow exhibiting hysteretic phase changes, Phys. Fluids, 26, Johnson, M., and Segalman, D. (1977). A model for viscoelastic fluid behavior which allows non-anne deformation, J. Non-Newtonian Fluid Mech., 2, Joseph, D.D., and Renardy, Y. (1992). Fundamentals of Two-Fluid Dynamics, Parts 1 and 2, Springer-Verlag, New York. Kalika, D.S., and Denn, M.M. (1987). Wall slip and extrudate distortion in linear low-density polyethylene, J. Rheol., 31, Kolkka, R., and Ierley, G. (1989). Spurt phenomena for the Giesekus viscoelastic liquid model, J. Non-Newtonian Fluid. Mech., 33, Kolkka, R., Malkus, D., Hansen, M., Ierley, G., and Worthing, R. (1988). Spurt phenomena of the Johnson Segalman fluid and related models, J. Non-Newtonian Fluid Mech., 29, Larson, R.G. (1992). Instabilities in viscoelastic flows, Rheol. Acta, 31, Leonov, A.I. (1984). A linear model of the stick-slip phenomena in polymer flow in rheometers, Rheol. Acta, 23, Leonov, A.I. (1990). On the dependence of friction force on sliding velocity in the theory of adhesive friction of elastomers, Wear, 141, Leonov, A.I., and Srinivasan, A. (1993). Self-oscillations of an elastic plate sliding over a smooth surface, Internat. J. Engrg. Sci., 31, Lira, FJ. (1988). Wall slip in narrow molecular weight distribution polybutadienes, Ph.D. Thesis, Princeton University. Lim, F.J., and Schowalter, W.R. (1989). Wall slip of narrow molecular weight distribution polybutadienes, J. Rheol., 33, Malkus, D.S., Nobel, J.A., and Plohr, B.J., (1990). Dynamics of shear flow of a non-newtonian fluid, J. Comput. Phys., 87(2), Malkus, D.S., Nohel, J.A., and Plohr, B.J. (1991). Analysis of new phenomena in shear flow of non-newtonian fluids, SlAM J. Appl. Math., 51(4), Malkus, D.S., Nohel, J.A., and Plohr, B.J., (1994). Oscillations in piston-driven shear flow of a non-newtonian fluid, Forschungsinstitut fiir Mathematik, ETH Ziirich, preprint. Moynihan, R.H., Barid, D.G., and Ramanathan, R. (1990). Additional observations on the surface melt fracture behavior of linear low-density polyethylene, J. Non-Newtonian Fluid Mech., 36, Nohel, J.A., Pego, R.L., and Tzavaras, A.E. (1990). Stability of discontinuous steady states in shearing motions of a non-newtonian fluid, Proc. Roy. Soe. Edinburoh Sect. A, 115, Nohel, J.A., and Pego, R.L. (1993). Nonlinear stability and asymptotic behavior of shearing motions of a non-newtonian fluid, preprint. Orszag, S. (1971). Accurate solutions of the Orr-Sommerfeld stability equation, J. Fluid Mech., 50, Renardy, Y. (1988). Stability of the interface in two-layer Couette flow of upper convected Maxwell liquids, J. Non-Newtonian Fluid Mech., 28, Renardy, Y. (1996). An instability of plane Couette flow of the Johnson-Segalman liquid, in Advances in Multi-Fluid Flows, Y. Renardy, D. Papageorgiou, and S.M. Sun, eds., SIAM, Philadelphia, PA, to appear. Renardy, M., and Renardy, Y. (1986). Linear stability of plane Couette flow of an upper convected Maxwell fluid, J. Non-Newtonian Fluid Mech., 22, Su, Y.-Y., and Khomami, B. (1991). Stability of multilayer power law and second-order fluids in plane Poiseuille flow, Chem. Engr 9. Comm., 109, Su, Y.-Y., and Khomami, B. (1992a). Interracial stability of multilayer viscoelastic fluids in slit and converging channel die geometries, J. Rheol., 36, Su, Y.-Y., and Khomami, B. (1992b). Purely elastic interfacial instabilities in superposed flow of polymeric fluids, Rheol. Acta, 31, Tirrell, M., and Malone, M.F. (1977). Stress-induced diffusion of macromolecules, J. Polym. Sci., 15, Yerushalmi, J., Katz, S., and Shinnar, R. (1970). The stability of steady shear flows of some viscoelastic fluids, Chem. Engrg. Sci., 25,
TWO-DIMENSIONAL SIMULATIONS OF THE EFFECT OF THE RESERVOIR REGION ON THE PRESSURE OSCILLATIONS OBSERVED IN THE STICK-SLIP INSTABILITY REGIME
