Analysis on creeping channel flows of compressible fluids subject to wall slip

Transcription

1 heol Acta 01 51: DOI /s y OIGINA CONTIBUTION Analysis on creeping channel flows of compressible fluids subject to wall slip Hansong Tang eceived: 5 July 011 / evised: 1 December 011 / Accepted: 9 January 01 / Published online: 16 February 01 Springer-Verlag 01 Abstract Creeping channel flows of compressible fluids subject to wall slip are widely encountered in industries. This paper analyzes such flows driven by pressure in planar as well as circular channels. The analysis elucidates unsteady flows of Newtonian fluids subject to the Navier slip condition followed by steady flows of viscoplastic fluids in particular Herschel Bulkley fluids and their simplifications including power law and Newtonian fluids that slip at wall with a constant coefficient or a coefficient inversely proportional to pressure. Under the lubrication assumption analytical solutions are derived validated and discussed over a wide range of parameters. Analysis based on the derived solutions indicates that unsteadiness alters crosssection velocity profiles. It is demonstrated that compressibility of the fluids gives rise to a concave pressure distribution in the longitudinal direction whereas wall slip with a slip coefficient that is inversely proportional to pressure leads to a convex pressure distribution. Energy dissipation resulting from slippage can be a significant portion in the total dissipation of such a flow. A distinctive feature of the flow is that in case of the pressure-dependent slip coefficient the slip velocity increases rapidly in the flow direction and the flow can evolve into a pure plug flow at the exit. Keywords Wall slip Compressibility Creeping channel flow Herschel Bulkley fluid Analytical solution H. S. Tang B Department of Civil Eng. City College City University of New York New York NY USA htang@ccny.cuny.edu Introduction There has been increasing interest in creeping flows especially creeping non-newtonian flows. Such flows happen in various backgrounds such as material processing in chemical engineering and hydraulic fracturing in oil recovery Coussot 1994; Huang and Garcia 1998; Fujiyama and Inata 00; Ishida et al. 004; Huo and Kassab 006. One of the most commonly encountered creeping flows is pressure-driven channel flow or the Poiseuille flow which is important in flow mechanism study parameter identification and industrial applications Hatzikiriakos and Dealy 199a; Allende and Kalyon 000; Bird et al. 00; Tang and Kalyon 004. Although pressure-driven channel flows are usually unidirectional and have been studied earlier they still attract special attention in a number of emerging problems Pozrikidis 1998; Gratton et al. 1999; Engqvist and Hogg 004; Brochard-Wyart and de Gennes 007; Mcginty et al Creeping channel flows become complicated as well as interesting but remain unclear when both wall slip and compressibility are involved which are two common features of fluids with viscoplasticity that is an important characteristic of non-newtonian fluids especially concentrated suspensions and dispersions of polymeric fluids incorporated with rigid or soft particles Bird et al. 1983; Kalyon 005. Wall slip of fluids was generally overlooked in many studies but has recently gained an increasing attention in both experimental and theoretical investigations Uhland 1976; de Gennes 1979; Chenetal.1993; Aral and Kalyon 1994; Denn 001; Gevgilili and Kalyon 001; Granick et al. 003; Netoetal.005; Ballesta et al Slippage at wall leads to an increase in

2 4 heol Acta 01 51: flow rate in a channel flow affects flow profiles and the surface smoothness of extrudates emerging from extrusion dies and plays a role in the development of various flow instabilities Yilmazer et al. 1989; Yilmazer and Kalyon 1991; Hatzikiriakos and Dealy 199a; Hatzikiriakos 1994; Kalyon and Gevgilili 003. Wall slip is also known to play negative roles in mixing in shear-shinning and shear-thickening fluids awal and Kalyon 1994; Kalyon et al. 1999; Pereira 009. In gaseous microchannel flows wall slip has to be considered when viscous dissipation is not negligible and the ratio between molecular mean free path and size of the channel the Knudsen number is sufficiently large Gad-El-Hak The behavior of wall slip velocity is usually linked to shear stress by a power law function with a temperature- and pressure-dependent slip coefficient Yilmazer and Kalyon 1989 Hatzikiriakos and Dealy 199a Zhang et al Person and Denn 1997Parketal.008 Tang and Kalyon 008a b. Another complication in the analysis on creeping flow is compressibility of the fluid. As noted by upton and egester 1965 the evidence of variation of extrudate flow rate in spite of the constant piston speed indicates clearly the compressibility of the melt in a capillary flow. In flows of viscoplastic polymer suspensions it is believed that the compressibility principally results from air bubbles that are entrained into the fluids and also lubricate the wall Kalyon et al ; Aral and Kalyon 1995; de Gennes 00. Although its value is small it is now widely recognized that compressibility together with wall slip plays an important role for the transient stage and onset and development of instability phenomena in channel flows Hatzikiriakos and Dealy 199b 1994 Den Doelder et al Ovaici et al 1998 Georgiou 003 Tangand Kalyon 008a b. The compressibility is usually taken into account by considering density as a function of pressure Hatzikiriakos and Dealy 199a b. Analytical solutions have been presented for flows of compressible Newtonian fluids in microchannels Prudhomme et al. 1986; andau and ifshitz 1987; Venerus and Bugajsky 010. Flow rate expressions are also available when such fluids slip at wall with slip coefficients inversely proportional to pressure Kennard 1938; Knudsen ecently more analytical expressions for compressible gaseous flows with wall slippage have been obtained for both circular and non-circular channels Zohar et al. 00; Jang and Wereley 004; Duan and Muzychka 007a. Investigations show that compressibility makes pressure gradient in the flow direction to deviate from a constant and exhibit a concave pressure distribution in the longitudinal direction Jang and Wereley 004; Duan and Muzychka 007b. In addition recently limited effort has been made in seeking analytical expressions for channel flows of non-newtonian fluids. A pressure distribution in longitudinal direction is solved for an incompressible power law fluid subject to wall slip with a pressure-dependent wall slip coefficient Person and Denn An analytical solution is presented for a compressible flow of a Carreau fluid without slip Georgiou 003. Semi-analytical solutions are obtained for steady channel flows of compressible Herschel Bulkley fluids without slippage Taliadorou et al A closely related work is the analysis on creeping mud flows over inclined surfaces iu and Mei 1989; Wieland et al To the best knowledge of the author currently there are no analytical solutions for flows of compressible non-newtonian fluids subject to wall slippage. The objective of this paper is to understand the flow behavior of weakly compressible fluids subject to wall slip. Particularly it aims at developing explicit solutions and analyzing various features of pressuredriven flows between two parallel planes as well as in circular tubes. The paper considers isothermal simple shear flows of compressible Newtonian power law and Herschel Bulkley fluids subject to wall slip. Two types of wall slippage are considered: one has a constant slip coefficient and the other has a pressure-dependent slip coefficient. Governing equations Consider a planar two-dimensional D pressuredriven creeping flow of a fluid between two plates as shown in Fig. 1 which is expressed as u = u x y t v= 0 p = p x t 1 where u and v are the velocity in x and y direction respectively and p is the absolute pressure. The continuity and momentum equations of the flow read as Bird et al. 00 ρ t + ρu = 0 x ρ u t = p x + τ xy y 3 0 < x < H < y < H 4 where ρ is the density and τ xy is the shear stress. The corresponding boundary conditions read as u = βτ α xy y =±H 5

