- Email: [email protected]
Effects of suction or blowing on the velocity and temperature distribution in the flow past a porous flat plate of a power-law fluid
Fluid Dynamics Research 32 (2003) 283 – 294
Eects of suction or blowing on the velocity and temperature distribution in the $ow past a porous $at plate of a power-law $uid A.S. Gupta∗ , J.C. Misra, M. Reza Department of Mathematics, Indian Institute of Technology, Kharagpur 721 302, India Received 11 November 2002; received in revised form 6 May 2003; accepted 7 May 2003 Communicated by S. Kida
Abstract An analysis is made of the steady $ow of a non-Newtonian $uid past an in5nite porous $at plate subject to suction or blowing. The incompressible $uid obeys Ostwald-de Waele power-law model. It is shown that steady solutions for velocity distribution exist only for a pseudoplastic (shear-thinning) $uid for which the power-law index n satis5es 0 ¡ n ¡ 1 provided that there is suction at the plate. Velocity at a point is found to increase with increase in n. No steady solution for velocity distribution exists when there is blowing at the plate. The solution of the energy equation governing temperature distribution in the $ow of a pseudoplastic $uid past an in5nite porous plate subject to uniform suction reveals that temperature at a given point near the plate increases with n but further away, temperature decreases with increase in n. A novel result of the analysis is that both the skin-friction and the heat $ux at the plate are independent of n. c 2003 Published by The Japan Society of Fluid Mechanics and Elsevier B.V. All rights reserved. Keywords: Power-law $uid; Flow past a porous plate; Heat transfer
1. Introduction The increasing emergence of non-Newtonian $uids, such as molten plastics, emulsions, pulps, polymer melts, fossil fuels in saturated underground beds as important raw materials and products in a wide variety of industrial processes, has stimulated a considerable amount of interest in the behaviour of such $uids when in motion. In particular what has been studied intensely for obvious ∗
Corresponding author. Tel.: +91-03222-282275; fax: +91-03222-2755303/282700 E-mail address: [email protected] (A.S. Gupta)
c 2003 Published by The Japan Society of Fluid Mechanics and Elsevier B.V. 0169-5983/03/$30.00 All rights reserved. doi:10.1016/S0169-5983(03)00068-6
284
A.S. Gupta et al. / Fluid Dynamics Research 32 (2003) 283 – 294
practical reasons is how momentum and heat are transferred to a moving non-Newtonian $uid under more common $ow con5gurations usually met in practice. The term non-Newtonian $uid is one of very great generality and includes all $uids for which the equations of motion of Newtonian $uid (i.e., viscous $uid characterized by the property that during its motion the stress is proportional to the rate of strain) do not apply. One type of non-Newtonian $uid is the power-law $uid. These $uids are characterized by the property that during its motion the stress is a nonlinear function of the rate of strain. Such $uids with anomalous viscosity are inelastic in so far as they neither show stress relaxation nor normal stress dierences in shear $ow. Flows of many plastics melts may be described in terms of this power-law. Using this non-Newtonian model, the pressure drop and heat transfer in channels and pipes, the $ow between two rotating concentric cylinders, the Couette $ow between two parallel plates, etc. were investigated and the results of these studies are given in review articles by Metzner (1965), Lyche and Bird (1956), and Mishiyoshi (1964). Two- and three-dimensional boundary layer equations governing momentum and heat transfer in the $ow of power-law $uids were developed by Schowalter (1960), Acrivos et al. (1965), and Shulman (1980). Acrivos et al. (1960) presented a theoretical analysis of forced convection momentum and heat transfer in laminar boundary layer $ows of power-law $uids past external surfaces. It is well known that the $ow in a boundary layer separates in regions of adverse pressure gradient. The occurrence of separation has several undesirable eects in so far as it leads to increase in the drag on the body immersed in the $ow and adversely aects heat transfer from the surface of the body. Separation can be prevented by suction since the low-energy $uid in the boundary layer is removed. By contrast, the wall shear stress and hence the friction drag is reduced by blowing. The stability of the boundary layer and transition to turbulence are also signi5cantly in$uenced by continuous suction and blowing. In fact suction tends to stabilize the boundary layer $ow. The steady $ow of a uniform stream of an incompressible viscous $uid over an in5nite porous $at plate subject to uniform suction was investigated by GriJth and Meredith (1936). Comprehensive summaries of research in the area of boundary layer control by suction or injection in the $ow of a viscous $uid are to be found in Lachmann (1961) and Chang (1976). Gersten and KKorner (1968) investigated momentum and heat transfer in the steady $ows of an incompressible viscous $uid in the laminar boundary layer over a porous wedge subject to suction or blowing. However, to the best of the authors’ knowledge, the study of $ow and heat transfer in motions of a power-law $uid past bodies subject to suction or blowing has not received any attention. In this paper we investigate the steady $ow of a power-law $uid past an in5nite porous $at plate subject to suction or blowing. The heat transfer in this $ow is also analyzed in the case when the plate is held at a constant temperature. The motivation and the implication of this study are to explore the in$uence of suction or blowing on the control of $ow separation as well as heat transfer in $ow of power-law $uids.
