ELASTIC INSTABILITIES IN CONE{AND{PLATE FLOW: SMALL GAP THEORY. David O. Olagunju. University of Delaware. Newark, DE 19716

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1 ELASTIC INSTABILITIES IN CONE{AND{PLATE FLOW: SMALL GAP THEORY David O. Olagunju Department of Mathematical Sciences University of Delaware Newark, DE June 15, 1995

2 Abstract Consider the axisymmetric, inertialess cone{and{plate ow of a viscoelastic uid. A perturbation method is used to obtain more tractable equations that describe the ow when the gap angle is small. A linear stability analysis of the base viscometric ow shows that there is a loss of stability when an elasticity parameter E=DeWe, increases past a critical value. This purely elastic instability is of the oscillatory type. We obtain expressions for the critical elasticity number, frequency and wave number. The critical Deborah number varies as p, the wave length as and the wave speed as p, where is the gap angle. The most unstable mode exhibits innitely many logarithmically{spaced roll cells which propagate inward towards the apex of the cone. These results are in agreement with experimental and numerical results.

3 1 Introduction In this paper we consider the cone{and{plate ow of a viscoelastic uid. This ow has important applications in rheology. One of the most commonly used instruments for the measurement of viscosity and normal stresses coecients is based on this ow. A sample of a material is placed in the gap between an inverted cone and a plate. The cone is then rotated at a constant angular speed. The torque and normal force on the plate are then measured and used to determine the properties of the material. In so doing certain assumptions are made regarding the ow: for example, it is assumed that the ow is stationary and axisymmetric; and that there is no radial secondary ow. Experimental studies however have shown that some of these assumptions do not always hold. The work of Magda and Larson [3] and McKinley et al. [4, 5] showed that beyond a critical value of the shear rate the ow becomes time{dependent. In particular, McKinley et al., showed through detailed experiments that at the critical shear rate a Hopf bifurcation occurs. This conclusion was conrmed by their numerical solution of the linear stability problem. Phan{Thien [11] solved analytically the stability problem for cone{plate ow of an Oldroyd{B uid. Assuming that the ow is inertialess, and that the gap angle is small he showed that the base ow became unstable when 1

4 the Deborah number De exceeded a critical value given by (1.1) De P T c s 2 = 5 ; where is the retardation parameter. Olagunju and Cook[6] found that the critical Deborah number is (1.2) s 2 De c = (2 + 3) : As the Deborah number passes through this critical value a real eigenvalue crosses the imaginary axis into the right half{plane. In [10], Olagunju showed that at the critical Deborah number there is a supercritical pitchfork bifurcation to another stationary solution. These analytical results do not however explain the kind of instabilities found in the experiments cited in which the observed instabilities are time dependent and oscillatory. One of the deciencies of the approximation used in [6, 11] is the assumption that certain derivatives are small and therefore negligible. In the current work, that deciency is corrected by a special choice of coordinate transformation. The results obtained from this new asymptotic theory agree with both the experimental and numerical results cited above. In addition, we are able to nd analytical expressions for the critical parameters at the onset of instability including the frequency, wave number and elasticity number. The most unstable mode has an innite number of roll cells. Our analysis also shows that the critical Deborah number varies as the square root of the 2

5 gap angle while the critical radial wave number varies as the reciprocal of the gap angle. Although we consider the inertialess ow of an Oldroyd{B uid, the method used can be extended to other constitutive models and to ows with inertia. It can also be applied to the parallel{plate ow with small aspect ratio. Some of these will be discussed in future work. For further discussion on these and other shear ows with curved streamlines see the review paper by Larson [2]. 2 Governing Equations Consider the ow of a viscoelastic uid conned to the gap between an inverted cone and a at plate in which the cone rotates at a constant angular speed $ and the gap angle is. In rheometric devices the gap angle is very small usually less than 4 o. In the following discussion a spherical coordinate system (r; ; ) will be used. The equations of motion are: (2.1) r ~v = 0; (2.2) ~ D~v D~ t =?r~p + r ~ T: For the extra stress we use the Oldroyd{B constitutive model given by (2.3) ~T = 2 s D + ~ ; where ~, the Maxwell stress, satises the equation (2.4) ~ + ( D~ D~ t? L~? ~ L T ) = 2 p D: 3

