Evan Mitsoulis. School of Mining Engineering and Metallurgy. National Technical University of Athens. R.R. Huilgol

Transcription

1 Cessation of Couette and Poiseuille ows of a Bingham plastic and nite stopping times Maria Chatzimina, Georgios C. Georgiou Department of Mathematics and Statistics, University of Cyprus P.O. Box 2537, 678 Nicosia, CYPRUS Tel.: Fax: georgios@ucy.ac.cy Evan Mitsoulis School of Mining Engineering and Metallurgy National Technical University of Athens Heroon Polytechniou 9, 57 8 Zografou, Athens, GREECE R.R. Huilgol School of Informatics and Engineering Flinders University of South Australia G.P.O. Box 2, Adelaide, SA 5, AUSTRALIA Short title: CESSATION OF BINGHAM FLOWS

2 Abstract We solve the one-dimensional cessation Couette and Poiseuille ows of a Bingham plastic using the regularized constitutive equation proposed by Papanastasiou and employing nite elements in space and a fully implicit scheme in time. The numerical calculations conrm previous theoretical ndings that the stopping times are nite when the yield stress is nonzero. The decay of the volumetric ow rate, which is exponential in the Newtonian case, is accelerated and eventually becomes linear as the yield stress is increased. In all ows studied, the calculated stopping times agree very well with the theoretical upper bound estimates. KEYWORDS : Couette Flow Poiseuille Flow Bingham Plastic Papanastasiou Model Cessation. 2

3 Introduction In viscometric ows, one can bring the uid to a halt by setting the moving boundary to rest, in the case of Couette ows, or by reducing the applied pressure gradient tozeroinpoiseuille ows. In a Newtonian uid, the corresponding velocity elds decay to zero in an innite amount of time []. In a Bingham plastic, the velocity elds go to zero in a nite time, which emphasizes the role of the yield stress [2]. Glowinski [3] and Huilgol and co-workers [2, 4] provided explicit theoretical nite upper bounds on the time for a Bingham material to rest in various ows, such as the plane and circular Couette ows and the plane and axisymmetric Poiseuille ows. In each case, the theoretical bound depends on the density, the viscosity, the yield stress, and the least eigenvalue of the Laplacian operator on the ow domain. More recently, Huilgol [5] has also derived upper bounds for the cessation of round Poiseuille ow of more general viscoplastic uids. The objective of the present work is to compute numerically the stopping times and make comparisons with the theoretical upper bounds provided in the literature for the cessation of three ows of a Bingham uid: (a) the plane Couette ow (b) the plane Poiseuille ow and (c) the axisymmetric Poiseuille ow. Instead of the ideal Bingham-plastic constitutive equation, we employ the regularized equation proposed by Papanastasiou [6], in order to avoid the need of determining a priori the yielded and unyielded regions in the ow domain. It should be noted that preliminary results for the case of the plane Poiseuille ow can also be found in Ref. [7]. The paper is organized as follows. In section 2, we discuss the regularized Papanastasiou equation for a Bingham plastic. In section 3, we present the dimensionless forms of the governing equations for the three ows of interest along with the corresponding theoretical 3

4 stopping times. In section 4, we present and discuss representative numerical results for all ows. The numerical stopping times agree very well with the theoretical upper bounds. Some discrepancies are observed only for low Bingham numbers when the growth parameter in the Papanastasiou model is not suciently high. Finally, section 5 contains the conclusions of this work. 2 Constitutive equation Let u and denote the velocity vector and the stress tensor, respectively, and _ denote the rate-of-strain tensor, _ ru +(ru) T () where ru is the velocity-gradient tensor, and the superscript T denotes its transpose. The magnitudes of _ and are respectively dened as follows: _ = r 2 II _ = r 2 _ : _ and = r 2 II = r 2 : (2) where II stands for the second invariant of a tensor. In tensorial form, the Bingham model is written as follows: 8 >< >: _ = = _ + _ (3) where is the yield stress, and is a constant viscosity. In anyow of a Bingham plastic, determination of the yielded ( ) and unyielded ( ) regions in the ow eld is necessary, which leads to considerable computational diculties in the use of the model. These are overcome by using the regularized constitutive equation 4