2. Momentum and heat transfer in the ow of a power-law uid past an innite porous plate Consider the steady $ow of an incompressible power-law $uid past an in5nite porous $at plate subject to suction or blowing. The free stream velocity is uniform and the plate is subject to uniform suction or blowing. The $ow con5guration is shown in Fig. 1.
A.S. Gupta et al. / Fluid Dynamics Research 32 (2003) 283 – 294
285
Fig. 1. A sketch of the physical problem.
The rheological equation of state for an incompressible power-law $uid (Bird et al., 1960) is given by 1=2 n−1 1 = k ( : ) ; (1) 2 where is the deviatoric part of the stress tensor and is the symmetrical rate-of-strain tensor 9v with Cartesian components Nij = 99xvji + 9xji , vi being the velocity components. Further, k(¿ 0) is a rheological constant and n(¿ 0) is a rheological index known as the power-law index. In (1), ( : ) is an invariant of given by ( : ) = Nij Nji : (2) i
j
Eq. (1) represents Ostwald-de Waele model of a non-Newtonian inelastic $uid with anomalous viscosity. It follows from (1) and (2) that in a two-dimensional $ow of this $uid, the shear stress xy is given by 2 2 1=2 n−1 2 9v 9v 9u 9u 9v 9u + + ; (3) xy = k 2 +2 + 9x 9y 9y 9x 9y 9x where x and y are Cartesian coordinates, u and v are the velocity components in the x and y directions, respectively. When n = 1, Eq. (1) reduces to the constitutive equation for a viscous (Newtonian) $uid with k = , being the coeJcient of viscosity. Thus the deviation of n from unity indicates the degree of deviation from Newtonian behaviour. The $uids for which n ¡ 1 are called pseudoplastic $uids. It can be seen from (3) that for such $uids, the apparent viscosity given by the ratio of the stress xy and the strain-rate (9u=9y + 9v=9x) decreases with increasing rate of shear. On the other hand, the $uids for which n ¿ 1 are called dilatant $uids and for such $uids, increases with increasing rate of shear.
286
A.S. Gupta et al. / Fluid Dynamics Research 32 (2003) 283 – 294
We take x-axis along the plate, the y-axis being normal to it. Since the plate is in5nite, in the steady state the physical variables depend on y only. The equation of continuity then gives dv =0 dy
(4)
and its solution is v = −v0 ;
(5)
where v0 is the constant velocity at the plate with v0 ¿ 0 for suction and v0 ¡ 0 for blowing. We 5rst consider the case of suction at the plate so that v0 ¿ 0. The x-component of the equation of momentum gives, from (3) and (5), that k d du n−1 du 1 9p du = − ; (6) − v0 dy dy dy dy 9x where is the density of the $uid and p is the $uid pressure. From (1)–(3), the y-component of the momentum equation gives 9p=9y = 0, which means that p is a function of x only. It then follows from (6) that 9p=9x is at most a constant. Since far away from the plate, the free stream velocity is uniform, it follows from (6) that 9p=9x = 0. Hence, the pressure is uniform throughout the $ow. The boundary conditions are u=0
at y = 0;
u→U
as y → ∞;
where U is the uniform velocity at in5nity. In terms of the dimensionless variables y u = ; uO = ; L U Eq. (6) with p = constant is written as d uO d d uO n−1 d uO −V = ; d d d d where V and the characteristic length L are given by n−2 1=n kU v0 V = ; L= : U
(7)
(8)
(9)
(10)
It is well known that in the $ow of an incompressible viscous $uid past an in5nite porous $at plate subject to uniform suction, the $ow near the plate has a boundary layer structure inside which there is no velocity overshoot (see Schlichting, 1978). Since a power-law $uid is a viscous $uid with shear-dependent viscosity, one would expect that the $ow near the plate would also exhibit boundary layer behaviour. To see if this boundary layer $ow has a velocity distribution in which there is a velocity overshoot, we consider a velocity pro5le of the form shown in Fig. 2, which is consistent with boundary condition u(0) O = 0, u(∞) O = 1 (see (7)). In Fig. 2, the velocity u() O (=u(y)=U ) has a maximum at = 1 , which is a point inside the boundary layer and this maximum value exceeds unity since there is a velocity overshoot. It is clear from Fig. 2 that d u=d O ¿ 0 in the region
A.S. Gupta et al. / Fluid Dynamics Research 32 (2003) 283 – 294
287
Fig. 2. Boundary layer velocity pro5le exhibiting velocity overshoot.