6 Here ~v is the velocity, ~p is the pressure, ~ is the density, is the relaxation time, while s and p are solvent and polymer viscosities respectively; L is the velocity gradient tensor and D its symmetric part. The above system of equations is to be solved subject to no slip conditions at the solid boundaries: (2.5) (2.6) (2.7) ~v = 0 on = 2 ; ~v r = ~v = 0; ~v = ~r$ cos on = 2? ; ~v = 0 at ~r = 0: In addition we must prescribe conditions at the uid{air interface. In the present analysis we will assume that the domain is innite and instead only require that ~v = O(~r) as ~r! 1. We nondimensionalize as follows: (2.8) (2.9) ~r = ar; ~ t = $t; ~v = a$v; ~p = _p; ~ = _ ; where _ = $= is the shear rate and a is a typical length say the radius of the plate. Dene a new independent variable (2.10) = (=2? )=; so that 0 1. Then = 0 on the plate and = 1 on the cone. We also introduce the following coordinate transformation (2.11) r = (or = r 1= ) 4

7 so that we can retain higher order derivatives with respect to r. Lastly we let (2.12) and (2.13) = 0 = v = (ru; rv; rw): 1 C A ; Let q = (u; v; w; p; ; ; ;?; ; ): We now seek an expansion of the form (2.14) q = q 0 + q 1 + : 3 Leading order theory In this paper we shall consider only the leading order equations in the limit! 0. For axisymmetric, creeping (i. e. ~ = 0) ow the leading order equations, after dropping the subscripts, are: (3.1) (3.2) @? + (1? @? + (1? )r2 v; + (1? )r2 w; 5

8 (3.5) (3.6) @ );? +? ) (3.7) (3.8) ? ; (3.9) and (3.10) )?We(? + +? )? v ): The no{slip boundary conditions in this case are (3.11) u = v = w = 0; on = 0; and (3.12) u = v = 0; w = 1; on = 1: In addition, we require u, v, and w to be bounded at = 0 and as! 1. The laplacian is dened as r @ 2 : 6

9 The dimensionless quantities appearing above are the Deborah number De=$, the Weissenberg number We= _, and the retardation parameter = p =( s + p ). This system of equations may be described as a short wave approximation (radial wave number O(1=)). In contrast, the analysis of Phan{Thien [11], Olagunju and Cook [7],and Olagunju [10] all assume that the radial wave number is O(1). In that respect the two models represent dierent asymptotic limits. Consequently neither is recoverable as a special case of the other. 4 Base ow and its stability The problem dened by (3.1){(3.12) admits a viscometric base ow given by (4.1) (4.2) u = v = 0; w = = = =? = 0; =?; = 2De: To determine the stability of the base ow we dene small perturbations (4.3) q = q + ^q where q is the base ow given by (4.1){(4.2), and ^q is the perturbation. Substituting (4.3) in (3.1){(3.12) and linearizing we obtain the following equations in which the hats have been = 0; 7

10 (4.5) (4.6) (4.7) (4.8) (4.9) (4.10) (4.11) @? + (1? @? + (1? )r2 @t + (1? )r2 @t ); + =?We(? @t ; @ ; ): Introduce the stream function ; ; and seek separated solutions of the form (; w) = ik e t (q 1 ; q 2 ); (; ; ;?; ; ) = ik e t (q 3 ; q 4 ; q 5 ; q 6 ; q 7 ; q 8 ) where q i = q i ( ), k is positive and may be complex. First solve equations (4.8){(4.13) for q 3 q 8, substitute these in (4.5){(4.7) and eliminate p to 8