5 proposed by Papanastasiou [6]: = [ ; exp(;m _)] _ + _ (4) where m is a stress growth exponent. For suciently large values of the regularization parameter m, the Papanastasiou model provides a satisfactory approximation of the Bingham model, while at the same time the need of determining the yielded and the unyielded regions is eliminated. The model has been used with great success in solving various steady and time-dependent ows (see, for example, [8, 9] and references therein). 3 Flow problems and governing equations The governing equations along with the boundary and initial conditions of the three timedependent, one-dimensional Bingham-plastic ows of interest are discussed below. The theoretical upper bounds for the stopping times are also presented. 3. Cessation of plane Couette ow The geometry of the plane Couette ow is shown in Fig. a. The steady-state solution is given by u s x(y) = ; y V (5) H where V is the speed of the lower plate (the upper one is kept xed) and H is the distance between the two plates. We assume that at t=, the velocity u x (y t) isgiven by the above prole and that at t= + the lower plate stops moving. To non-dimensionalize the x-momentum equation, we scale the lengths by H, the velocity by V, the stress components by V=H, and the time by H 2 =, where is the constant density of the uid. With these 5

6 scalings, the x-momentum equation : (6) The dimensionless form of the Papanastasiou model is reduced to yx = Bn [ ; exp(;m _)] (7) where _=j@u x =@yj, is the Bingham number, and is the dimensionless growth parameter. Bn H V M mv H (8) (9) The dimensionless boundary and initial conditions are as follows: u x ( t)= t> u x ( t)= t u x (y ) = ; y y 9 >= > : () In the case of a Newtonian uid (Bn=), the analytical solution of the time-dependent ow, governed by Eqs. (6), (7) and (), is known []: u x (y t) = 2 X k= k sin (ky) e;k2 2 t : () Hence, the ow ceases theoretically at innite time. If the uid is a Bingham plastic (Bn > ), however, the ow comes to rest in a nite amount of time, as demonstrated by Huilgol et al. [2], who provide the following upper bound for the dimensionless stopping time: " # T f = jju x (y )jj 2 2 Bn (2) 6

7 where jju x (y )jj = Z u 2 x(y ) dy =2 : (3) 3.2 Cessation of plane Poiseuille ow The geometry of the plane Poiseuille ow is depicted in Fig. b. The steady-state solution is given by u s x (y) = 8 >< >: 2 s (H ; y ) 2 y s (H 2 ; y 2 ) ; where (;@p=@x) s is the pressure gradient, and (H ; y) y y H (4) y = <H (5) (;@p=@x) s denotes the point at which the material yields. Note that ow occurs only if (;@p=@x) s > H. The volumetric ow rate is given by Q = 2W s H 3 " ; 3 2 y H + y 3 # (6) 2 H where W is the width of the plates (in the z-direction). We assume that at t= the velocity u x (y t)isgiven by the steady-state solution (4) and that at t= + the pressure gradient isvanished, or reduced to (;@p=@x) < (;@p=@x) s, in which case the ow is expected to stop. The evolution of the velocity is again governed by the x-momentum equation. Using the same scales as in the plane Couette ow, with V denoting now the mean velocity in the slit, the dimensionless form of the x-momentum equation is = f (7) 7

8 where f denotes the dimensionless pressure gradient. The dimensionless form of the constitutive equation is given by Eq. (7). The dimensionless steady velocity prole becomes: u s x(y) = 8 >< 2 f s ( ; y ) 2 y y (8) >: 2 f s ( ; y 2 ) ; Bn ( ; y) y y where y = Bn f s : (9) It turns out that y is the real root of the cubic equation: y 3 ; y +2=: (2) Bn It is clear that a steady ow in the channel occurs only if f s >Bn. The dimensionless boundary and initial conditions for the time-dependent problem ( t)= t u x ( t)= t u x (y ) = u s x(y) y 9 >= > : (2) In the case of Newtonian ow (Bn=), the time-dependent solution is given by [] u x (y t) = 48 3 X k= (;) k+ (2k (2k ; ) cos ; ) y exp 3 2 " ; (2k ; )2 2 4 t # (22) which indicates that the ow stops only after an innite amount of time. In the case of a Bingham plastic (Bn > ), Huilgol et al. [2] provide the following estimate for the stopping time: T f = 4 2 " # jju x (y )jj f<bn: (23) Bn ; f The above estimate is valid when f < Bn(or, equivalently, when f < f s ) otherwise, the ow will not stop. 8