1 TWO-DIMENSIONAL SIMULATIONS OF THE EFFECT OF THE RESERVOIR REGION ON THE PRESSURE OSCILLATIONS OBSERVED IN THE STICK-SLIP INSTABILITY REGIME Eleni Taliadorou and Georgios Georgiou * Department of Mathematics
More informationJ. Non-Newtonian Fluid Mech. 91 (2000) Received in revised form 1September1999
J. Non-Newtonian Fluid Mech. 91 (2000 85 104 An experimental/theoretical investigation of interfacial instabilities in superposed pressure-driven channel flow of Newtonian and well-characterized viscoelastic
More informationTime-dependent plane Poiseuille flow of a Johnson±Segalman fluid
J. Non-Newtonian Fluid Mech. 82 (1999) 105±123 Abstract Time-dependent plane Poiseuille flow of a Johnson±Segalman fluid Marios M. Fyrillas a,b, Georgios C. Georgiou a,*, Dimitris Vlassopoulos b a Department
More informationElastic short wave instability in extrusion flows of viscoelastic liquids. KangPing Chen 1 & Daniel D. Joseph 2
KangPing Chen 1 & Daniel D. Joseph 1 Department of Mechanical and Aerospace Engineering Arizona State University Tempe, AZ 8587-6106 Department of Aerospace Engineering and Mechanics University of Minnesota
More informationStructure of the spectrum in zero Reynolds number shear flow of the UCM and Oldroyd-B liquids
J. Non-Newtonian Fluid Mech., 80 (1999) 251 268 Structure of the spectrum in zero Reynolds number shear flow of the UCM and Oldroyd-B liquids Helen J. Wilson a, Michael Renardy b, Yuriko Renardy b, * a
More informationThe time-dependent extrudate-swell problem of an Oldroyd-B fluid with slip along the wall
The time-dependent extrudate-swell problem of an Oldroyd-B fluid with slip along the wall Eric Brasseur Unité de Mécanique Appliquée, Université Catholique de Louvain, Bâtiment Euler, 4 6 Avenue Georges
More informationInertial effect on stability of cone-and-plate flow Part 2: Non-axisymmetric modes
J. Non-Newtonian Fluid Mech., 78 (1998) 27 45 Inertial effect on stability of cone-and-plate flow Part 2: Non-axisymmetric modes Yuriko Renardy *, David O. Olagunju 1 Department of Mathematics and ICAM,
More informationStability of two-layer viscoelastic plane Couette flow past a deformable solid layer
J. Non-Newtonian Fluid Mech. 117 (2004) 163 182 Stability of two-layer viscoelastic plane Couette flow past a deformable solid layer V. Shankar Department of Chemical Engineering, Indian Institute of Technology,
More informationRHEOLOGY Principles, Measurements, and Applications. Christopher W. Macosko
RHEOLOGY Principles, Measurements, and Applications I -56081-5'79~5 1994 VCH Publishers. Inc. New York Part I. CONSTITUTIVE RELATIONS 1 1 l Elastic Solid 5 1.1 Introduction 5 1.2 The Stress Tensor 8 1.2.1
More informationNUMERICAL ANALYSIS OF THE VISCOELASTIC FLUID IN PLANE POISEUILLE FLOW
NUMERICAL ANALYSIS OF THE VISCOELASTIC FLUID IN PLANE POISEUILLE FLOW N. Khorasani and B. Mirzalou 1 Department of Mechanical and Aerospace engineering, Science and Research Branch, Islamic Azad University,
More informationELASTIC INSTABILITIES IN CONE{AND{PLATE FLOW: SMALL GAP THEORY. David O. Olagunju. University of Delaware. Newark, DE 19716
ELASTIC INSTABILITIES IN CONE{AND{PLATE FLOW: SMALL GAP THEORY David O. Olagunju Department of Mathematical Sciences University of Delaware Newark, DE 19716 June 15, 1995 Abstract Consider the axisymmetric,
More informationExcerpt from the Proceedings of the COMSOL Users Conference 2006 Boston
Using Comsol Multiphysics to Model Viscoelastic Fluid Flow Bruce A. Finlayson, Professor Emeritus Department of Chemical Engineering University of Washington, Seattle, WA 98195-1750 finlayson@cheme.washington.edu
More informationStability analysis of constitutive equations for polymer melts in viscometric flows
J. Non-Newtonian Fluid Mech. 103 (2002) 221 250 Stability analysis of constitutive equations for polymer melts in viscometric flows Anne M. Grillet 1, Arjen C.B. Bogaerds, Gerrit W.M. Peters, Frank P.T.