3 heol Acta 01 51: power law fluid model when τ 0 = 0 and a Newtonian fluid model when n = 1 and τ 0 = 0. The pressure is formulated as a function of density: ρ = γ p + γ 1 9 Fig. 1 A schematic representation of a pressure-driven flow between two plates p = p 0 x = 0; p = p x =. 6 Equation 5 is the wall slip condition in which β is the slip coefficient and α is an empirical number. The slip coefficient is frequently treated as a constant Zhang et al. 1995; Bird et al. 00. Generally speaking however the slip coefficient depends on pressure temperature etc. Hill et al. 1990; Hatzikiriakos and Dealy 199a. In this study the slip coefficient is taken as p > 0 β = β 0 pa p κ 7 where ρ a is the atmosphere pressure β 0 is a constant and κ is either κ = 0 or κ = 1 corresponding to a constant and a pressure-dependent slip coefficient respectively. When κ = 0 and α = 1 Eq. 5 reduces to the Navier slip condition Navier and Sur 187. Equation 7 with κ = 0 is based on experimental evidence and proposed for low-density compressible flow Kennard 1938; Knudsen 1950; Bird et al. 00 aswell as flow of polymer melts Tang and Kalyon 008a b. In view that p > 0 wall slip behavior described by Eq. 7 with κ = 1 is similar to the wall behavior described by β β 0 exp β s p β s being a small positive constant as proposed by Hill et al and Person and Denn The fluid is considered to be a generalized Newtonian fluid in which stress tensor is directly proportional to the rate of deformation tensor with the scalar shear viscosity as the parameter of the proportionality. In particular the Herschel Bulkley fluid is considered Herschel and Bulkley 196; Tangand Kalyon 004: τ xy = m u n 1 u y ± τ 0 τ xy >τ 0 y 8 u y = 0 τ xy τ 0 where m is the consistency index n is the power law index and τ 0 is the yield stress. Equation 8 reduces to a where γ and γ 1 are constants. Hereafter γ is referred as to the compressibility of the fluid. Equations 9 comprise the governing equations of the flow. It is known from the dimensionless forms of these equations that the flow is described by constants n α κ together with the Euler number dimensionless entrance pressure yield stress compressibility and slip coefficient defined as follows Appendix A: Er = p 0 m /n p γ 1 H 0 Hp = p 0 τ0 0 p = τ 0 Hp 0 γ = γ p 0 β = β 0m 1/n κ α 1/n pa Hp0. 10 γ 1 H Unsteady Newtonian flow p 0 For an unsteady pressure-driven fully developed flow of an incompressible Newtonian fluid subject to the Navier slip condition Eqs. and 3 can be simplified as u t = 1 p ρ x + m u ρ y. 11 Consider problem 11 and5 6. Assume the pressure gradient is prescribed as follows: 1 p h t = at + b 1 ρ x where a b = const. et the solution of the problem be dh t u = u 0 y h t + u 1 y. 13 dt Plugging 13 into11 and noticing a b andt are arbitrary real numbers one has the following two equations equivalent to 11: 1 + m d u 0 ρ dy = 0 m d u 1 ρ dy u 0 = Solving Eq. 14 and slip condition 5 withκ = 0 and α = 1 yields u = at + b ρy m + ρ H 8m + β 0ρ H + a ρ 3H y / y 4 5H 4 /16 4m + β 0ρ Hy 4m 5β 0ρ H 3 48m β0 ρ H. 15