0 ¡ ¡ 1 . Denoting the velocity distribution in this region by uO 1 (), the governing equation (9) for uO 1 () becomes
d uO 1 n d uO 1 d −V : (11) = d d d Integration of (11) leads to d uO 1 n − V uO 1 = + c1 ; d
(12)
where c1 is a constant. Clearly d uO 1 ()=d = 0 at = 1 since uO 1 () has a maximum at = 1 . This gives − V uO 1 |=1 = c1 ;
(13)
since n ¿ 0. Denoting the velocity distribution in the region 1 ¡ ¡ ∞ by uO 2 (), we 5nd from Fig. 2 that d uO 2 ()=d ¡ 0 in this region. Hence the governing equation (9) for uO 2 () is given by d d uO 2 n−1 d uO 2 d uO 2 = − : (14) −V d d d d Integration of (14) results in d uO 2 n−1 d uO 2 d uO 2 n − V uO 2 = − + c2 = − − + c2 ; d d d
(15)
where c2 is a constant. Now, since uO 2 () → 1 and d uO 2 =d → 0 as → ∞ (see (7)), it follows from (15) that − V = c2 ;
(16)
since n ¿ 0. Further d uO 2 =d = 0 at = 1 since uO 2 has a maximum at = 1 . This gives similarly from (15) that − V uO 2 |=1 = c2 :
(17)
288
A.S. Gupta et al. / Fluid Dynamics Research 32 (2003) 283 – 294
Hence from (16) and (17), we 5nd that uO 2 |=1 = 1:
(18)
Now from continuity of velocity at = 1 , and (18) we must have uO 1 |=1 = uO 2 |=1 = 1:
(19)
This contradicts the assumption that there is a velocity overshoot in the boundary layer such that uO 1 (1 ) (=uO 2 (1 )) exceeds unity. So we conclude that there is no velocity overshoot and the velocity u() O is everywhere less than unity inside the boundary layer. This in turn implies that d u=d O ¿0 inside the boundary layer. Hence the governing equation for u() O is obtained from (9) as n
d uO d d uO −V = (20) d d d with boundary conditions u(0) O = 0;
u() O →1
as → ∞:
Integrating (20) and noting that d u=d O → 0 as → ∞, we obtain from (21) that n d uO : V (1 − u) O = d
(21)
(22)
Eq. (22) gives, on integration along with the boundary condition uO = 0 at = 0 (see (21)), the velocity distribution u() O as
(1 − n) 1=n n=(n−1) V : (23) u() O =1− 1+ n Notice that this solution is consistent with the boundary condition uO → 1 as → ∞ only when 0 ¡ n ¡ 1. We thus arrive at the novel result that steady asymptotic velocity distribution for the $ow past an in5nite porous plate subject to uniform suction exists only for a pseudoplastic (shear-thinning) $uid for which 0 ¡ n ¡ 1. Evidently for a dilatant (shear-thickening) $uid with n ¿ 1 (e.g., quicksand), no steady solution for velocity distribution exists. It is found from experiments that 23:3% Illinois clay in water (n = 0:229), 1:5% carboxymethylcellulose solution in water (n = 0:554), 33% lime in water (n = 0:171) and 4% paper pulp in water (n = 0:575) behave like pseudoplastic $uid (Bird et al., 1960). Dierenting (23) with respect to , we get
d uO (1 − n) 1=n 1=(n−1) 1=n =V V 1+ ; (24) d n which shows that d u=d O is indeed positive everywhere in any $ow for 0 ¡ n ¡ 1. Fig. 3 shows the variation of u() O with for various values of the suction parameter V when n = 0:5. It can be seen that the velocity at any point increases with increase in V . The $ow near the plate shows a boundary layer structure for large suction velocity. Fig. 4 shows the velocity distribution for several values of the power-law index n with V = 0:5. It can be seen that velocity at any point increases with increasing n, and the $ow near the plate shows a tendency towards a boundary layer structure as n increases with 0 ¡ n ¡ 1. Further the thickness of this boundary layer decreases with increase in n.
