Criticality and transition for a steady reactive plane couette flow of a viscous fluid

MECHANICS RESEARCH COMMUNICATIONS

Mechanics Research Communications 34 (2007) 130–135 www.elsevier.com/locate/mechrescom

Criticality and transition for a steady reactive plane couette flow of a viscous fluid Samuel S. Okoya

*

Department of Mathematics, Obafemi Awolowo University, Ile-Ife 220005, Nigeria

Abstract Numerical integration is used to determine critical and transitional values of parameters for steady, reactive, viscous, one dimensional plane Couette flow of an incompressible, homogeneous fluid of third-grade with the lower plate at rest while the upper is in uniform motion. The solutions are found for the following cases: (i) Bimolecular (ii) Arrhenius and (iii) Sensitized temperature dependence. Specifically, it is shown that the parameter K controlling the non-Newtonian fluid does not affect the flow velocity in any sense while the influence on the viscous dissipation parameter C is examined. The results obtained are then compared with similar results in the literature. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Plane couette flow; Non-Newtonian thermal explosions; Variable pre-exponential factor

1. Introduction Heat transfer to incompressible viscous non-Newtonian fluids is a problem of considerable practical significance involving heat exchange process either during preparation or in their applications. These include food processing, polymer processing, and biochemical industries (see Wei and Luo (2003) as well as the references contained there-in for many other industrial flows). Due to increasing importance of non-Newtonian fluids in industry, the influence of chemical reaction on heat transfer has more recently been extended to fluids obeying non-Newtonian constitutive equations. Plane Poiseuille flow has been analyzed recently in this type of reactive flow (Okoya, 2006) and further extensions in the directions of other simple reactive flows is still under investigation. Much of the earlier work on the third-grade fluids has been limited to cases as follows: Fosdick and Rajagopal (1980) made a complete thermodynamical analysis of the third-grade fluids and showed the restrictions on the stress constitutive equation. Rajagopal (1980) investigated some stability characteristics of third-grade fluids and show that they exhibit features different from that of Newtonian and second-grade fluids. The flow of third-grade fluids between heated parallel plates was treated by Szeri and Rajagopal (1985) and Kacou et al. *

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S.S. Okoya / Mechanics Research Communications 34 (2007) 130–135

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(1987) studied the flow of a third-grade fluid in a journal bearing and constructed a perturbative solution. Rajagopal and Na (1985) studied the flow between two vertical flat plates and heat transfer analysis of a thermodynamically compatible third-grade fluid. Passerini et al. (2000) analyzed the steady motions due to a pressure drop of second- and third-grade incompressible fluids through a hole in a wall. Lubrication of a slider bearing with a special third-grade fluid was considered by Yurusoy and Pakdemirli (2002). Remarks on the creeping flow of a third-grade fluid was provided by Rajagopal and Gupta (1981). Rajagopal (1982) carried out in detail the general thermodynamics of the differential type in which the fluid of third-grade is a special case. It is worth noting that the early work on the influence of reaction rate due to an exothermic zero-order chemical reaction has been limited to cases of Newtonian fluid. In fact, reactive–diffusive equations that incorporates the effect of viscous dissipation (Adler, 1975; Dubrulle and Zahn, 1991; Schlichting, 1968; Shonhiwa and Zaturska, 1986; Shonhiwa and Zaturska, 1987; Zaturska, 1981) (and several papers contained there-in) and those which excludes viscous dissipation (Okoya, 2004; Okoya, 2002; Tam, 1980) (and the relevant references given in the studies) provide a substantial contribution to energy transfer. In this paper, we discuss the solution set {b, d and hmax} of an incompressible third-grade fluid which includes variable pre-exponential factor, m. Here, b, d and hmax represent the activation energy parameter, Frank–Kamenetskii parameter and dimensionless central temperature excess at the center of the plates, respectively. We obtain critical and transitional values for particular cases: the Bimolecular temperature dependence (m = 1/2) (e.g., see Zaturska (1982)); The Arrhenius or zero-order reaction (m = 0) (e.g., see Tam (1980)); Sensitized temperature dependence (m =  2) (e.g., see Okoya (2004) and Okoya (2006)); the Frank–Kamenetskii approximation (m = 0 and b  1) (e.g., see Adler (1975), Okoya (2002), Zaturska (1981) and the literature contained there-in). As it stands, the Frank–Kamenetskii case grows exponentially with h. However, the Arrhenius case is bounded by exp(1/b). More importantly, the Sensitized source term has a maximum of (2b)2exp((1  2b)/b) at h = (1  2b)/2b2, while the Bimolecular source term has a maximum of (2/b)1/2exp((2  b)/2b) at h = (2  b)/b2. Although the procedures described in this paper are standard, we feel that it is necessary to explain some of the specific details for this particular application. This in turn would further our understanding of the non-Newtonian behavior of this flow. 2. Basic equations An incompressible, homogeneous fluid of grade 3 that is thermodynamically compatible is characterized by a stress tensor T which is related to the kinematical variables by T ¼ pI þ lA1 þ a1 A2 þ a2 A21 þ b3 ðtrA21 ÞA1 ;