11 obtain the following equations (4.14) ( + 1) 2 (?? 1)(D 2? k 2 ) 2 q 1 = +2ikDeWe( + 3)D 2 q 1? 2De( + 1)( + 2)D 2 q 2 ; and (4.15) (D 2? k 2 )[( + 1)(?? 1)q 2 + ikweq 1 ] = 0; with boundary conditions (4.16) q 1 = q 0 1 = 0; q 2 = 0 for = 0; 1 where D d d and = De. From (4.15) and (4.16), we have (4.17) ( + 1)(?? 1)q 2 + ikweq 1 = 0: Note that =?1 is an eigenvalue. The corresponding eigensolution is stable. Using (4.17) in (4.14) and assuming that 6=?1, we obtain for q 1 the following eigenvalue problem (4.18) (D 2? k 2 ) 2 q 1 + D 2 q 1 = 0: and the boundary conditions (4.19) q 1 = q 0 1 = 0 for = 0; 1 where (4.20) = 2ikE[(1? )(2 + 4) + 3? 2] ( + 1) 2 (?? 1) 2 : 9

12 We have introduced a new elasticity parameter E = DeWe: We now show that the eigenvalue must be necessarily positive. If we multiply (4.18) by q 1, the complex conjugate of q 1 and integrate by parts we obtain (4.21) = R 1 0 j d2 q 1 j 2 + d 2 2k 2 R 1 0 j dq 1 d j2 d + k 4 R 1 0 jq 1j 2 d R 1 0 j dq 1 d j2 d and the assertion follows. The eigenvalue problem (4.18){(4.19) can be solved explicitly. First introduce a change of independent variables x =? 1=2, then we have to solve (4.18) on the interval?1=2 x 1=2 with boundary conditions the q 1 = q 0 1 = 0 for x = 1=2. In that case the eigenfunctions separate into even and odd functions. The even eigenfunctions are given by (4.22) where ' e (x) = cos( 2 =2) cos( 1 x)? cos( 1 =2) cos( 2 x); 1 = 1 2 ( + q 2? 4k 2 ); 2 = 1 2 (? q 2? 4k 2 ); are positive and = p satises the transcendental equation (4.23) 1 tan( 1 =2) = 2 tan( 2 =2): The odd eigenfunctions are (4.24) ' o (x) = sin( 2 =2) sin( 1 x)? sin( 1 =2) sin( 2 x); where (4.25) 2 tan( 1 =2) = 1 tan( 2 =2): 10

13 Equations (4.23) and (4.25) possess an increasing sequence of positive roots which we denote by f n g. From (4.20) we then solve for. If <(), the real part of, is negative then the base ow is linearly stable. On the other hand if <() > 0 then it is unstable while it is neutrally stable if <() = 0. If De=0, then = 0 and there is no non{trivial solution with <() 0 therefore the base ow is stable. Thus in the absence of elasticity the ow is stable. In particular, this shows that the inertialess ow of a Newtonian uid is stable. Note also that if = 0 in (4.20) then is purely imaginary and so there can be no non{trivial solution. This implies that there is no exchange of stability from the base ow to another stationary ow. In order to determine the critical condition for the onset of instability we solve (4.20) for. This equation reduces to (4.26) A A A A 1 + A 0 = 0; where (4.27) (4.28) (4.29) (4.30) (4.31) A 0 = 2ikE(3? 2)? ; A 1 = 8ikE(1? ) + 2(? 2); A 2 = 2ikE(1? )? ( 2? 6 + 6); A 3 =?2(? 1)(? 2); A 4 =?(? 1) 2 : 11