More informationThis article was originally published in a journal published by Elsevier, and the attached copy is provided by Elsevier for the author s benefit and for the benefit of the author s institution, for non-commercial
More informationvs. Chapter 4: Standard Flows Chapter 4: Standard Flows for Rheology shear elongation 2/1/2016 CM4650 Lectures 1-3: Intro, Mathematical Review
CM465 Lectures -3: Intro, Mathematical //6 Chapter 4: Standard Flows CM465 Polymer Rheology Michigan Tech Newtonian fluids: vs. non-newtonian fluids: How can we investigate non-newtonian behavior? CONSTANT
More informationAnalysis of the flow instabilities in the extrusion of polymeric melts Aarts, A.C.T.
Analysis of the flow instabilities in the extrusion of polymeric melts Aarts, A.C.T. DOI: 10.6100/IR475535 Published: 01/01/1997 Document Version Publisher s PDF, also known as Version of Record (includes
More informationCompressible viscous flow in slits with slip at the wall
Compressible viscous flow in slits with slip at the wall Georgios C. Georgiou Department of Mathematics and Statistics, University of Cyprus, Kallipoleos 75, F!O. Box 537, Nicosia, Cyprus Marcel J. Crochet
More informationINTERFACIAL WAVE BEHAVIOR IN OIL-WATER CHANNEL FLOWS: PROSPECTS FOR A GENERAL UNDERSTANDING
1 INTERFACIAL WAVE BEHAVIOR IN OIL-WATER CHANNEL FLOWS: PROSPECTS FOR A GENERAL UNDERSTANDING M. J. McCready, D. D. Uphold, K. A. Gifford Department of Chemical Engineering University of Notre Dame Notre
More informationDirect Simulation of the Motion of Solid Particles in Couette and Poiseuille Flows of Viscoelastic Fluids
Direct Simulation of the Motion of Solid Particles in Couette and Poiseuille Flows of Viscoelastic Fluids by P. Y. Huang 1, J. Feng 2, H. H. Hu 3 and D. D. Joseph 1 1 Department of Aerospace Engineering
More informationIn the name of Allah the most beneficent the most merciful
In the name of Allah the most beneficent the most merciful Transient flows of Maxwell fluid with slip conditions Authors Dr. T. Hayat & Sahrish Zaib Introduction Non-Newtonian Newtonian fluid Newtonian
More informationCENG 501 Examination Problem: Estimation of Viscosity with a Falling - Cylinder Viscometer
CENG 501 Examination Problem: Estimation of Viscosity with a Falling - Cylinder Viscometer You are assigned to design a fallingcylinder viscometer to measure the viscosity of Newtonian liquids. A schematic
More informationIntroduction to Marine Hydrodynamics
1896 1920 1987 2006 Introduction to Marine Hydrodynamics (NA235) Department of Naval Architecture and Ocean Engineering School of Naval Architecture, Ocean & Civil Engineering First Assignment The first
More informationTheory of melt fracture instabilities in the capillary flow of polymer melts
PHYSICAL REVIEW E VOLUME 55, NUMBER 3 MARCH 1997 Theory of melt fracture instabilities in the capillary flow of polymer melts Joel D. Shore, 1,* David Ronis, 1,2 Luc Piché, 1,3 and Martin Grant 1 1 Department
More informationLes Houches School of Foam: Rheology of Complex Fluids
Les Houches School of Foam: Rheology of Complex Fluids Andrew Belmonte The W. G. Pritchard Laboratories Department of Mathematics, Penn State University 1 Fluid Dynamics (tossing a coin) Les Houches Winter
More informationChapter 9: Differential Analysis
9-1 Introduction 9-2 Conservation of Mass 9-3 The Stream Function 9-4 Conservation of Linear Momentum 9-5 Navier Stokes Equation 9-6 Differential Analysis Problems Recall 9-1 Introduction (1) Chap 5: Control