A.S. Gupta et al. / Fluid Dynamics Research 32 (2003) 283 – 294
V=2 V = 1.5 V=1 V = 0.8 V = 0.6
u(η)
1.5
1.0
0.5
0.0 0
5
10
15
η Fig. 3. Variation of velocity u() O for several values of the suction parameter V with n = 0:5.
n= n= n= n=
1.5
0.2 0.4 0.6 0.8
u(η)
1.0
0.5
0.0 0
5
10
15
η Fig. 4. Variation of u() O for several values of the power-law index n with V = 0:5.
289
290
A.S. Gupta et al. / Fluid Dynamics Research 32 (2003) 283 – 294
The shear stress w at the plate y = 0 is obtained from (3) as n
du ; w = k dy y=0
(25)
which gives from (10), (22) and u(0) O = 0 that w = v0 U:
(26)
It might appear paradoxical that w is independent of both the viscosity parameter k and the power-law index n. Strictly speaking, this is not a friction drag, but rather is the so-called sink drag which every body in a $ow where a certain mass is sucked in experiences. This can be easily shown from the balance of momentum. It emerges from (26) that $ow separation can be prevented by suction. Let us next consider the case of blowing v0 ¡ 0 at the plate. In this case V ¡ 0. Clearly Eq. (22) cannot then hold since the left-hand side is negative because uO ¡ 1, while the right-hand side is positive. We thus reach the conclusion that no steady asymptotic solution exists for blowing at the plate. The physical reason for this is that vorticity generated at the plate due to no-slip condition continually diuses out from the plate by viscosity and is also continually convected away from the plate by blowing. On the other hand, the steady asymptotic solution in the case of suction at the plate which we have already found stems from the fact that the vorticity generated at the plate due to no-slip condition is prevented from diusing away from the plate by convection of vorticity towards the plate by suction. To determine the temperature distribution in the above $ow, we solve the energy equation n+1 dT du d2 T − cp v0 ; (27) = 2 +k dy dy dy where cp , T and denote the speci5c heat of the $uid, temperature and thermal conductivity of the $uid, respectively. The last term in (27) represents the viscous dissipation in the $ow. Since the steady $ow solution exists only for the case of suction at the plate, it follows that a steady distribution of temperature can be found only for suction at the plate. We assume v0 ¿ 0 in Eq. (27). The boundary conditions are T = Tw
at y = 0;
T → T∞
as y → ∞;
where the wall temperature Tw is given a priori and Tw and T∞ are constants. Introducing the dimensionless temperature as T − T∞ = Tw − T ∞ and using (8) and (24) in (27), we get
d2 (1 − n) 1=n (n+1)=(n−1) d (n+1)=n 1+ + VPe ; = −EPeV V d2 d n
(28)
(29)
(30)
where a=
; cp
Pe =
UL ; a
E=
U2 : cp (Tw − T∞ )
(31)
A.S. Gupta et al. / Fluid Dynamics Research 32 (2003) 283 – 294
291
The boundary conditions for follow from (28) and (29) as =1
at = 0;
→0
as → ∞:
(32)
Integration of (30) along with the boundary conditions → 0 and d =d → 0 as → ∞ gives