14 For a Maxwell uid ( = 1), both A 3 and A 4 are zero and (4.26) reduces to the quadratic equation (4.32) 2 + 2? (2ikE? ) = 0: For neutral stability set = i 0 where 0 is real. In that case (4.32) gives (4.33) 0 = 1; and (4.34) E = k : The last equation gives the neutral stability curve in the E{k plane. For 6= 1, we set = i 0 in (4.26) to obtain (4.35) 8k(? 1) 0 E? f[(? 1) ] 2? (2? ) 2 2 0g = 0; and (4.36) 2k[(? 1) ? 2]E (? 2)[(? 1) ] = 0: From (4.36) we have (4.37) E = k 0 (2? )[(? 1) ] [(? 1) ? 2] : Substituting for E in (4.35) we get (4.38) (1? ) (1? )(5 2? ) 4 0 +(3? ? 2 3 ) ? 3 = 0: 12

15 For a given viscosity ratio 6= 1, we solve (4.38) to obtain 0, substitute in (4.37) to obtain the critical elasticity number E as a function of the wave number k. For = 1=2, the equations give 0 = 1 and E = =k the same as the Maxwell case = 1. For a given wave number k, we obtain from (4.37) Figure 1: Neutral stability curves in E{k plane for selected values of the retardation parameter. a critical elasticity number E (corresponding to = 1 ), above which the base ow loses stability. As E passes through this critical number a pair of complex conjugate eigenvalues crosses the imaginary axis into the right half{plane. Thus at criticality the ow becomes time{dependent. This is in 13

16 Figure 2: Frequency 0 vs retardation parameter. agreement with the experimental and numerical results in [3, 4, 5]. Plots of the neutral stability curve for selected values of are given in g 1. In g. 2 we plot 0 vs. Note that the frequency c is given by c = i 0 De c : McKinley et al.[5] have solved numerically the linear stability problem for the inertialess cone{and{plate ow. They assumed a base ow which is valid for small but otherwise they solved the full equations using a Galerkin{ Chebychev method. In order to facilitate comparison with their work we 14

17 plot the neutral stability curves in?1 { k ~ plane (g. 3) where the physical wave number, which we denote by k, ~ is related to the wave number k by ~k = k=. Comparing this with McKinley et al. ( [5],g. 9) we see that agreement with their numerical results is excellent. Also our results show that the physical wave number k ~ scales like 1= so that the graph of k ~ vs is a rectangular hyperbola. This also agrees with McKinley et al. ( [5], g. 14a). Figure 3: Neutral stability curves in?1 { ~ k plane for = 0:41 and selected values of the Deborah number De. For each value of there is a minimum value of E above which the 15

18 base ow loses stability. This critical elasticity number depends only on the viscosity ratio and is given by (4.39) E c = (2? ) 0[(? 1) ] [(? 1) ? 2)] ; where 0 is a real root of (4.38) and = min k = 21: This minimum is attained at the critical wave number k c = 3:0905. In their numerical calculations McKinley et al. [5] obtained ~ k c 3:1 which gives k c correct to one decimal place. Figure 4: Critical elasticity number E c as a function of the retardation parameter. 16

19 Figure 5: Streamlines of radial secondary ow showing the existence of multiple toroidal vortices. Fig. 4 gives a plot of E c as a function of. For = 0, E c is innite and the ow is stable. As increases E c decreases until it reaches a minimum. Thereafter, E c increases to a local maximum and then decreases steadily as approaches unity. Note that since E = De 2 =, the critical Deborah number varies as p. The stream function given by taking the even eigenfunction is (4.40) = cos(k c ln + c t)' e (? 1=2): 17