More informationExperimental Investigation of the Development of Interfacial Instabilities in Two Layer Coextrusion Dies
SCREW EXTRUSION R. Valette 1 *, P. Laure 2, Y. Demay 1, J.-F. Agassant 1 1 Centre de Mise Forme des MatØriaux, Ecole Nationale des Mines de Paris, Sophia Antipolis, France 2 Institut Non-LinØaire de Nice,
More informationCorrection of Lamb s dissipation calculation for the effects of viscosity on capillary-gravity waves
PHYSICS OF FLUIDS 19, 082105 2007 Correction of Lamb s dissipation calculation for the effects of viscosity on capillary-gravity waves J. C. Padrino and D. D. Joseph Aerospace Engineering and Mechanics
More informationOldroyd Viscoelastic Model Lecture Notes
Oldroyd Viscoelastic Model Lecture Notes Drew Wollman Portland State University Maseeh College of Engineering and Computer Science Department of Mechanical and Materials Engineering ME 510: Non-Newtonian
More informationSTABILITY ANALYSIS FOR BUOYANCY-OPPOSED FLOWS IN POLOIDAL DUCTS OF THE DCLL BLANKET. N. Vetcha, S. Smolentsev and M. Abdou
STABILITY ANALYSIS FOR BUOYANCY-OPPOSED FLOWS IN POLOIDAL DUCTS OF THE DCLL BLANKET N. Vetcha S. Smolentsev and M. Abdou Fusion Science and Technology Center at University of California Los Angeles CA
More informationInterfacial waves in steady and oscillatory, two-layer Couette flows
Interfacial waves in steady and oscillatory, two-layer Couette flows M. J. McCready Department of Chemical Engineering University of Notre Dame Notre Dame, IN 46556 Page 1 Acknowledgments Students: M.
More informationSOME DYNAMICAL FEATURES OF THE TURBULENT FLOW OF A VISCOELASTIC FLUID FOR REDUCED DRAG
SOME DYNAMICAL FEATURES OF THE TURBULENT FLOW OF A VISCOELASTIC FLUID FOR REDUCED DRAG L. Thais Université de Lille Nord de France, USTL F9 Lille, France Laboratoire de Mécanique de Lille CNRS, UMR 817
More informationvery elongated in streamwise direction
Linear analyses: Input-output vs. Stability Has much of the phenomenology of boundary la ut analysis of Linearized Navier-Stokes (Cont.) AMPLIFICATION: 1 y. z x 1 STABILITY: Transitional v = (bypass) T
More informationChapter 3: Newtonian Fluid Mechanics. Molecular Forces (contact) this is the tough one. choose a surface through P
// Molecular Constitutive Modeling Begin with a picture (model) of the kind of material that interests you Derive how stress is produced by deformation of that picture Write the stress as a function of
More information4. The Green Kubo Relations
4. The Green Kubo Relations 4.1 The Langevin Equation In 1828 the botanist Robert Brown observed the motion of pollen grains suspended in a fluid. Although the system was allowed to come to equilibrium,
More informationLast update: 15 Oct 2007
Unsteady flow of an axisymmetric annular film under gravity HOUSIADAS K, TSAMOPOULOS J -- PHYSICS OF FLUIDS --10 (10)2500-2516 1998 Yang XT, Xu ZL, Wei YM Two-dimensional simulation of hollow fiber membrane
More informationViscoelastic Flows in Abrupt Contraction-Expansions
Viscoelastic Flows in Abrupt Contraction-Expansions I. Fluid Rheology extension. In this note (I of IV) we summarize the rheological properties of the test fluid in shear and The viscoelastic fluid consists
More informationON THE EFFECTIVENESS OF HEAT GENERATION/ABSORPTION ON HEAT TRANSFER IN A STAGNATION POINT FLOW OF A MICROPOLAR FLUID OVER A STRETCHING SURFACE
5 Kragujevac J. Sci. 3 (29) 5-9. UDC 532.5:536.24 ON THE EFFECTIVENESS OF HEAT GENERATION/ABSORPTION ON HEAT TRANSFER IN A STAGNATION POINT FLOW OF A MICROPOLAR FLUID OVER A STRETCHING SURFACE Hazem A.