9 3.3 Cessation of axisymmetric Poiseuille ow The geometry of the axisymmetric Poiseuille ow is depicted in Fig. c. The steady-state solution is given by u s z(r) = 8 >< >: 4 s (R ; r ) 2 r r (R 2 ; r 2 ) ; (R ; r) r r R (24) where (;@p=@z) s is the pressure gradient, and the yield point isgiven by r = 2 <R: (25) (;@p=@z) s The volumetric ow rate is given by Q = s R 4 " ; 4 3 r R + r 4 # : (26) 3 R We assume that at t= the velocity u z (r t)isgiven by the steady-state solution and that at t= + the pressure gradient isvanished, or reduced to (;@p=@z) < (;@p=@z) s. Scaling the lengths by the tube radius R, the velocity by the mean velocity V, the pressure and the stress components by V=R, and the time by R 2 =, we obtain the dimensionless form of the z-momentum = f (r rz) (27) where f is the dimensionless pressure gradient. The dimensionless form of the constitutive equation is given by rz = Bn [ ; exp(;m _)] (28) where _=j@u z =@rj, Bn R V (29) 9

10 and M mv R : (3) The dimensionless steady velocity prole takes the form 8 u s z(r) = >< 4 f s ( ; r ) 2 r r (3) >: 4 f s ( ; r 2 ) ; Bn ( ; r) r r where r satises r = 2Bn f s (32) and r 4 ; r +3=: (33) Bn Note that a steady ow in the tube occurs only if f s > 2Bn. The growth of r with Bn is illustrated in Fig. 2, in which steady-state velocity proles calculated for various Bingham numbers are shown. The dimensionless boundary and initial conditions ( t)= t u z ( t)= t u z (r ) = u s z(r) r 9 >= > : (34) The time-dependent solution for Newtonian ow (Bn=) is given by [] u z (r t)=6 X k= J (a k r) a 3 k J (a k ) e;a2 k t (35) where J and J are respectively the zeroth- and rst-order Bessel functions of the rst kind, and a k, k=, 2, :::are the roots of J. In the case of a Bingham plastic (Bn > ), Glowinski

11 [3] provides the following estimate for the stopping time: T f = jju z (r )jj + 2Bn ; f f<2bn : (36) where jju z (r )jj = 2 Z u 2 z(r ) rdr =2 (37) and is the smallest (positive) eigenvalue of the problem: d r dw r dr dr + w = w () = w() = : (38) It is easily found that =a 2 ' 5:783, where a is the least root of J (x), with the corresponding eigenfunction being given by w (x)=j (a x). Therefore, T f = a 2 +a 2 jju z (r )jj f<2bn : (39) 2Bn ; f The estimate (39) holds only when f<2bn (or, equivalently, when f<f s ) otherwise, the ow will not stop. 4 Numerical results Since there are no analytical solutions to the ows under study, in the case of the Bingham plastic or the Papanastasiou model, we have used a numerical method, namely the nite element method with quadratic (P 2 -C ) elements for the velocity. For the spatial discretization, we used the Galerkin form of the momentum equation. For the time discretization, we used the standard fully-implicit (Euler backward-dierence) scheme. At each time step, the nonlinear system of discretized equations was solved using the Newton method. In the case of Couette ow, a 2-element mesh rened near the two plates has been used. In Poiseuille ows, the mesh consisted of 8 elements and was rened near the wall. The code has been