ð1Þ

with the following restrictions: l P 0;

a1 P 0;

b3 P 0;

j a1 þ a2 j6

pffiffiffiffiffiffiffiffiffiffiffiffi 24lb3 ;

ð2Þ

where l is the coefficient of viscosity; a1, a2 and b3 are the material moduli and all are functions of temperature in general. Here p denotes the pressure,  I is the identity tensor and tr the trace of a matrix. In the above representation, the kinematical tensors A1 and A2 are defined by T

A1 ¼ ðrvÞ þ ðrvÞ ; d T A2 ¼ A1 þ A1 ðrvÞ þ ðrvÞ A1 ; dt

ð3Þ ð4Þ

where v denotes the velocity vector, $ is the gradient operator, the superscript ‘‘T’’ is the transpose and d/dt denotes the material time differentiation. A detailed thermodynamic analysis of the model, represented by several material variables in Eq. (1), is given in (Fosdick and Rajagopal, 1980). In the following, we considered steady flow of a grade 3 fluid between parallel plates located at y ¼ y 0 and y ¼ y 0 with the lower plate at rest while the upper is in uniform motion, respectively. The type of flow to be considered here is driven by the motion of the upper plate alone, without any externally imposed pressure gradient (plane Couette device) and has a unidirectional velocity field (Okoya, 2006; Szeri and Rajagopal, 1985). The balance of linear momentum reduces to

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S.S. Okoya / Mechanics Research Communications 34 (2007) 130–135

 2  3 !  2 ! d d u d d u d du l b3 ð2a1 þ a2 Þ ¼ ¼ 0; þ2 dy dy dy dy dy dy

ð5Þ

in the absence of body forces. Mathematically, Eq. (5) when l, a1, a2 and b3 are independent of temperature is formally identical to the velocity for a newtonian fluid. The problem for the velocity is now defined by Eq. (5) while the energy equation remain unaltered as in (Okoya, 2006). Using the dimensionless variables u, b, h, C, K and y of the new variable approach of Okoya (2006) and the references contained there-in, the reactive viscous liquid in steady flow can be written as  2 !   d2 u du d2 u du 1 þ 6K ¼ 0; ð6Þ ¼ dy 2 dy dy 2 dy  2  2 ! d2 h du du þC 1 þ 2K þ df ðh; b; mÞ ¼ 0; ð7Þ 2 dy dy dy subject to the boundary conditions uð1Þ ¼ 0;

uð1Þ ¼ 1;

ð8Þ

hð1Þ ¼ 0;

hð1Þ ¼ 0;

ð9Þ

and

where 2 2 y  RT0 lU ðT  T0 ÞE b3 U u 0 ; C ¼ 0 ; h ¼ ; u¼  ; b¼ ; K ¼ ; y 0 E ly 20 RT0 2 U0 K T 0b     QEAy 20 C 0 k m T0 m2 E h m d¼ exp  exp ; and f ðh; b; mÞ ¼ ð1 þ bhÞ 1 þ bh mm hm RK RT0 y¼