1 (1 − n) 1=n 2n=(n−1) d + VPe = VEPe 1 + V : (33) d 2 n It is clear from (33) that the temperature distribution is characterized by four dimensionless parameters: the suction velocity V , the Peclet number Pe, the power-law index n and the Eckert number E. Note that the parameter E is a measure of viscous dissipation in the $ow. The rate of heat transfer at the plate is derived from (8), (29), and (31)–(33) as dT v0 U 2 − : (34) = cp v0 (Tw − T∞ ) − dy y=0 2 It is indeed remarkable that the wall heat $ux is independent of k, n and the thermal conductivity . Eq. (33) subject to the boundary condition (0) = 1 (see (32)) can be analytically integrated. The solution is
2n=(n−1) 1 1 − n () = e−VPe 1 + VEPe V 1=n " 1+ eVPe" d" : (35) 2 n 0 Fig. 5 shows the temperature distribution for various values of suction with E = 5, Pe = 20 and n = 0:5. It can be seen that at any point, temperature decreases with increase in suction parameter V . Notice that there is a temperature overshoot near the plate. This, of course, is a consequence of the fact that for E = 5, there is signi5cant generation of heat due to viscous dissipation near the plate so that the temperature in the region very close to the plate exceeds Tw . Fig. 6 shows the temperature distribution for several values of the power-law index n with V = 2; Pe = 20 and E = 1:5. It is observed that temperature at any point near the plate increases with increase in n but further away from the plate temperature decreases with increase in n. Fig. 7 displays the temperature distribution for various values of the Eckert number E with V = 2, Pe = 20 and n = 0:5. As expected, temperature at any point increases with increase in E. Notice the temperature overshoot (i.e. T ¿ Tw ) very near the plate for E ¿ 2. This is consistent with the expression for heat $ux given by (34) which shows that heat $ows from the $uid to the wall when E ¿ 2. Fig. 8 shows the variation of () with for several values of the Peclet number Pe with V = n = 0:5 and E = 1:5. It is observed that temperature at any point decreases with increase in Pe. 3. Results and discussion An investigation is made of the steady $ow of a power-law $uid past an in5nite porous $at plate subject to suction or blowing. It turns out that steady solutions for velocity distribution exist only for a pseudoplastic (shear-thinning) $uid for which the power-law index n satis5es 0 ¡ n ¡ 1 provided that there is suction at the plate. Velocity at any point increases with increase in n. No steady solution for velocity distribution exists when there is blowing at the plate. It is observed that
292
A.S. Gupta et al. / Fluid Dynamics Research 32 (2003) 283 – 294
2.0
V V V V V
θ(η)
1.5
=4 =2 =1 = 0.8 = 0.6
1.0
0.5
0.0 0
5
10
15
η Fig. 5. Variation of temperature () for several values of V with E = 5, Pe = 20 and n = 0:5.
1.0
θ(η)
0.8
n= n= n= n=
0.6
0.8 0.6 0.4 0.2
0.4
0.2
0.0 0
1
2
3
4
η Fig. 6. Variation of () for several values of n with E = 1:5, Pe = 20 and V = 2.
A.S. Gupta et al. / Fluid Dynamics Research 32 (2003) 283 – 294
293
3.0 E E E E E
2.5
=1 =2 =4 =6 =8
θ(η)
2.0
1.5
1.0
0.5
0.0 0.0
0.5
1.0
1.5
2.0
2.5
η Fig. 7. Variation of () for several values of Eckert number E with n = 0:5, Pe = 20 and V = 2.
1.0
θ(η)
0.8 Pe = 5 Pe = 10 Pe = 15 Pe = 20
0.6
0.4
0.2
0.0 0.0
0.5
1.0
1.5
2.0
2.5
η Fig. 8. Variation of () for several values of Peclet number Pe with n = 0:5, E = 1:5 and V = 0:5.