20 A contourplot of the stream function (at t = 0) is shown in g. 5. We see that the secondary ow at the onset of instability consists of innitely many logarithmically{spaced toroidal vortices. This agrees with experimental results reported in [5]. Secondary ows with multiple roll cells have also been reported in the experimental work of Savins and Metzner [12] and the numerical results of DuPont and Crochet [1]. In fact there is an innite number of these recirculating cells. The equations of the dividing streamlines between cells are (4.41) (2n + 1) = exp[? ]; n = 0; 1; 2 : 2k c In physical variables the stream function is given by (4.42) = cos( k c ln r + ct)' e (? 1=2): If we let = ln r, then is a 0 wave in the {t plane with wave speed (4.43) c = c k c = 0 De c k c : Therefore since De c varies as p it follows that the wave speed also varies as p. 5 Conclusion We have derived a set of short wave equations which govern the axisymmetric inertialess ow of an Oldroyd{B uid when the gap angle is small. The equations admit a viscometric solution. A linear stability analysis shows 18

21 that when the elasticity number dened by E =DeWe increases past a critical value, the base ow loses stability. De and We are the Deborah number and Weissenberg number respectively. The instability is of the oscillatory type. We obtained analytical expressions for the critical elasticity number, the critical wave number and the frequency as functions of the retardation parameter. The critical wave number in physical coordinates is given by ~k = k c =, where k c = 3:0905. Thus we see that the radial wavelength is indeed small of order the gap angle. The critical Deborah number varies as p, the frequency as 1= p and the wave speed as p. The stream function of the ow at the onset of instability has innitely many roll cells which are logarithmically{spaced along the radial direction. These results are in very good agreement with experimental and numerical results. 19

22 References [1] Dupont S., and Crochet, M. J., Swirling ows of viscoelastic uids of integral type in rheogoniometers, Chem. Engng. Comm., 53 (1987) 199{221. [2] R. G. Larson, Instabilities in viscoelastic ows, Rheol. Acta, 31 (1992) 213{263. [3] Magda, J. J., and Larson, R. G., A transition occurring in ideal elastic liquids during shear ows, J. Non{New. Fluid Mech., 30 (1988) 1{19. [4] McKinley, G. H., Byars, J. A., Brown, R. A., and Armstrong, R. C., Observations on the inelastic instability in cone{and{plate ow of a polyisobutylene Boger uid, J. Non{New. Fluid Mech., 40, 201{229 (1991). [5] McKinley, G. H., Oztekin, A., Byars, J A., and Brown, R. A., Self{ similar spiral instabilities in elastic ows between a cone and a plate, J. Fliud Mech., 285 (1995) 123{164. [6] Olagunju, D. O., and Cook, L. P., Secondary ows in cone and plate ow of an Oldroyd{B uid, J. Non{New. Fliud Mech., 46 (1993) 29{47. [7] Olagunju, D. O., and Cook, L. P., Linear stability analysis of cone and plate ow of an Oldroyd{B uid, J. Non{New. Fluid Mech., 47 (1993) pp. 93{

23 [8] Olagunju, D. O., Asymptotic analysis of the nite cone{and{plate ow of a non{newtonian uid, J. Non{New. Fluid Mech., 50 (1993) 289{303. [9] Olagunju, D. O., Eect of free surface and inertia on viscoelastic parallel plate ow, J. Rheol., 38(1) (1994) 151{168. [10] Olagunju, D. O., Instabilities and bifurcations of von{karman similarity solutions in swirling viscoelastic ow, J. Appl. Math. Phy. (ZAMP), 46 (1995) 224{238. [11] Phan{Thien, N., Cone{and{plate ow of the Oldroyd{B uid is unstable, J. Non{New. Fluid Mech., 17 (1985) 37{44. [12] Savins, J. G., and Metzner, A. B., Radial (secondary) ows in rheogoniometric devices, Rheol. Acta, 3 (1970) 365{

24 FIGURE CAPTIONS Fig. 1 Neutral stability curves in E{k plane for selected values of the retardation parameter. Fig. 2 Frequency 0 vs retardation parameter. Fig. 3 Neutral stability curves in?1 { k ~ plane for = 0:41 and selected values of the Deborah number De. Fig. 4 Critical elasticity number E c as a function of the retardation parameter. Fig. 5 Streamlines of radial secondary ow showing existence of multiple toroidal vortices. 22

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Inertial 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,