12 tested by solving rst the Newtonian ows and making comparisons with the analytical solutions. In all three problems, the agreement between the theory and the calculations was excellent. Cessation of plane Couette ow Figures 3-5, show the evolution of the velocity forbn= (Newtonian uid), 2 and 2, respectively. The growth parameter has been taken to be M =2. The numerical solution in Figure 3 compares very well with the analytical solution () for the Newtonian ow. The numerical solutions for Bingham ow (Figs. 4 and 5), show that a small unyielded region, where the velocity is at, appears near the moving plate. Note that for high Bingham numbers (i.e., Bn > 5), very small time steps (of the order of ;9 )were necessary in order to get convergence in the early stages of cessation. The size of the unyielded region increases as the time proceeds. Its left limit initially moves to the right but at higher times starts moving to the left, as the ow approaches complete cessation. Figure 6 shows the evolution of the volumetric ow rate, Q(t) = Z u x (y t) dy (4) for various Bingham numbers. These curves show the dramatic eect of the yield stress, which accelerates the cessation of the ow. In the Newtonian case (Bn=) and for small Bingham numbers the decay of the volumetric ow rate is exponential, at least initially. At higher Bingham numbers, the decay ofq becomes polynomial and eventually linear. The times at which Q= ;3 and ;5 are plotted as functions of the Bingham number in Fig. 7. The two times coincide for moderate or large Bingham numbers, which indicates that the ow indeed stops at a nite time. In order to make comparisons with the theoretical upper bound (2) we consider as numerical stopping time the time at which Q= ;5. The comparison between 2

13 calculations and theory, provided in Fig. 8, shows a verygoodagreement for moderate and higher Bingham numbers. The small discrepancies observed for low Bingham numbers are due to the fact that the value of M is not suciently high, as discussed below. For very small Bn, the eect of M is not crucial, since the material is practically Newtonian, which explains why the calculations fall again below the theoretical upper bound, as they should. Cessation of plane Poiseuille ow Figures 9-2 show the evolution of u x (y t) forbn= (Newtonian case),, 5 and 2. In Fig. 3, we see the evolution of the calculated volumetric ow rate for various Bingham numbers. As in plane Couette ow, the decay of the volumetric ow rate is exponential for small Bingham numbers and becomes polynomial at higher Bn values. Figure 4 shows plots of the times required for the volumetric ow ratetobecome ;3 and ;5 versus the Bingham number for both plane and round Poiseuille ows (with M =2). Before proceeding to the comparisons with the theoretical estimate (23), let us investigate the eect of the growth parameter M on the calculated stopping times. As demonstrated in Fig. 5, which shows results obtained with M=2 and M=5, the calculated stopping times are not so sensitive tom when the Bingham number is moderate or high, i.e. Bn. For smaller Bingham numbers, i.e. ;3 Bn, the time required for the volumetric ow rate to become ;5 is reduced as M is increased. For very small Bingham numbers, the uid is essentially Newtonian, and therefore the value of M has no eect on the calculations. Hence, in order to get converged results in the range ;3 Bn, the value of M has to be increased further. However, our studies showed that when M =, convergence diculties are observed of Bn > :. One way toresolve the problem is to reduce the time step. This might be good for extending the calculations to a slightly higher Bn. Beyond this critical Bn value, the required time step is very small and the accumulated round-o errors are so high so 3

14 that the error in the calculated stopping time is higher than that corresponding to a smaller value of M. As a conclusion, decreasing the time step is not the best way to obtain converged results for M>5. If results for small (but not vanishingly small) Bingham numbers and large M are necessary, then continuation in M must be used at each time step. According to our numerical experiments, using such a continuation will increase the computational time by at least times. Since we arenotinterested in so small values of Bn, such calculations have not been pursued. A comparison between theory and calculations is provided in Fig. 6 for the case f= (i.e., when the imposed pressure gradient is set to zero). Again, the agreement between theory and computations is excellent for moderate and high Bingham numbers. Again, the small discrepancies observed for small values of the Bingham number are due to the fact that M is not suciently high. We have also examined the case in which the imposed pressure gradient f is not zero. An ideal Bingham plastic stops after a nite time if f Bn and reaches a new steady-state if f>bn(with the volumetric ow rate corresponding to the new value of f). This is not the case with a regularized Bingham uid. Since M is nite, the ow will reach a new steady-state in which the volumetric ow ratemay be small but not zero. To illustrate this eect, we considered the case in which Bn= and M =5 and carried out simulations for dierent values of f. In Fig. 7a, we see the evolution of the volumetric ow rate for dierent values of f. Figure 7b is a zoom of the previous gure showing that indeed the volumetric ow rate reaches a nite value when f 6=. Thisvalue may be reduced further by increasing the value of M. The new volumetric ow rate is plotted against f in Fig. 8. Finally, in Fig. 9 we compare the times required to reach Q= ;3 with the theoretical estimate (23). For smaller values of Q, the numerical results move closer to the theoretical curve, but in a smaller range of f, as it is easily deduced from Fig. 23. The deviations between theory and 4