294
A.S. Gupta et al. / Fluid Dynamics Research 32 (2003) 283 – 294
temperature at any point near the plate increases with n but further away from the plate, temperature decreases with increase in n. When the plate temperature Tw exceeds the free stream temperature T∞ , heat $ows from the plate to the $uid if the Eckert number E (characterizing viscous dissipation) is less than 2. But if E ¿ 2, heat $ows from the $uid to the plate. Acknowledgements One of the authors (A.S.G) acknowledges the 5nancial support of Indian National Science Academy for carrying out this work. We thank the referees for their comments which enabled us to make an improved presentation of the paper. References Acrivos, A., Shah, M.I., Petersen, E.E., 1960. Momentum and heat transfer in laminar boundary-layer $ows of non-Newtonian $uids past external surfaces. A.I.Ch.E. J. 6, 312–317. Acrivos, A., Shah, M.I., Petersen, E.E., 1965. On the solution of the two-dimensional boundary-layer $ow equations for a non-Newtonian power-law $uid. Chem. Eng. Sci 20, 301–305. Bird, R.B., Stewart, W.E., Lightfoot, E.N., 1960. Transport Phenomena. Wiley, New York. Chang, P.K., 1976. Control of Flow Separation. Hemisphere Publ. Corp., Washington, DC. Gersten, H., KKorner, H., 1968. WKarmeKubergang unter BerKucksichtigung der ReibungswKarme bei laminaren KeilstrKomungen mit verKanderlicher Temperatur und Normalgeschwindigkeit entlang der Wand. Int. J. Heat Mass Transfer 11, 655–673. GriJth, A.A., Meredith, F.W., 1936. The possible improvement in aircraft performance due to the use of boundary layer suction. Report No. 2315, Aeronautical Research Council, London. Lachmann, G.V. (Ed.), 1961. Boundary layer and $ow control, Vols. I and II. Pergamon Press, London. Lyche, B.C., Bird, R.B., 1956. The Graetz–Nusselt problem for a power-law non-Newtonian $uid. Chem. Eng. Sci. 6, 35–41. Metzner, A.B., 1965. Heat transfer in non-Newtonian $uids. In: Advances in Heat Transfer, Vol. 2. Academic Press, New York. Mishiyoshi, M.R., 1964. Heat transfer in slurry and $ow with internal heat generation. Bull. JSME 7, 376–384. Schlichting, H., 1978. Boundary Layer Theory. McGraw-Hill, New York. Schowalter, W.R., 1960. The application of boundary layer theory to power-law pseudoplastic $uids: similar solutions. A.I.Ch.E. J. 6, 24–28. Shul’man, Z.P., 1980. Heat transfer to non-Newtonian boundary layers. Heat Transfer-Sov. Res. 12, 17–30.
Eects of suction or blowing on the velocity and temperature distribution in the $ow past a porous $at plate of a power-law $uid A.S. Gupta∗ , J.C. Misra, M. Reza Department of Mathematics, Indian Institute of Technology, Kharagpur 721 302, India Received 11 November 2002; received in revised form 6 May 2003; accepted 7 May 2003 Communicated by S. Kida
Abstract An analysis is made of the steady $ow of a non-Newtonian $uid past an in5nite porous $at plate subject to suction or blowing. The incompressible $uid obeys Ostwald-de Waele power-law model. It is shown that steady solutions for velocity distribution exist only for a pseudoplastic (shear-thinning) $uid for which the power-law index n satis5es 0 ¡ n ¡ 1 provided that there is suction at the plate. Velocity at a point is found to increase with increase in n. No steady solution for velocity distribution exists when there is blowing at the plate. The solution of the energy equation governing temperature distribution in the $ow of a pseudoplastic $uid past an in5nite porous plate subject to uniform suction reveals that temperature at a given point near the plate increases with n but further away, temperature decreases with increase in n. A novel result of the analysis is that both the skin-friction and the heat $ux at the plate are independent of n. c 2003 Published by The Japan Society of Fluid Mechanics and Elsevier B.V. All rights reserved. Keywords: Power-law $uid; Flow past a porous plate; Heat transfer
1. Introduction The increasing emergence of non-Newtonian $uids, such as molten plastics, emulsions, pulps, polymer melts, fossil fuels in saturated underground beds as important raw materials and products in a wide variety of industrial processes, has stimulated a considerable amount of interest in the behaviour of such $uids when in motion. In particular what has been studied intensely for obvious ∗