4 44 heol Acta 01 51: Solution 15 indicates that generally speaking wall slip and unsteadiness a = 0 have a direct effect on flow velocity; the former makes the velocity to be non-zero at wall y =±H/ and the latter modifies velocity profile within the channel H/ < y < H/. Only when the flow is steady a = 0 the velocity profile is a parabola in the cross-section direction. The evolution of the velocity profile in dimensionless time [ bγ 1 H/ 4m ] t and effects of wall slip β 0 that is defined in Eq. 10 are illustrated in Fig.. Solution 15 degrades to those of Gupta and Goyal 1971 and Muzychka and Yovanovich 010 when β 0 = 0. Cross-section velocity of steady flow With the aid of the momentum Eq. 3 the wall slip condition 5 and the constitutive Eq. 8 the steady cross-section velocity profile is readily derived as u = T 1+1/n H H/ dp m 1/n 1 + 1/n T 1+1/n H H/ m 1/n 1 + 1/n dp 1 κ pa H α + β 0 p α 1 T1+1/n H y m 1/n 1 + 1/n dp dp α y λ 1 κ pa H α + β 0 p α dp α λ < y H/ 16 where λ = τ 0 / dp/ and T H y = max τ 0 ydp/ 0. The last terms in the two branches on right hand side of Eq. 16 arise from wall slip. Solution 16 indicates that the flow consists of a plug flow y λ sandwiched within two deformed layers λ < y H/ next to the wall. Using Eq. 16 the cross-section averaged velocity V=1/H H/ ux ydy is determined as T +1/n H H/ V = Hm 1/n 1 + 1/n + 1/n H/ dp + T1+1/n H H/ dp 1 m 1/n 1 + 1/n κ pa H α + β 0 dp α. 17 p α When T H H/ > 0 or dp/ > τ 0 /H the fluid yields. When T H H/ = 0 or dp/ τ 0 /H the fluid does not yield and the flow becomes a plug flow. In case of a plug flow the cross-section averaged velocity equals the slip velocity and Eq. 17 becomes κ pa H α V = u s = β 0 dp α. 18 p α For a power law fluid Eq. 17 reduces to V = H/1+1/n dp 1/n κ pa H α +β m 1/n 0 + 1/n p α dp α 19 and for a Newtonian fluid it is further simplified as V = H dp κ pa H α + β 0 dp α. 0 1m p α Equations 16 0 indicate that both the cross-section velocity profile and its average are functions of pressure Fig. Unsteady Newtonian flow with the Navier slip condition

5 heol Acta 01 51: gradient. It is also noticed that these velocity expressions are in forms similar to those of steady fully developed incompressible flows Tang and Kalyon 004. However unlike that in a fully developed flow in which pressure gradient is a constant the pressure gradient changes in the flow direction and thus the cross-section velocity profile and its average will also change in the flow direction. In general the pressure can be expressed as p = P x + p 0 + f x 1 f 0 = f = 0 where P p 0 p /Eq.1 decomposes the pressure into two parts: P x + p 0 and f x. The former corresponds to the pressure of a steady fully developed incompressible flow and the latter reflects the deviation of the pressure from it. In view of Eq. 1 the pressure gradient can also be split into two parts: dp = P df x +. 3 Here a small perturbation of pressure is considered; it is assumed that the deviation of the pressure gradient from that of the corresponding to the fully developed incompressible flow P is small: df x / P << 1. 4 In addition it is assumed the contribution of f x in wall slip is negligible. Using Taylor expansion and ignoring second- and higher-order terms of dfx/ the averaged velocity of a Herschel Bulkley fluid Eq. 17 can be approximated as V = A 1 + A 4 A x + A 3 κ + A 6 df x A 5 + A x + A 3 κ in which ˆT +1/n H H/ P A 1 = m 1/n 1 + 1/n + 1/n H 5 1+1/n ˆT H H/P 1 + m 1/n 1 + 1/n A = P A 3 = p 0 A 4 = β 0 H α P α 6 p a p a α 4 ˆT +1/n H H/ P 3 A 5 = m 1/n 1 + 1/n + 1/n H + ˆT 1+1/n H H/ P m 1/n 1 + 1/n H ˆT 1/n H H/ P 1 m 1/n A 6 = αβ 0 H α P α 1 α and ˆT H H/ is obtained using P to replace dp/ in T H H/. In expression 5 the terms associated with A 4 and A 6 are contributions from wall slip. In case of a power law or a Newtonian fluid A 3 A 4 A 5 anda 6 remain the same however A 1 and A 5 are simpler. For a power law fluid A 1 = H 1+1/n P 1/n 1+1/n m 1/n + 1/n H 1+1/n P 1/n 1 A 5 = 1+1/n m 1/n n + 1/n. 7 For a Newtonian fluid A 1 = H P 1m A 5 = H 1m. 8 Fig. 3 Flows with constant slip coefficients. Herschel Bulkley fluid: p = τ 0 = β = and γ = 6 10 and n = 0.4. Power fluid flow: p = 98.7 β = and γ = 5 10 and n = 0.4. Newtonian fluid: p = 79.0 β =.9 andγ = In all of the three flows α =.1 and κ = 0

6 46 heol Acta 01 51: Fig. 4 Velocity distributions of the Herschel Bulkley fluid with constant slip coefficient in Fig. 3. eft analytical solution right numerical solution In addition in case of a plug flow A 1 A 5 =0.Inorder to obtain flow distributions in both cross-section and longitudinal directions that is Eqs. 1 3and5 f x will be solved in the following section. Approximate solution Integrating the continuity Eq. over y and then x direction and using Eqs. 6 and 9 give rise to a boundary problem as follows { γ p + γ1 V = Ṁ 0 < x < 9 p = p 0 x = 0; p = p x = where Ṁ is the mass flow rate which is a constant. In general problem 9 would be solved numerically. In the following problem 9 is analytically solved with the aid of the pressure decomposition described in the previous section. Constant slip coefficient Consider a Herschel Bulkley fluid subject to wall slip with a constant slip coefficient. In view of the pressure decomposition 1 and velocity 5 problem 9 is expressed as follows: A 1 + A 4 + A 5 + A 6 f 0 = f = 0 df x = Ṁ A x + A 7 30 where A 7 = p 0 + γ γ p a and Ṁ = Ṁ/ γ p a. In view of weak compressibility contribution of f x in density is ignored in the Table 1 Perturbation of pressure in Figs. 3 and 4 Newtonian Power Herschel Bulkley law f x / P x +p df x / /P derivation of Eq. 30. The problem 30 is readily solved as f x = A 1 + A 4 n 1 + A /A 7 x A 5 + A 6 n 1 + A /A 7 x. 3 Therefore in view of Eqs. 1 and 5 the solution for pressure becomes p= p 0 + P x+ A 1 + A 4 n 1 + A /A 7 x A 5 + A 6 n 1 + A /A 7 x. 33 The solution for the cross-section averaged velocity reads as A /A 7 A 1 + A 4 V = 1 + A /A 7 x n 1 + A /A The slip velocity at the wall becomes A 1 + A 4 u s = A 4 + A 6 A 5 + A 6 A /A A x/a 7 n 1 + A /A According to Eqs. 3 and 3 the pressure gradient is evaluated as dp = P + A 1 + A 4 A 5 + A 6 A /A A /A 7 x n 1 + A /A Equations and3 with κ = 0 provide the cross-section velocity profile. The above analytical solutions could be validated against numerical solutions that are not restricted by the small perturbation assumption 4. The numerical solutions are obtained by solving Eq. 9 in which the Table Error as a function of compressibility. ρ 0 = 100 τ 0 = β = κ = 0 γ f x / P x + p df x / /P