15 experiment become larger as the value of the imposed pressure gradient is increased. These, however, can be further reduced by increasing the value of M. Cessation of axisymmetric Poiseuille ow The results for the axisymmetric Poiseuille ow arevery similar to those obtained for the planar case. Figures 2-23 show the evolution of u x for M=2 and Bn= (Newtonian case),, 5 and 2. In Fig. 24, we zoom near the wall in order to see how the velocity prole changes when Bn=2. It is clear that a second unyielded region of a smaller size appears near the wall, in which the velocity is zero. In Fig. 25, we see the evolution of the calculated volumetric ow rate (scaled by 2), Q(t) = Z u z (r t) rdr (4) for M =2 and various Bingham numbers. As in plane Poiseuille ow, the cessation of the ow is accelerated as the Bingham number is increased. The calculated stopping times for Q= ;5 and f=, plotted versus the Bingham number in Fig. 26, agree well with the theoretical estimate (39), with the small discrepancies observed for low Bn excepted. 5 Conclusions The Papanastasiou modication of the Bingham model has been employed in order to solve numerically the cessation of plane Couette, plane Poiseuille, and axisymmetric Poiseuille ows of a Bingham plastic. The nite element calculations showed that the volumetric ow rate decreases exponentially for low, polynomially for moderate, and linearly for high Bingham numbers. Unlike their counterparts in a Newtonian uid, the corresponding times for complete cessation are nite, in agreement with theory. The numerical stopping times are 5

16 found to be in very good agreement with the theoretical upper bounds provided in Refs. [3, 2], for moderate and higher Bingham numbers. Some minor discrepancies observed for rather low Bingham numbers can be reduced by increasing the regularization parameter introduced by the Papanastasiou model. A noteworthy dierence between the predictions of the ideal and the regularized Bingham model is revealed when the imposed pressure gradient is nonzero and below the critical value at which a nonzero steady-state Poiseuille solution exists. In contrast to the ideal Bingham ow, which reaches complete cessation at a nite time, the regularized ow reaches a velocity prole corresponding to a small but nonzero volumetric ow rate. The value of the latter may be reduced by increasing the value of the regularization parameter M but will always be nonzero. Acknowledgement This research was partially supported by the Research Committee of the University of Cyprus. References [] T. Papanastasiou, G. Georgiou and A. Alexandrou, Viscous Fluid Flow, CRC Press, Boca Raton, 999. [2] R.R. Huilgol, B. Mena, and J.M. Piau, Finite stopping time problems and rheometry of Bingham uids, J. Non-Newtonian Fluid Mech. 2 (22) 97. [3] R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer-Verlag, New York,

17 [4] R.R. Huilgol, and B. Mena, On kinematic conditions aecting the existence and nonexistence of a moving yield surface in unsteady unidirectional ows of Bingham uids, J. Non-Newtonian Fluid Mech. 23 (24) 25. [5] R.R. Huilgol, Variational inequalities in the ows of yield stress uids including inertia: Theory and applications, Phys. Fluids 4 (22) 269. [6] T.C. Papanastasiou, Flows of materials with yield, J. Rheology 3 (987) 385. [7] M. Chatzimina, G.C. Georgiou, E. Mitsoulis, and R.R. Huilgol, Finite stopping times in Couette and Poiseuille ows of viscoplastic uids, in: Proceedings of the XIVth Int. Cong. Rheol., Seoul, Korea, pp. NFF22--NF22-4. [8] Y. Dimakopoulos and J. Tsamopoulos, Transient displacement of a viscoplastic material by air in straight and suddenly constricted tubes, J. Non-Newtonian Fluid Mech. 2 (23) 43. [9] E. Mitsoulis and R.R. Huilgol, Entry ows of Bingham plastics in expansions, J. Non- Newtonian Fluid Mech. 22 (24) 45. 7