More information2. FLUID-FLOW EQUATIONS SPRING 2019
2. FLUID-FLOW EQUATIONS SPRING 2019 2.1 Introduction 2.2 Conservative differential equations 2.3 Non-conservative differential equations 2.4 Non-dimensionalisation Summary Examples 2.1 Introduction Fluid
More informationModeling of Anisotropic Polymers during Extrusion
Modeling of Anisotropic Polymers during Extrusion Modified on Friday, 01 May 2015 10:38 PM by mpieler Categorized as: Paper of the Month Modeling of Anisotropic Polymers during Extrusion Arash Ahmadzadegan,
More informationPressure corrections for viscoelastic potential flow analysis of capillary instability
ve-july29-4.tex 1 Pressure corrections for viscoelastic potential flow analysis of capillary instability J. Wang, D. D. Joseph and T. Funada Department of Aerospace Engineering and Mechanics, University
More informationTime dependent finite element analysis of the linear stability of viscoelastic flows with interfaces
J. Non-Newtonian Fluid Mech. 116 (23 33 54 Time dependent finite element analysis of the linear stability of viscoelastic flows with interfaces Arjen C.B. Bogaerds, Martien A. Hulsen, Gerrit W.M. Peters,
More informationBoundary-Layer Theory
Hermann Schlichting Klaus Gersten Boundary-Layer Theory With contributions from Egon Krause and Herbert Oertel Jr. Translated by Katherine Mayes 8th Revised and Enlarged Edition With 287 Figures and 22
More informationThe Reynolds experiment
Chapter 13 The Reynolds experiment 13.1 Laminar and turbulent flows Let us consider a horizontal pipe of circular section of infinite extension subject to a constant pressure gradient (see section [10.4]).
More informationCONTRIBUTION TO EXTRUDATE SWELL FROM THE VELOCITY FACTOR IN NON- ISOTHERMAL EXTRUSION
Second International Conference on CFD in the Minerals and Process Industries CSIRO, Melbourne, Australia 6-8 December 1999 CONTRIBUTION TO EXTRUDATE SWELL FROM THE VELOCITY FACTOR IN NON- ISOTHERMAL EXTRUSION
More informationp + µ 2 u =0= σ (1) and
Master SdI mention M2FA, Fluid Mechanics program Hydrodynamics. P.-Y. Lagrée and S. Zaleski. Test December 4, 2009 All documentation is authorized except for reference [1]. Internet access is not allowed.
More informationA new numerical framework to simulate viscoelastic free-surface flows with the finitevolume
Journal of Physics: Conference Series PAPER OPEN ACCESS A new numerical framework to simulate viscoelastic free-surface flows with the finitevolume method Related content - Gravitational collapse and topology
More informationINDEPENDENT CONFIRMATION THAT DELAYED DIE SWELL IS A HYPERBOLIC TRANSITION. D. D. Joseph and C. Christodoulou
INDEPENDENT CONFIRMATION THAT DELAYED DIE SWELL IS A HYPERBOLIC TRANSITION Abstract by D. D. Joseph and C. Christodoulou The University of Minnesota, Minneapolis, Minnesota January 1993 We measured shear
More informationx j r i V i,j+1/2 r Ci,j Ui+1/2,j U i-1/2,j Vi,j-1/2
Merging of drops to form bamboo waves Yuriko Y. Renardy and Jie Li Department of Mathematics and ICAM Virginia Polytechnic Institute and State University Blacksburg, VA -, U.S.A. May, Abstract Topological
More informationVISCOELASTIC PROPERTIES OF POLYMERS
VISCOELASTIC PROPERTIES OF POLYMERS John D. Ferry Professor of Chemistry University of Wisconsin THIRD EDITION JOHN WILEY & SONS New York Chichester Brisbane Toronto Singapore Contents 1. The Nature of
More informationUNIT IV BOUNDARY LAYER AND FLOW THROUGH PIPES Definition of boundary layer Thickness and classification Displacement and momentum thickness Development of laminar and turbulent flows in circular pipes
More informationBoundary Layer Flow and Heat Transfer due to an Exponentially Shrinking Sheet with Variable Magnetic Field
International Journal of Scientific Research Engineering & Technology (IJSRET), ISSN 78 088 Volume 4, Issue 6, June 05 67 Boundary ayer Flow and Heat Transfer due to an Exponentially Shrinking Sheet with
More information1. Comparison of stability analysis to previous work
. Comparison of stability analysis to previous work The stability problem (6.4) can be understood in the context of previous work. Benjamin (957) and Yih (963) have studied the stability of fluid flowing
More informationExperiments at the University of Minnesota (draft 2)
Experiments at the University of Minnesota (draft 2) September 17, 2001 Studies of migration and lift and of the orientation of particles in shear flows Experiments to determine positions of spherical
More informationFriction Factors and Drag Coefficients
Levicky 1 Friction Factors and Drag Coefficients Several equations that we have seen have included terms to represent dissipation of energy due to the viscous nature of fluid flow. For example, in the