7 heol Acta 01 51: Table 3 Error as a function of slip coefficient. γ = 0.1 ρ 0 = 100 τ 0 = κ = 0 β f x / P x + p df x / /P Table 4 Error as a function of ratio of entrance pressure to exit pressure. γ = 0.1 τ 0 = β = κ = 0 p f x / P x + p df x / /P Table 5 Error as a function of yield stress. γ = 0.1 p 0 = 100 β = κ = 0 τ f x / P x + p df x / /P Fig. 5 Plane flows with pressure-dependent slip coefficients. Herschel Bulkley fluid: p = τ 0 = β = γ = 6 10 andn = 0.4 power law fluid: p = 98.7 β = γ = 5 10 andn = 0.4 Newtonian fluid: p = 79.0 β = andγ = In all of the three flows α =.1 and κ = 1 Fig. 6 D velocity distributions of the Herschel Bulkley fluid with pressure-dependent slip coefficient in Fig. 5. eft Analytical solution right numerical solution

8 48 heol Acta 01 51: Table 6 Perturbation of pressure in Figs. 5 and 6 Newtonian Power law Herschel Bulkley f x / P x + p df x / /P cross-section averaged velocity is determined by one of Eqs and0 depending on the type of the fluid. Equation 9 is discretized using a uniform grid and central difference and the mass flow rate in it is determined using backward difference at the exit. The discretization is second-order accurate and the resulting nonlinear algebraic system is solved using the Newton iterative method Press et al In obtaining the numerical solutions a mesh with 31 grid nodes is chosen and mesh refinement indicates that the numerical solutions at this mesh size can be taken as mesh independent solutions. For convenience to show the comparisons between analytical and numerical solutions entrance pressures of Pa 10 7 Pa and Pa are used for a Newtonian fluid a power law fluid and a Herschel Bulkley fluid respectively. Figure 3 presents the analytical solutions 3 34 and 35 together with their corresponding numerical solutions. In the figure pressure deviation f x rather than pressure pisplotted to clearly illustrate the pressure distribution. The D analytical and numerical velocity solutions for the Herschel Bulkley fluid are given in Fig. 4. As indicated previously the analytical solutions are valid under small perturbation assumption 4. The perturbation or pressure deviation from a linear distribution of the solutions presented in Fig. 3 is given in Table 1. The figures show that with these amounts of perturbation the analytical solutions are basically identical to the numerical solutions at scales of the figures. The validity of the analytical solutions holds for a wide range of the parameters involved. The error of the analytical solution for the Herschel Bulkley fluid in the sense of its difference from the corresponding numerical solution is listed in Tables 3 4 and 5. In the tables fx and df x / are errors for f x and dfx/ respectively. It is seen from the tables that the errors of the analytical solutions mainly change with compressibility but remain roughly the same regardless variation of other parameters. The tables indicate that the error of f x is about one order of magnitude smaller than that of dfx/. Pressure-dependent slip coefficient For a Herschel Bulkley fluid subject to wall slip with a pressure-dependent slip coefficient the boundary value problem 9 is derived as A 4 A 1 + A x + A 3 A 6 df x + A 5 + = A x + A 3 f 0 = f = 0 Ṁ A x + A 7 37 The boundary problem 37 is solved by the following solution f x = Ã1 à x à 1 x 38 à where à 1 x= A 1 A4 x+ A 1 A 6 A 5 A A 5 A A n 1+ A A 5 x 5 A 3 A 5 + A 6 39 A 6 à x = A A 5 A 5 A 3 A 7 + A 6 n 1+ A A 5 x A 3 A 5 + A 6 A 3 A 7 + A A 5 A 3 A 7 + A 6 n 1+ A x. 40 A 7 As a result the solutions for pressure velocity and slip velocity are derived as p = p 0 + p p 0 x + Ã1 à x à 1 x 41 à A 4 A 6 V = A A 5 + A x + A 3 A x + A 3 Ã1 dã x dã 1 x 4 à A 4 A 6 Ã1 dã x u s = + dã 1 x A x+ A 3 A x+ A 3 à 43 Table 7 Error as a function of slip coefficient. γ = 0.01 p 0 = 100 τ 0 = 0.andκ = 1 β f x / P x + p df x / /P