18 FIGURE CAPTIONS. Flow problems under study: (a) cessation of plane Couette ow (b) cessation of plane Poiseuille ow and (c) cessation of axisymmetric Poiseuille ow. 2. Steady velocity distributions for various Bingham numbers in round Poiseuille ow M=2. 3. Evolution of the velocity in cessation of plane Couette ow of a Newtonian uid. Comparison of the analytical (solid lines) with the numerical (dashed lines) solutions. 4. Evolution of the velocity in cessation of plane Couette ow of a Bingham uid with Bn=2 and M=2. 5. Evolution of the velocity in cessation of plane Couette ow of a Bingham uid with Bn=2 and M=2. 6. Evolution of the volumetric ow rate during the cessation of plane Couette ow of a Bingham uid with M=2 and various Bingham numbers. 7. Calculated times for Q= ;3 and ;5 in cessation of plane Couette ow of a Bingham uid with M=2. 8. Comparison of the computed stopping time (Q= ;5 ) in cessation of plane Couette ow of a Bingham uid with the theoretical upper bound M=2. 9. Evolution of the velocity in cessation of plane Poiseuille ow of a Newtonian uid.. Evolution of the velocity in cessation of plane Poiseuille ow of a Bingham uid with Bn= and M=2. 8

19 . Evolution of the velocity in cessation of plane Poiseuille ow of a Bingham uid with Bn=5 and M=2. 2. Evolution of the velocity in cessation of plane Poiseuille ow of a Bingham uid with Bn=2 and M=2. 3. Evolution of the volumetric ow rate during the cessation of plane Poiseuille ow of a Bingham uid with M=2 and various Bingham numbers. 4. Calculated times for Q= ;3 and ;5 in cessation of plane and round Poiseuille ows of Bingham uids with M=2. 5. Calculated times for Q= ;3 and ;5 in cessation of plane Poiseuille ow of Bingham uids with M=2 (dashed) and M=5 (solid). 6. Comparison of the computed stopping time (Q= ;5 ) in cessation of plane Poiseuille ow of a Bingham uid with the theoretical upper bound f= and M=5. 7. (a) Evolution of the volumetric ow rate for various values of the imposed pressure gradient f (b) Zoom of the same plot showing that a nite volumetric ow rate is reached when f> plane Poiseuille ow with Bn= and M=5. 8. Volumetric ow rates reached with the regularized Papanastasiou model versus the imposed pressure gradient f plane Poiseuille ow, Bn= and M=5. 9. Comparison of the times required to reach Q= ;3 in cessation of plane Poiseuille ow of a regularized Bingham uid with the theoretical estimate of Huilgol et al. (22) for an ideal Bingham uid Bn= and M=5. 2. Evolution of the velocity in cessation of round Poiseuille ow of a Newtonian uid. 9

20 2. Evolution of the velocity in cessation of round Poiseuille ow of a Bingham uid with Bn= and M= Evolution of the velocity in cessation of round Poiseuille ow of a Bingham uid with Bn=5 and M= Evolution of the velocity in cessation of round Poiseuille ow of a Bingham uid with Bn=2 and M= Evolution of the velocity in cessation of round Poiseuille ow of a Bingham uid with Bn=2 and M=2 (zoom near the wall). 25. Evolution of the volumetric ow rate during the cessation of round Poiseuille ow of a Bingham uid with M=2 and various Bingham numbers. 26. Comparison of the computed stopping time (Q= ;5 ) in cessation of round Poiseuille ow of a Bingham uid with the theoretical upper bound f= and M=5. 2

21 u x = u x = H u x (y ) u x (y t) y u x =V u x = x t= t> (a) H u x (y ) u x (y t) y = t= t> (b) R u z (r ) u z (r t) r = t= t> (c) Figure : Flow problems under study: (a) cessation of plane Couette ow (b) cessation of plane Poiseuille ow and (c) cessation of axisymmetric Poiseuille ow. 2

22 2.5 u z 2.5 Bn= (Newtonian uid) Bn= Bn=3 Bn= Bn= (Solid) r Figure 2: Steady velocity distributions for various Bingham numbers in round Poiseuille ow M =2..2 u x t= y Figure 3: Evolution of the velocity in cessation of plane Couette ow of a Newtonian uid. Comparison of the analytical (solid lines) with the numerical (dashed lines) solutions. 22