More informationFundamentals of Fluid Dynamics: Elementary Viscous Flow
Fundamentals of Fluid Dynamics: Elementary Viscous Flow Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pan.pl/ tzielins/ Institute of Fundamental Technological Research
More informationShear rheology of polymer melts
Shear rheology of polymer melts Dino Ferri dino.ferri@versalis.eni.com Politecnico Alessandria di Milano, 14/06/2002 22 nd October 2014 Outline - Review of some basic rheological concepts (simple shear,
More informationChristel Hohenegger A simple model for ketchup-like liquid, its numerical challenges and limitations April 7, 2011
Notes by: Andy Thaler Christel Hohenegger A simple model for ketchup-like liquid, its numerical challenges and limitations April 7, 2011 Many complex fluids are shear-thinning. Such a fluid has a shear
More informationLecture 2: Constitutive Relations
Lecture 2: Constitutive Relations E. J. Hinch 1 Introduction This lecture discusses equations of motion for non-newtonian fluids. Any fluid must satisfy conservation of momentum ρ Du = p + σ + ρg (1) Dt
More informationOn fully developed mixed convection with viscous dissipation in a vertical channel and its stability
ZAMM Z. Angew. Math. Mech. 96, No. 12, 1457 1466 (2016) / DOI 10.1002/zamm.201500266 On fully developed mixed convection with viscous dissipation in a vertical channel and its stability A. Barletta 1,
More informationChapter 9: Differential Analysis of Fluid Flow
of Fluid Flow Objectives 1. Understand how the differential equations of mass and momentum conservation are derived. 2. Calculate the stream function and pressure field, and plot streamlines for a known
More informationFlow Transition in Plane Couette Flow
Flow Transition in Plane Couette Flow Hua-Shu Dou 1,, Boo Cheong Khoo, and Khoon Seng Yeo 1 Temasek Laboratories, National University of Singapore, Singapore 11960 Fluid Mechanics Division, Department
More informationMYcsvtu Notes HEAT TRANSFER BY CONVECTION
www.mycsvtunotes.in HEAT TRANSFER BY CONVECTION CONDUCTION Mechanism of heat transfer through a solid or fluid in the absence any fluid motion. CONVECTION Mechanism of heat transfer through a fluid in
More informationTHE 3D VISCOELASTIC SIMULATION OF MULTI-LAYER FLOW INSIDE FILM AND SHEET EXTRUSION DIES
THE 3D VISCOELASTIC SIMULATION OF MULTI-LAYER FLOW INSIDE FILM AND SHEET EXTRUSION DIES Kazuya Yokomizo 1, Makoto Iwamura 2 and Hideki Tomiyama 1 1 The Japan Steel Works, LTD., Hiroshima Research Laboratory,
More informationStress Overshoot of Polymer Solutions at High Rates of Shear
Stress Overshoot of Polymer Solutions at High Rates of Shear K. OSAKI, T. INOUE, T. ISOMURA Institute for Chemical Research, Kyoto University, Uji, Kyoto 611-0011, Japan Received 3 April 2000; revised
More informationResearch Article Innovation: International Journal of Applied Research; ISSN: (Volume-2, Issue-2) ISSN: (Volume-1, Issue-1)
Free Convective Dusty Visco-Elastic Fluid Flow Through a Porous Medium in Presence of Inclined Magnetic Field and Heat Source/ Sink 1 Debasish Dey, 2 Paban Dhar 1 Department of Mathematics, Dibrugarh University,
More informationFiber spinning and draw resonance: theoretical background
Fiber spinning and draw resonance: theoretical background dr. ir. Martien Hulsen March 17, 2005 1 Introduction The fiber spinning process is nicely described in the book by Dantig & Tucker ([1], Chapter
More informationHEAT TRANSFER OF SIMPLIFIED PHAN-THIEN TANNER FLUIDS IN PIPES AND CHANNELS
HEAT TRANSFER OF SIMPLIFIED PHAN-THIEN TANNER FLUIDS IN PIPES AND CHANNELS Paulo J. Oliveira Departamento de Engenharia Electromecânica, Universidade da Beira Interior Rua Marquês D'Ávila e Bolama, 600
More informationCESSATION OF VISCOPLASTIC POISEUILLE FLOW IN A RECTANGULAR DUCT WITH WALL SLIP
8 th GRACM International Congress on Computational Mechanics Volos, 2 July 5 July 205 CESSATION OF VISCOPLASTIC POISEUILLE FLOW IN A RECTANGULAR DUCT WITH WALL SLIP Yiolanda Damianou, George Kaoullas,
More informationA multiscale framework for lubrication analysis of bearings with textured surface
A multiscale framework for lubrication analysis of bearings with textured surface *Leiming Gao 1), Gregory de Boer 2) and Rob Hewson 3) 1), 3) Aeronautics Department, Imperial College London, London, SW7
More informationLecture 7: Rheology and milli microfluidic
1 and milli microfluidic Introduction In this chapter, we come back to the notion of viscosity, introduced in its simplest form in the chapter 2. We saw that the deformation of a Newtonian fluid under