9 heol Acta 01 51: Table 8 Error as a function of compressibility.p 0 = 100 τ 0 = 0. β = andκ = 1 γ f x / P x + p df x / /P In case of plug flow A 1 A 5 = 0 and the solutions become f x = A 4A 7 A 3 xna /A 7 +1 na /A 7 x+1 A 6 A 3 A 7 na /A 7 +1+A 44 and p= p 0 + p p 0 x + A 4A 7 A 3 xna /A 7 +1 na /A 7 x+1 A 6 A 3 A 7 na /A A 45 A 4 V = u s = A x + A 3 A 7 A A 3 A 7 n A /A A n A /A A 46 A x + A 7 Figures 5 and 6 show the analytical solutions 38 4 and 43 together with their corresponding numerical solutions summarized in Table 6. It is seen from the figures and the table that although perturbation of pressure f x/ P x + p 0 and df x / /P is now much larger in comparison with that in the previous situation of constant slip coefficient shown in Table 1 still the analytical solutions compare well with numerical solutions. The accuracy as well as limitation of the analytical solutions for the Herschel Bulkley fluid is illustrated in Tables and 10. It is seen from these tables that unlike the situation of constant slip coefficient the error varies slightly with the dimensionless compressibility but increases significantly with dimensionless slip coefficient ratio of entrance to exit pressure and yield stress. Again as illustrated in the tables the analytical solutions are valid in a relatively wide range of parameters involved. Special cases: exact solution The analytical solutions derived in the previous section are based on the small perturbation assumption 4. In certain situations exact solutions can be obtained without such assumption. For a Newtonian flow with the Navier slip condition the exact solution of boundary problem 9 is obtained as follows: p = γ 1 + γ1 + γ γ p 0 / + γ 1 p 0 1 x/ + γ p 1/ / + γ 1 p x/ γ H / 1m + β 0 H/ γ p 0 V = / + γ 1 p 0 / γ p / + γ 1 p / γ 1 + γ γ p 0 / + γ 1 p 0 1 x/ + γ p 1/ 47 / + γ 1 p x/ u s = β 0 H γ p 0 / + γ 1 p 0 / γ p / + γ 1 p a / γ 1 + γ γ p 0 / + γ 1 p 0 1 x/ + γ p / + γ 1 p x/ 1/. Table 9 Error as a function of ratio of entrance pressure to exit pressure. γ = 0.01 τ 0 = 0. β = andκ = 1 p f x / P x + p df x / /P Table 10 Error as a function of yield stress. γ = 0.01 p 0 = 100 τ 0 = 0. β = andκ = 1 τ f x / P x + p df x / /P

10 430 heol Acta 01 51: For plug flow with a constant wall slip coefficient problem 9 yields p = 1 x γ γ p + γ 1 1+1/α + 1 x γ p 0 + γ 1 1+1/α α/1+α γ 1 γ α αh γ p 0 + γ 1 1+1/α γ p + γ 1 1+1/α V = u s = β α γ x/γp + γ 1 1+1/α + 1 x/γp 0 + γ 1 1+1/α. 48 1/1+α An interesting fact given by Eqs. 47 and 48 should be noted: in case of a Newtonian flow and the Navier slip condition compressibility affects both pressure and velocity distribution profiles but wall slip only changes the shape of velocity profile but not that of pressure since the slip coefficient does not appear in the solutions for pressure. However as to be discussed later in general wall slip will affect the pressure profile. Condition γ = 0 is used in derivations of solutions 47 and 48. Solution for an incompressible fluid γ = 0 has been given by Mooney Approximate solution for circular channel flow This section considers solutions for an axis-symmetrical steady creeping flow in a circular tube. The governing equations for the flow are similar to those for the planar flow which are Eqs Using the procedures same to those in the planar flow the cross-section velocity profile is derived as follows: u = nt 1+1/n dp m 1/n 1 + 1/n nt 1+1/n m 1/n 1 + 1/n dp 1 + β 0 pa p κ 1 nt1+1/n r m 1/n 1 + 1/n dp α 0 r λ dp 1 pa + β 0 p κ dp α λ<r 49 where T r = max τ 0 r/dp/ 0. Furthermore the cross-section averaged velocity V = / 0 u x r rdr is determined as 16T 3+1/n V = dp 3 m 1/n 1 + 1/n3 + 1/n 16τ 0 T +1/n dp 3 m 1/n 1 + 1/n + 1/n + T 1+1/n dp 1 m 1/n 1 + 1/n κ pa α + β 0 dp α. 50 p α It is seen that both the cross-section velocity profile and the averaged velocity are very similar to those for the planar channel flows. Using the same procedures the pressure velocity and slip velocity distributions in the tube are solved and the solutions are presented in Appendix B. Discussion Effects of compressibility As seen in exact solutions 47 and 48 pressure distribution is only affected by compressibility. It is seen from Fig. 3 that d f x / < 0 or equivalently d p/ < 0 which indicates that the pressure distribution in x direction is concave. This conclusion can be arrived by considering a power law fluid with constant slip coefficient. Noticing that A 1 A 4 A 7 >0 A A 5 A 6 <0 and change of density is small in comparison with density at atmosphere or γ p 0 γ p << γ 1 one has A A = γ p p 0 7 γ p 0 + γ <<

11 heol Acta 01 51: Fig. 7 Effects of compressibility determined by Eqs and35. p = τ 0 = β = n= 0.4 κ = 0 α =.1 and κ = 0 It is derived from Eq. 3 that f x= A 1+ A 4 A 5 + A 6 and d p = d f x x = A 1 + A 4 A 5 + A 6 A A x +O >0 A 7 A 7 5 A A 1 + O < 0. A 7 A 7 53 Furthermore in view that A /A 7 γ Eq.53 shows that the larger the compressibility is the more concave the pressure distribution will be. This is clearly seen in Fig. 7. The concaveness of the pressure distribution is also reported in gaseous microchannel flows Jang and Wereley 004; Duan and Muzychka 007b. In Figs. 3 and 5 slip velocity is observed to increase in the flow direction. Again this conclusion can be explained by considering the flow of a power law fluid with a constant slip coefficient as follows. It is derived from Eq. 35 that du s = A 6 A 1 + A 4 A /A 7 A 5 + A A x/a 7 ln 1 + A /A 7 = A 6 A 1 + A 4 A A 1 + O > 0. A 5 + A 6 A 7 A 7 54 This also indicates that a larger compressibility a faster increases of slip velocity increases in x direction which is demonstrated in Fig. 7. Effects of wall slip As indicated above compressibility tends to make the pressure distribution to be concave down and the larger the compressibility is the more concave the pressure distribution will be. In case of constant slip coefficient longitudinal pressure distribution still remains concave Fig. 3 and more slip leads to a more concave pressure distribution Fig. 8. However when Fig. 8 Effects of wall slip with constant slip coefficients determined by Eqs and35. p = τ 0 = γ = n = 0.4 α =.1andκ = 0