and we have assumed negligible reactant consumption. In the above formulation, T is the absolute temperature, l the viscosity, b3 the material coefficient, T0 is the wall temperature, K is the thermal conductivity of the material, Q is the heat of reaction, A is the rate constant, E is the activation energy, R is the universal gas constant, C0 is the initial concentration of the reactant species, h is the Planck’s number, k is the Boltzmann’s constant, m is the vibration frequency, m is a numerical exponent, C is a viscous heating parameter, K is the dimensionless non-Newtonian coefficient, U0 is a reference velocity, y is perpendicular to the direction of parallel flow, u is the flow velocity, u is the dimensionless velocity, y is the dimensionless perpendicular distance, h the dimensionless temperature and y 0 is the half-width of the reactant mass for simple geometries. Of interest in this formulation are the values of b, d and hmax for fixed values of K, C and m at which the solution h ceases to be determined as a smooth function of b, d and hmax. In most cases, the value of hmax is fixed, the control parameter is b and the value of d is generated. It is worth pointing out that some previous and classical results can be considered as particular cases of our result. 3. Method of solution The problem is defined by Eqs. (6)–(9). Eqs. (6) and (8) can be solved for u and is seen to be given by u ¼ ðy þ 1Þ=2:

ð10Þ

In this case, the non-Newtonian parameter K does not affect the flow in any sense. Now that the velocity of motion has been evaluated, we turn to the energy equation. Substitute Eq. (10) in Eq. (7) we have     d2 h h C K m 1 þ þ dð1 þ bhÞ exp þ ¼ 0: ð11Þ dy 2 1 þ bh 4 2

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Table 1 Criticality under exponential approximation C(1 + K/2)

0.0

0.4

4

20

80

200

dcr hmax, cr

0.878458 1.186842

0.841440 1.235279

0.569182 1.672226

0.094558 3.631242

0.000078 11.075664

3.40701(11) 26.048217

dcr

hmax, cr

Table 2 Criticality under bimolecular, Arrhenius and sensitized reaction where C* = C(1 + K/2) m