Corresponding author. Tel.: +91-03222-282275; fax: +91-03222-2755303/282700 E-mail address: [email protected] (A.S. Gupta)
c 2003 Published by The Japan Society of Fluid Mechanics and Elsevier B.V. 0169-5983/03/$30.00 All rights reserved. doi:10.1016/S0169-5983(03)00068-6
284
A.S. Gupta et al. / Fluid Dynamics Research 32 (2003) 283 – 294
practical reasons is how momentum and heat are transferred to a moving non-Newtonian $uid under more common $ow con5gurations usually met in practice. The term non-Newtonian $uid is one of very great generality and includes all $uids for which the equations of motion of Newtonian $uid (i.e., viscous $uid characterized by the property that during its motion the stress is proportional to the rate of strain) do not apply. One type of non-Newtonian $uid is the power-law $uid. These $uids are characterized by the property that during its motion the stress is a nonlinear function of the rate of strain. Such $uids with anomalous viscosity are inelastic in so far as they neither show stress relaxation nor normal stress dierences in shear $ow. Flows of many plastics melts may be described in terms of this power-law. Using this non-Newtonian model, the pressure drop and heat transfer in channels and pipes, the $ow between two rotating concentric cylinders, the Couette $ow between two parallel plates, etc. were investigated and the results of these studies are given in review articles by Metzner (1965), Lyche and Bird (1956), and Mishiyoshi (1964). Two- and three-dimensional boundary layer equations governing momentum and heat transfer in the $ow of power-law $uids were developed by Schowalter (1960), Acrivos et al. (1965), and Shulman (1980). Acrivos et al. (1960) presented a theoretical analysis of forced convection momentum and heat transfer in laminar boundary layer $ows of power-law $uids past external surfaces. It is well known that the $ow in a boundary layer separates in regions of adverse pressure gradient. The occurrence of separation has several undesirable eects in so far as it leads to increase in the drag on the body immersed in the $ow and adversely aects heat transfer from the surface of the body. Separation can be prevented by suction since the low-energy $uid in the boundary layer is removed. By contrast, the wall shear stress and hence the friction drag is reduced by blowing. The stability of the boundary layer and transition to turbulence are also signi5cantly in$uenced by continuous suction and blowing. In fact suction tends to stabilize the boundary layer $ow. The steady $ow of a uniform stream of an incompressible viscous $uid over an in5nite porous $at plate subject to uniform suction was investigated by GriJth and Meredith (1936). Comprehensive summaries of research in the area of boundary layer control by suction or injection in the $ow of a viscous $uid are to be found in Lachmann (1961) and Chang (1976). Gersten and KKorner (1968) investigated momentum and heat transfer in the steady $ows of an incompressible viscous $uid in the laminar boundary layer over a porous wedge subject to suction or blowing. However, to the best of the authors’ knowledge, the study of $ow and heat transfer in motions of a power-law $uid past bodies subject to suction or blowing has not received any attention. In this paper we investigate the steady $ow of a power-law $uid past an in5nite porous $at plate subject to suction or blowing. The heat transfer in this $ow is also analyzed in the case when the plate is held at a constant temperature. The motivation and the implication of this study are to explore the in$uence of suction or blowing on the control of $ow separation as well as heat transfer in $ow of power-law $uids.
2. Momentum and heat transfer in the ow of a power-law uid past an innite porous plate Consider the steady $ow of an incompressible power-law $uid past an in5nite porous $at plate subject to suction or blowing. The free stream velocity is uniform and the plate is subject to uniform suction or blowing. The $ow con5guration is shown in Fig. 1.
A.S. Gupta et al. / Fluid Dynamics Research 32 (2003) 283 – 294
285
Fig. 1. A sketch of the physical problem.