12 43 heol Acta 01 51: Fig. 9 Effects of wall slip with pressure-dependent slip coefficient presented by Eqs and43. p = τ 0 = and γ = n = 0.4 α =.1andκ = 1 wall slip coefficient is pressure-dependent the fluid behaves very differently. Actually as seen in Fig. 5 the pressure distribution for a Newtonian fluid is interesting; as x increases first it is concave and then becomes convex d f x/ > 0. Compared with the flow with a constant slip coefficient a main difference in a flow with a pressure-dependent slip coefficient is that wall slip offsets the effect of compressibility on pressure distribution. The distribution may be no longer concave instead it becomes convex as the slip is strong enough Fig. 9. A pressure-dependent slip coefficient tends to make the pressure distribution to be convex. As the pressure is convex the shear stress at wall which is proportional to -dp/ will decrease in the flowing direction. This tendency is the same as that observed in other authors results Hatzikiriakos and Dealy 199a. et us discuss the ratio of slip velocity to the crosssection averaged velocity. Consider a power law fluid flow. Equations 18 and 19 yield u s V = 1 H 1+1/n α 1+1/n α m 1/n +1/nβ 0 p 0 /p κ dp 1/n α Equation 55 indicates that the ratio of slip velocity to the average velocity depends on n α pressure and its gradient. In view that dp/ increases in the flow direction Eq. 55 shows that the ratio increases in the flow direction when 1/n <α remains constant when 1/n = α and decreases when 1/n >α. An example of the third situation is shown in Fig. 8. Besides the ratio will be a constant in case of a Newtonian fluid with the Navier slip condition since n α = 1. However when the flow has a strong slip with a pressure-dependent slip coefficient the ratio always increases in the flow direction as shown in Fig. 9 in which it is seen that at the exit the ratio can reach 1 and the flow becomes a plug flow. Figure 10 clearly demonstrates how the cross-section velocity profile gradually evolves from an uneven to a uniform distribution the flow becoming a plug flow at the exit. As indicated earlier other expressions for wall slip are also available such as that by Hill et al and Person and Denn 1997 for which solutions for the flows are also available Appendix C. Since these expressions are similar in the sense that their slip coefficients decrease with local pressure it is expected that they will lead to similar flow behaviors. Energy dissipation effects from unsteadiness and mass flow rate Fig. 10 D velocity distribution of the flow of the Herschel Bulkley fluid with pressure-dependent slip coefficient in Fig. 9 obtained by Eq. 16. β 0 = In the channel flow the total energy dissipation consists of two parts; the first part comes from D vis the viscous dissipation within the fluid and the second part results from D slip the dissipation because of wall slip. Consider a fully developed incompressible flow of a power law fluid with a constant slip coefficient within a channel of

13 heol Acta 01 51: unit width. The ratio of wall slip dissipation to the total dissipation reads as Bird et al. 00 D slip D vis + D slip = m du/dy n y=h/ u s H/ H/ m du/dy n+1 dy + m du/dy n y=h/ u s β 0 H/ 1+α p 0 p 1+α / α = H/ +1/n p 0 p 1+1/n / m 1/n 1/n + 1/n + β 0 H/ 1+α p 0 p 1+α / α 1 = 1 + α 1/n H/ 1+1/n α p 0 p 1/n α / + 1/n β 0 m 1/n 56 in which because the flow is fully developed the strain rate in the longitudinal direction is taken as zero in evaluation of viscous dissipation. Equation 56 shows that the ratio is determined by entrance pressure rheology of the fluid wall slip parameters and geometry of the channel. When 1/n α>0 the ratio decreases with the difference between the entrance and the exit pressure and whereas when 1/n α<0 it increases with the difference. Table 11 presents the ratio of a power law fluid flow in a narrow channel and it indicates that the wall slip dissipation is a main portion in the total dissipation especially at low entrance pressure. It is seen from Eq. 15 that an unsteady solution consists of two terms: the first term on the right hand side of the equation is the result of the instantaneous pressure gradient and the second term reflects the effects of unsteadiness. The second term indicates that a larger unsteadiness or a larger value of a has a larger effect on the velocity distribution. However its effect is very small in the ranges of the parameters used Fig. 11. Mass flow rate can be determined according to Eq. 9. Figure 1 presents the flow rate using Eqs and4 with x = 0. The figure clearly indicates that flow rate increases rapidly with the entrance pressure. With the given fluid parameters and range of the entrance pressure the mass flow rates of the Herschel Bulkley and the power law fluids are about the same Fig. 11 Effects of unsteadiness on velocity distribution of the Newtonian flow with the Navier slip condition Table 11 atio of wall slip dissipation to viscous dissipation p 0 /p a D slip /D total H = 0.00 m = 0.04 m p = Pa. The fluid is highdensity polyethylene HDPE with m = Pa s 0.5 n = 0.5 β 0 = m s 1 Pa.345 and α =.345 Zhang et al Fig. 1 Mass flow rates at different entrance pressure of the fluid flows in Fig. 3