More informationInstability suppression in viscoelastic film flows down an inclined plane lined with a deformable solid layer
Instability suppression in viscoelastic film flows down an inclined plane lined with a deformable solid layer Aashish Jain and V. Shankar* Department of Chemical Engineering, Indian Institute of Technology,
More informationSymmetry Properties of Confined Convective States
Symmetry Properties of Confined Convective States John Guckenheimer Cornell University 1 Introduction This paper is a commentary on the experimental observation observations of Bensimon et al. [1] of convection
More informationANALYSIS ON PLANAR ENTRY CONVERGING FLOW OF POLYMER MELTS
Journal of Materials Science and Engineering with Advanced Technology Volume 2, Number 2, 2010, Pages 217-233 ANALYSIS ON PLANAR ENTRY CONVERGING FLOW OF POLYMER MELTS College of Industrial Equipment and
More informationState Space Solution to the Unsteady Slip Flow of a Micropolar Fluid between Parallel Plates
International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Volume 2, Issue 10, October 2014, PP 827-836 ISSN 2347-307X (Print) & ISSN 2347-3142 (Online) www.arcjournals.org State
More informationFALLING FILM FLOW ALONG VERTICAL PLATE WITH TEMPERATURE DEPENDENT PROPERTIES
Proceedings of the International Conference on Mechanical Engineering 2 (ICME2) 8-2 December 2, Dhaka, Bangladesh ICME-TH-6 FALLING FILM FLOW ALONG VERTICAL PLATE WITH TEMPERATURE DEPENDENT PROPERTIES
More information2 D.D. Joseph To make things simple, consider flow in two dimensions over a body obeying the equations ρ ρ v = 0;
Accepted for publication in J. Fluid Mech. 1 Viscous Potential Flow By D.D. Joseph Department of Aerospace Engineering and Mechanics, University of Minnesota, MN 55455 USA Email: joseph@aem.umn.edu (Received
More informationModel-based analysis of polymer drag reduction in a turbulent channel flow
2013 American Control Conference ACC Washington, DC, USA, June 17-19, 2013 Model-based analysis of polymer drag reduction in a turbulent channel flow Binh K. Lieu and Mihailo R. Jovanović Abstract We develop
More informationUnsteady Flow of a Newtonian Fluid in a Contracting and Expanding Pipe
Unsteady Flow of a Newtonian Fluid in a Contracting and Expanding Pipe T S L Radhika**, M B Srinivas, T Raja Rani*, A. Karthik BITS Pilani- Hyderabad campus, Hyderabad, Telangana, India. *MTC, Muscat,
More informationStudy of steady pipe and channel flows of a single-mode Phan-Thien Tanner fluid
J. Non-Newtonian Fluid Mech. 101 (2001) 55 76 Study of steady pipe and channel flows of a single-mode Phan-Thien Tanner fluid Manuel A. Alves a, Fernando T. Pinho b,, Paulo J. Oliveira c a Departamento
More informationLINEAR STABILITY ANALYSIS FOR THE HARTMANN FLOW WITH INTERFACIAL SLIP
MAGNETOHYDRODYNAMICS Vol. 48 (2012), No. 1, pp. 147 155 LINEAR STABILITY ANALYSIS FOR THE HARTMANN FLOW WITH INTERFACIAL SLIP N. Vetcha, S. Smolentsev, M. Abdou Fusion Science and Technology Center, UCLA,
More informationOn the displacement of two immiscible Stokes fluids in a 3D Hele-Shaw cell
On the displacement of two immiscible Stokes fluids in a 3D Hele-Shaw cell Gelu Paşa Abstract. In this paper we study the linear stability of the displacement of two incompressible Stokes fluids in a 3D
More informationRheology of cellulose solutions. Puu Cellulose Chemistry Michael Hummel
Rheology of cellulose solutions Puu-23.6080 - Cellulose Chemistry Michael Hummel Contents Steady shear tests Viscous flow behavior and viscosity Newton s law Shear thinning (and critical concentration)
More informationChemical and Biomolecular Engineering 150A Transport Processes Spring Semester 2017
Chemical and Biomolecular Engineering 150A Transport Processes Spring Semester 2017 Objective: Text: To introduce the basic concepts of fluid mechanics and heat transfer necessary for solution of engineering
More informationThis article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and
This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution
More informationA phenomenological model for shear-thickening in wormlike micelle solutions
EUROPHYSICS LETTERS 5 December 999 Europhys. Lett., 8 (6), pp. 76-7 (999) A phenomenological model for shear-thickening in wormlike micelle solutions J. L. Goveas ( ) and D. J. Pine Department of Chemical
More informationFrictional rheologies have a wide range of applications in engineering
A liquid-crystal model for friction C. H. A. Cheng, L. H. Kellogg, S. Shkoller, and D. L. Turcotte Departments of Mathematics and Geology, University of California, Davis, CA 95616 ; Contributed by D.