23 .2 u x.8.6 t= y Figure 4: Evolution of the velocity in cessation of plane Couette ow of a Bingham uid with Bn=2 and M =2..2 u x.8.6 t= y Figure 5: Evolution of the velocity in cessation of plane Couette ow of a Bingham uid with Bn=2 and M =2. 23

24 .6 Q Bn= t Figure 6: Evolution of the volumetric ow rate during the cessation of plane Couette ow of a Bingham uid with M =2 and various Bingham numbers. T f Q= ;5 Q= ;3.... Bn Figure 7: Calculated times for Q= ;3 and ;5 in cessation of plane Couette ow of a Bingham uid with M =2. 24

25 T f Theory. Numerical (Q= ;5 )... Bn Figure 8: Comparison of the computed stopping time (Q= ;5 ) in cessation of plane Couette ow of a Bingham uid with the theoretical upper bound M =2. 2 u x.5 t= y Figure 9: Evolution of the velocity in cessation of plane Poiseuille ow of a Newtonian uid. 25

26 2 u x.5.5 t= y Figure : Evolution of the velocity in cessation of plane Poiseuille ow of a Bingham uid with Bn= and M =2. 2 u x.5 t= y Figure : Evolution of the velocity in cessation of plane Poiseuille ow of a Bingham uid with Bn=5 and M =2. 26

27 2 u x.5.5 t= y Figure 2: Evolution of the velocity in cessation of plane Poiseuille ow of a Bingham uid with Bn=2 and M =2. Q Bn= t Figure 3: Evolution of the volumetric ow rate during the cessation of plane Poiseuille ow of a Bingham uid with M =2 and various Bingham numbers. 27

28 T f Q=.... Round Poiseuille ow Plane Poiseuille ow.... Bn Figure 4: Calculated times for Q= ;3 and ;5 in cessation of plane and round Poiseuille ows of Bingham uids with M =2. T f Q= ;5 Q= ;3.... Bn Figure 5: Calculated times for Q= ;3 and ;5 in cessation of plane Poiseuille ow of Bingham uids with M =2 (dashed) and M =5 (solid). 28

29 T f Theory. Numerical (Q= ;5 )... Bn Figure 6: Comparison of the computed stopping time (Q= ;5 ) in cessation of plane Poiseuille ow of a Bingham uid with the theoretical upper bound f = and M =5. 29

30 Q f = t.5 Q...5 f = t Figure 7: (a) Evolution of the volumetric ow rate for various values of the imposed pressure gradient f (b) Zoom of the same plot showing that a nite volumetric ow rate is reached when f> plane Poiseuille ow with Bn= and M =5. 3

31 .5 Q f f Figure 8: Volumetric ow rates reached with the regularized Papanastasiou model versus the imposed pressure gradient f plane Poiseuille ow, Bn= and M =5. 3 T f 2 Estimate for ideal Bingham uid Regularized Bingham uid (Q= ;3 ) f Figure 9: Comparison of the times required to reach Q= ;3 in cessation of plane Poiseuille ow of a regularized Bingham uid with the theoretical estimate of Huilgol et al. (22) for an ideal Bingham uid Bn= and M =5. 3

32 u x t= y Figure 2: Evolution of the velocity in cessation of round Poiseuille ow of a Newtonian uid. 2.5 u x t= y Figure 2: Evolution of the velocity in cessation of round Poiseuille ow of a Bingham uid with Bn= and M =2. 32

33 2.5 u x 2.5 t= y Figure 22: Evolution of the velocity in cessation of round Poiseuille ow of a Bingham uid with Bn=5 and M = u x t= y Figure 23: Evolution of the velocity in cessation of round Poiseuille ow of a Bingham uid with Bn=2 and M =2. 33

34 .25 u x t= y Figure 24: Evolution of the velocity in cessation of round Poiseuille ow of a Bingham uid with Bn=2 and M =2 (zoom near the wall). Q Bn= t Figure 25: Evolution of the volumetric ow rate during the cessation of round Poiseuille ow of a Bingham uid with M =2 and various Bingham numbers. 34

35 T f Theory. Numerical (Q= ;5 )... Bn Figure 26: Comparison of the computed stopping time (Q= ;5 ) in cessation of round Poiseuille ow of a Bingham uid with the theoretical upper bound f = and M =5. 35