dcr C*

0.5 0.0 2

hmax, cr

C*

=0

0.903816 0.928400 1.042336

1.290899 1.329203 1.510163

C* = 20 0.5 0.0 2

0.145797 0.158542 0.223959

dcr

hmax, cr

C*

= 0.4

0.868795 0.893445 1.007719

1.345235 1.383814 1.566155

C* = 80 4.055470 4.109157 4.369557

0.003206 0.004047 0.010464

=4

0.611936 0.635818 0.747599

1.836363 1.877457 2.072560

C* = 200 12.711276 12.841943 13.537658

0.000072 0.000114 0.000737

31.343701 31.862591 35.607956

Table 3 Transitional values for bimolecular, Arrhenius and sensitized reaction m

btr C*

0.5 0.0 2

dtr

hmax, tr

=0

0.33692 0.24578 0.13215

1.16777 1.30737 1.67153

5.10199 4.89655 5.19224

1.14422 1.28008 1.63881

5.29692 5.04841 5.30008

0.96207 1.06744 1.37648

6.98862 6.38103 6.26009

0.50809 0.53516 0.67410

13.83416 11.89654 10.37614

0.09415 0.08084 0.07706

36.92938 30.66711 24.88957

0.00921 0.00565 0.00290

80.25541 66.28153 52.40111

C* = 0.4 0.5 0.0 2

0.33228 0.24328 0.13141 C* = 4

0.5 0.0 2

0.29926 0.22443 0.12540 C* = 20

0.5 0.0 2

0.22751 0.17808 0.10725 C* = 80

0.5 0.0 2

0.14870 0.12054 0.07830 C* = 200

0.5 0.0 2

0.10434 0.08590 0.05789

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It is observed from Eq. (11) that the non-Newtonian parameter K with K = 0 corresponding to a Newtonian fluid affect the temperature. Furthermore, it is interesting to note that if K = m = 0 in Eqs. (11) and (9) we are in (Dubrulle and Zahn, 1991; Shonhiwa and Zaturska, 1987) and in addition if C = 0 we are in (Okoya, 2004; Okoya, 2002; Tam, 1980). In the case where K = d = 0 we are in (Schlichting, 1968). The authors of (Adler, 1975; Shonhiwa and Zaturska, 1987; Tam, 1980; Zaturska, 1981) studied the solution set {b, d and hmax} of the five simple newtonian reactive flows of viscous fluid, namely, plane Couette flow, plane Poiseuille flow, Poiseuille pipe flow, axial flow between concentric circular cylinders and rotating circular cylinders. They showed that there are transitional values btr, dtr and hmax, tr of b, d and hmax, respectively, such that, when b = btr, d = dtr and hmax = hmax, tr, there is a unique solution of h, and for b < btr, d < dtr and hmax < hmax, tr, there are multiple solutions, three for example in the case of parallel flows with corresponding two critical values. In the event that b > btr, d > dtr and hmax > hmax, tr, there is no solution for h. Therefore, our first aim is to compute the lower critical values corresponding to explosion for varying m, K and C with or without Frank–Kamenetskii approximation to the Arrhenius term. These values are given in Tables 1 and 2 for Eq. (11) subject to the boundary conditions (9). It is now possible to extend computation to transition (loss of criticality) for changing m, K and C. The results are contained in Table 3. 4. Results In order to get the physical insight, the plane Couette flow between parallel plates with the lower plate at rest while the upper is in uniform motion and held at temperature T0 is studied, where the effects of non-Newtonian coefficient K and the exponent m are both accounted for. We observe that the non-Newtonian coefficient do not affect the velocity. Furthermore, dcr and hmax, cr are decreasing functions of the exponent m. However, the values of dcr and hmax, cr for the non-Newtonian flow are smaller than the Newtonian flow. btr is a monotonically increasing function of the exponent m, while the parameter dtr is a monotonically decreasing function. hmax, tr is also of the same two fold; for small values of C, it is a concave function of the exponent m (see in particular the first to the third block of the results in Table 3), while for large values of C, hmax, tr is monotonically increasing (e.g., see in particular the last three blocks of the results in Table 3). These scenarios of hmax, tr might not be unconnected with the fact that circumstances normally described by thermal explosion theory indicates that C is small. Values of C around 200 are not usual but are included for completeness. Also that the larger K the smaller the dtr, btr and hmax, tr becomes. It is observed that our numerical approximation traces earlier results with an excellent agreement. Acknowledgements I gratefully acknowledge the financial support of the Swedish International Development Cooperation Agency (SIDA) to visit Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy where this research work was done and the paper was written. The kind hospitality of Abdus Salam ICTP is also gratefully acknowledged. I thank the referee for valuable comments. References Adler, J., 1975. Combust. Flame 24 (2), 151. Dubrulle, B., Zahn, J.-P., 1991. J. Fluid Mech. 231, 561. Fosdick, R.L., Rajagopal, K.R., 1980. Proc. R. Soc. Lond. A 369, 351. Kacou, A., Rajagopal, K.R., Szeri, A.Z., 1987. ASME J. Tribol. 109, 100. Okoya, S.S., 2002. Indian J. Pure Appl. Math. 33 (11), 1625. Okoya, S.S., 2004. Mech. Res. Commun. 31, 263. Okoya, S.S., 2006. Mech. Res. Commun. 33 (5), 728. Passerini, A., Patria, M.C., Thater, G., 2000. Math. Mod. Meth. Appl. Sci. 10 (5), 711. Rajagopal, K.R., 1980. Arch. Mech. 32, 867. Rajagopal, K.R., 1982. ACTA Ciencia Indica 8 (1), 28. Rajagopal, K.R., Gupta, A.S., 1981. J. Tech. 26 (1), 67. Rajagopal, K.R., Na, T.Y., 1985. ACTA Mech. 54, 239.

S.S. Okoya / Mechanics Research Communications 34 (2007) 130–135 Schlichting, H., 1968. Boundary Layer Theory. McGraw-Hill, NY. Shonhiwa, T., Zaturska, M.B., 1986. Z. Angew. Math. Phys. 37, 632. Shonhiwa, T., Zaturska, M.B., 1987. Combust. Flame 67, 175. Szeri, A.Z., Rajagopal, K.R., 1985. Int. J. Non-Linear Mech. 20 (2), 91. Tam, K.K., 1980. Z. Angew. Math. Phys. 31 (6), 762. Wei, D., Luo, H., 2003. Int. J. Heat Mass Transfer 46 (16), 3097. Yurusoy, M., Pakdemirli, M., 2002. Int. J. Non-Linear Mech. 37 (2), 187. Zaturska, M.B., 1981. Combust. Flame 41, 201. Zaturska, M.B., 1982. Z. Angew. Math. Phys. 33 (3), 379.

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