More informationFluid Mechanics Abdusselam Altunkaynak
Fluid Mechanics Abdusselam Altunkaynak 1. Unit systems 1.1 Introduction Natural events are independent on units. The unit to be used in a certain variable is related to the advantage that we get from it.
More informationSimilarity Approach to the Problem of Second Grade Fluid Flows over a Stretching Sheet
Applied Mathematical Sciences, Vol. 1, 2007, no. 7, 327-338 Similarity Approach to the Problem of Second Grade Fluid Flows over a Stretching Sheet Ch. Mamaloukas Athens University of Economics and Business
More informationCentrifugal Destabilization and Restabilization of Plane Shear Flows
Centrifugal Destabilization and Restabilization of Plane Shear Flows A. J. Conley Mathematics and Computer Science Division Argonne National Laboratory Argonne, L 60439 Abstract The flow of an incompressible
More informationRheometry. II.1 Introduction
II Rheometry II.1 Introduction Structured materials are generally composed of microstructures dispersed in a homogeneous phase [30]. These materials usually have a yield stress, i.e. a threshold stress
More informationRelaxation methods and finite element schemes for the equations of visco-elastodynamics. Chiara Simeoni
Relaxation methods and finite element schemes for the equations of visco-elastodynamics Chiara Simeoni Department of Information Engineering, Computer Science and Mathematics University of L Aquila (Italy)
More informationChapter 6 Molten State
Chapter 6 Molten State Rheology ( 流變學 ) study of flow and deformation of (liquid) fluids constitutive (stress-strain) relation of fluids shear flow shear rate ~ dγ/dt ~ velocity gradient dv 1 = dx 1 /dt
More informationBoundary Conditions in Fluid Mechanics
Boundary Conditions in Fluid Mechanics R. Shankar Subramanian Department of Chemical and Biomolecular Engineering Clarkson University The governing equations for the velocity and pressure fields are partial
More informationChapter 1 Introduction
Chapter 1 Introduction This thesis is concerned with the behaviour of polymers in flow. Both polymers in solutions and polymer melts will be discussed. The field of research that studies the flow behaviour
More informationTemperature dependence of critical stress for wall slip by debonding
J. Non-Newtonian Fluid Mech. 94 (2000) 151 157 Temperature dependence of critical stress for wall slip by debonding Yogesh M. Joshi a, Prashant S. Tapadia a, Ashish K. Lele a, R.A. Mashelkar b, a Chemical
More informationDiffusion and Adsorption in porous media. Ali Ahmadpour Chemical Eng. Dept. Ferdowsi University of Mashhad
Diffusion and Adsorption in porous media Ali Ahmadpour Chemical Eng. Dept. Ferdowsi University of Mashhad Contents Introduction Devices used to Measure Diffusion in Porous Solids Modes of transport in
More information(2.1) Is often expressed using a dimensionless drag coefficient:
1. Introduction Multiphase materials occur in many fields of natural and engineering science, industry, and daily life. Biological materials such as blood or cell suspensions, pharmaceutical or food products,
More information7 The Navier-Stokes Equations
18.354/12.27 Spring 214 7 The Navier-Stokes Equations In the previous section, we have seen how one can deduce the general structure of hydrodynamic equations from purely macroscopic considerations and
More informationExercise: concepts from chapter 10
Reading:, Ch 10 1) The flow of magma with a viscosity as great as 10 10 Pa s, let alone that of rock with a viscosity of 10 20 Pa s, is difficult to comprehend because our common eperience is with s like
More informationInertial migration of a sphere in Poiseuille flow
J. Fluid Mech. (1989), vol. 203, pp. 517-524 Printed in Great Britain 517 Inertial migration of a sphere in Poiseuille flow By JEFFREY A. SCHONBERG AND E. J. HINCH Department of Applied Mathematics and
More information