14 434 heol Acta 01 51: Fig. 13 Slit flows of PDMS. flow I: p 0 = Pa p = Pa; flow II: p 0 = Pa p = Pa. m = 1700 Pa s 0.39 n = 0.39 τ 0 = 0 Pa γ = s /m and γ 1 = kg/m 3 with β 0 = ms 1 Pa.5 κ = 1 andα =.5 whereas the mass flow rate of the Newtonian fluid is much smaller. Application Slit die experiments have been carried out for polydimethylsiloxane PDMS. In the experiment four transducers are installed along the flow direction the distance from the first to the last transducers is m the gap of the die H is m and the ratio between the gap and the width of the die is The accuracy of a transducer is 0.5% of its measured pressure. The pressure was measured at the four transducers and the cross-section velocity was recorded at the first transducer the most upstream one. All of the parameters involved are obtained from previous studies. m n and τ 0 are obtained using an inverse solution approach for capillary and squeeze flows γ and γ 1 are determined from a P V T experiment and β 0 κandα are chosen as those that fit the experimental data from the measurement Tang and Kalyon 004; Kalyon and Tang 007; Tang and Kalyon 008b. The analytical solutions and 43 together with experimental data are presented in Fig. 13. Inthe figures the coordinate is set as zero at the location of the first transducer. From the figure it is seen that the analytical solutions and experimental data compare favorably for pressure distribution velocity and slip velocity. The experimental data indeed indicate that pressure distributions are convex and the analytical solutions reproduce the feature. The discrepancy between measurement and analytical solutions is attributed to insufficient accuracy of the transducers and parameters of rheology and wall slip adopted from previous sources. Concluding remarks This paper studies pressure-driven creeping flows of the Herschel Bulkley fluid and its simplifications and it demonstrates that both compressibility and wall slippage affect behaviors of the flows such as longitudinal pressure distribution cross-section velocity and energy dissipation. Besides producing a quick estimate for the flows in practical problems the derived solutions can be used to formulate inverse problems to identify material shearing properties and wall slip coefficients

15 heol Acta 01 51: involved. In addition the analytical solutions may also provide information and hint for issues such as onset of unsteadiness and instability phenomena in creeping channel flows. Acknowledgements This work was supported by PSC CUNY. The author thanks Drs. D. M. Kalyon and H. Gevgilili for valuable inputs and Ms. E. Birinci for collecting experimental data. The author is grateful to the anonymous reviewers for their suggestions. Comments from Drs. M. M. Denn and J. Morris are acknowledged. Appendix A. Dimensionless governing equations With the following dimensionless variables t = H 1/n Hp0 t x = x m m ρ = ρ γ 1 p = p p 0 u = 1 H y = y H Hp 0 1/n uτ xy = Hp 0 τ xy 57 the continuity and momentum equations of the planar flow Eqs. 3 4 and 6 are transformed into the following dimensionless forms: ρ + ρ u = 0 t x 58 1 u ρ = p + τ xy = 0 Er t x y 59 0 < x < 1 1 < y < p = p 0 x = 0; p = 1 x = where Er = p 0 /γ 1 H m/hp 0 /n andp 0 = p 0/p. The wall slip relation 5 becomes u = β τ α xy y =± 1 6 where β = β 0 m 1/n /H p a /p 0 κ Hp 0 / α 1/n. The Herschel Bulkley model 8 is expressed as τ xy = u n 1 u y ± τ y 0 τ xy >τ 0 u = 0 τ y xy τ 0 in which τ 0 = τ 0/ Hp The relation between density and pressure 9 becomes ρ = γ p in which γ = γ p 0 /γ 1. Appendix B. Circular channel flow Approximate solution The solutions for the flow in a circular tube are in forms same to those for the planar flow. The solutions for the circular tube flow with a constant slip coefficient are in forms of Eqs and35 and those with pressuredependent slip coefficient are in forms of Eqs and 43. The constants in the solutions are given as follows: 16 ˆT 3+1/n P 3 A 1 = m 1/n 1 + 1/n3 + 1/n 16τ 0 ˆT +1/n P 3 m 1/n 1 + 1/n + 1/n + ˆT 1+1/n P 1 m 1/n 1 + 1/n A = P p a A 3 = p 0 p a A 4 = β 0 α P α α 48 ˆT 3+1/n P 4 A 5 = m 1/n 1 + 1/n3 + 1/n + 8 ˆT +1/n P 3 m 1/n 1 + 1/n 48τ 0 ˆT +1/n P 4 m 1/n 1 + 1/n + 1/n + 8τ 1+1/n 0 ˆT P 3 m 1/n 1 + 1/n + ˆT 1+1/n P m 1/n 1 + 1/n ˆT 1/n P 1 m 1/n A 6 = αβ 0 α P α 1 A α 7 = p 0 + γ γ p a In the above ˆT is obtained from T with dp/ replaced by -P. For a power law fluid A 1 and A 5 will be simplified as 1+1/n P 1/n A 1 = m 1/n 3 + 1/n A 5 = 1+1/n 1/n 1 P m 1/n n 3 + 1/n 66

16 436 heol Acta 01 51: and for a Newtonian fluid A 1 = P 8m A 5 = 8m. 67 For a plug flow A 1 A 5 = 0. Exact solution For a Newtonian fluid with the Navier slip condition the solutions are as follows p = γ 1 + γ1 + γ γ p 0 / + γ 1 p 0 1 x/+ γ p 1/ / + γ 1 p x/ γ / 8m + β 0 / γ p 0 / + γ 1 p 0 / γ p / + γ 1 p / V = γ 1 + γ γ p 0 / + γ 1 p 0 1 x/ + γ p 1/ 68 / + γ 1 p x/ β 0 γ p 0 u s = / + γ 1 p 0 / γ p a / + γ 1 p a / γ1 + γ γ p 0 / + γ 1 p 0 1 x/ + γ p 1/. a / + γ 1 p a x/ For a plug flow of the Herschel Bulkley fluid one has p = 1 γ x γ p + γ 1 1+1/α + 1 x γ p 0 + γ 1 1+1/α α/1+α γ 1 γ 69 α α γ p 0 + γ 1 1+1/α γ p + γ 1 1+1/α V = u s = β αγ 1/1+α x/γ p + γ 1 1+1/α + 1 x/ γ p 0 + γ 1 1+1/α. The above solutions hold for compressible flows γ = 0. Appendix C. Channel flow subject to an alternative wall slip expression Another wall slip formula is proposed by Hill et al and Person and Denn 1997asfollows β = β 0 e β s p 70 where β s 10 9.etβ s p << 1 the formula is approximated as β = β 0 1 β s p. 71 Consider a planar channel flow. Corresponding to the pressure decomposition 1 3 it is assumed that both f x and dfx/ are small perturbations f x df x / << 1 7 P x + p 0 P and ignoring their second- and higher-order terms simplify the problem 9as A 8 + A 9 Ṁ + A 10 + A 11 Ṁ x + A 1 + A 13 Ṁ f + A 5 + A 14 df = 0 f 0 = f = 0 73

17 heol Acta 01 51: where H α A 8 = A 1 + β 0 1 β s p 0 P α A 9 = 1 + γ p 0 γ 1 γ1 H α A 10 = β 0 β s P α+1 A 11 = γ H γ1 P A 1 = β 0 β s A 14 = αβ 0 H α P α A 13 = γ γ1 α P α 1 74 and A 1 and A 5 are given in Eq. 6. Problem 73 is solved by f = A 5+ A 14 A 10 + A 11 Ṁ A 8 + A 9 Ṁ A 1 + A 13 Ṁ A1 + A 13 Ṁ 1 e A 1+A 13 Ṁx/A 5 +A 14 A10 + A 11 Ṁ A1 + A 13 Ṁ x. 75 in which the mass flow rate Ṁ is determined by A 5 + A 14 A 10 + A 11 Ṁ A 8 + A 9 Ṁ A 1 + A 13 Ṁ A1 + A 13 Ṁ 1 e A 1+A 13 Ṁ/A 5 +A 14 A10 + A 11 Ṁ A1 + A 13 Ṁ =0. 76 Ṁ may be solved numerically from Eq. 76. etṁ s be the mass flow rate for the corresponding fully developed creeping flow and is determined by Eq. 17 with dp/ replaced by -P Ṁ can be also approximated as follows: Ṁ = B ± B 4B 1 B 3 77 B 1 in which Ṁ will be selected as that larger than Ṁ s and B 1 = A 8 A 13 e A 1+A 13 Ṁ s/a 5 +A 14 1 A 11 A 13. B = A 5 + A 14 A 11 A 9 A 1 A 8 A 13 1 e A 1+A 13 Ṁ s/a 5 +A 14 A 11 A 1 + A 10 A B 3 = A 5 + A 14 A 10 A 8 A 1 1 e A 1+A 13 Ṁ s /A 5 +A 14 A 10 A 1. Unlike solutions 3 and38 the analytical solution 75 includes effects of f x in the wall slip and density. The solutions for pressure velocity and slip velocity can be obtained using solution 75. In case of a circular channel flow still the solution is Eq. 75withA 1 and A 5 as given in Eq. 65 and other constants as given in 74 but H is replaced by. eferences Allende M Kalyon DM 000 Assessment of particle-migration effects in pressure-driven viscometric flows. J heol 441:79 90 Aral B Kalyon DM 1994 Effects of temperature and surface roughness on time-dependent development of wall slip in torsional flow of concentrated suspensions. J heol 38: Aral B Kalyon DM 1995 heology and extrudability of very concentrated suspensions: effects of vacuum imposition. Plast ubber Compos Process Appl 4:01 10 Ballesta P Besseling Isa Petekidis G Poon WCK 008 Slip and flow of hard-sphere colloidal glasses. Phys ev ett 101: Bird B Dai GC Yarusso BJ 1983 The rheology and flow of viscoplastic materials. ev Chem Eng 1:1 70 Bird B Stewart WE ightfoot EN 00 Transport phenomena nd ed. Wiley Brochard-Wyart F De Gennes PG 007 Naive model for stickslip processes 3: Chen Y Kalyon DM Bayramli E 1993 Effects of surface roughness and the chemical structure of materials of construction on wall slip behavior of linear low density polyethylene in capillary flow. J Appl Polym Sci 50: Coussot P 1994 Steady laminar flow of concentrated mud suspensions in open-channel. J Hydrol es 3: de Gennes P-G 1979 Viscometric flows of tangled polymers. C Acad Sci Ser B 8814:19 0 de Gennes P-G 00 On fluid/wall slippage. angmuir 18: Den Doelder C J Koopmans J Molenaar J 1998 Quantitative modeling of HDPE spurt experiments using wall slip and generalized Newtonian flow. J Non-Newton Fluid Mech 79: Denn MM 001 Extrusion instabilities and wall slip. Annu ev Fluid Mech 33:65 87 Duan ZP Muzychka YS 007a Slip flow in non-circular microchannels. Microfluidics Nanofluidics 3: Duan ZP Muzychka YS 007b Compressibility effect on slip flow in non-circular microchannels. Nanoscale Microscale Thermophys Eng 11:59 7 Engqvist A Hogg AM 004 Unidirectional stratified flow through a non-rectangular channel. J Fluid Mech 509:83 9 Fujiyama M Inata H 00 Melt fracture behavior of polypropylene-type resins with narrow molecular weight dis-

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