The rheological equation of state for an incompressible power-law $uid (Bird et al., 1960) is given by 1=2 n−1 1 = k ( : ) ; (1) 2 where is the deviatoric part of the stress tensor and is the symmetrical rate-of-strain tensor 9v with Cartesian components Nij = 99xvji + 9xji , vi being the velocity components. Further, k(¿ 0) is a rheological constant and n(¿ 0) is a rheological index known as the power-law index. In (1), ( : ) is an invariant of given by ( : ) = Nij Nji : (2) i
j
Eq. (1) represents Ostwald-de Waele model of a non-Newtonian inelastic $uid with anomalous viscosity. It follows from (1) and (2) that in a two-dimensional $ow of this $uid, the shear stress xy is given by 2 2 1=2 n−1 2 9v 9v 9u 9u 9v 9u + + ; (3) xy = k 2 +2 + 9x 9y 9y 9x 9y 9x where x and y are Cartesian coordinates, u and v are the velocity components in the x and y directions, respectively. When n = 1, Eq. (1) reduces to the constitutive equation for a viscous (Newtonian) $uid with k = , being the coeJcient of viscosity. Thus the deviation of n from unity indicates the degree of deviation from Newtonian behaviour. The $uids for which n ¡ 1 are called pseudoplastic $uids. It can be seen from (3) that for such $uids, the apparent viscosity given by the ratio of the stress xy and the strain-rate (9u=9y + 9v=9x) decreases with increasing rate of shear. On the other hand, the $uids for which n ¿ 1 are called dilatant $uids and for such $uids, increases with increasing rate of shear.
286
A.S. Gupta et al. / Fluid Dynamics Research 32 (2003) 283 – 294
We take x-axis along the plate, the y-axis being normal to it. Since the plate is in5nite, in the steady state the physical variables depend on y only. The equation of continuity then gives dv =0 dy
(4)
and its solution is v = −v0 ;
(5)
where v0 is the constant velocity at the plate with v0 ¿ 0 for suction and v0 ¡ 0 for blowing. We 5rst consider the case of suction at the plate so that v0 ¿ 0. The x-component of the equation of momentum gives, from (3) and (5), that k d du n−1 du 1 9p du = − ; (6) − v0 dy dy dy dy 9x where is the density of the $uid and p is the $uid pressure. From (1)–(3), the y-component of the momentum equation gives 9p=9y = 0, which means that p is a function of x only. It then follows from (6) that 9p=9x is at most a constant. Since far away from the plate, the free stream velocity is uniform, it follows from (6) that 9p=9x = 0. Hence, the pressure is uniform throughout the $ow. The boundary conditions are u=0
at y = 0;
u→U
as y → ∞;
where U is the uniform velocity at in5nity. In terms of the dimensionless variables y u = ; uO = ; L U Eq. (6) with p = constant is written as d uO d d uO n−1 d uO −V = ; d d d d where V and the characteristic length L are given by n−2 1=n kU v0 V = ; L= : U
(7)
(8)
(9)
(10)
It is well known that in the $ow of an incompressible viscous $uid past an in5nite porous $at plate subject to uniform suction, the $ow near the plate has a boundary layer structure inside which there is no velocity overshoot (see Schlichting, 1978). Since a power-law $uid is a viscous $uid with shear-dependent viscosity, one would expect that the $ow near the plate would also exhibit boundary layer behaviour. To see if this boundary layer $ow has a velocity distribution in which there is a velocity overshoot, we consider a velocity pro5le of the form shown in Fig. 2, which is consistent with boundary condition u(0) O = 0, u(∞) O = 1 (see (7)). In Fig. 2, the velocity u() O (=u(y)=U ) has a maximum at = 1 , which is a point inside the boundary layer and this maximum value exceeds unity since there is a velocity overshoot. It is clear from Fig. 2 that d u=d O ¿ 0 in the region
A.S. Gupta et al. / Fluid Dynamics Research 32 (2003) 283 – 294
287
Fig. 2. Boundary layer velocity pro5le exhibiting velocity overshoot.
0 ¡ ¡ 1 . Denoting the velocity distribution in this region by uO 1 (), the governing equation (9) for uO 1 () becomes
d uO 1 n d uO 1 d −V : (11) = d d d Integration of (11) leads to d uO 1 n − V uO 1 = + c1 ; d
(12)
where c1 is a constant. Clearly d uO 1 ()=d = 0 at = 1 since uO 1 () has a maximum at = 1 . This gives − V uO 1 |=1 = c1 ;
(13)
since n ¿ 0. Denoting the velocity distribution in the region 1 ¡ ¡ ∞ by uO 2 (), we 5nd from Fig. 2 that d uO 2 ()=d ¡ 0 in this region. Hence the governing equation (9) for uO 2 () is given by d d uO 2 n−1 d uO 2 d uO 2 = − : (14) −V d d d d Integration of (14) results in d uO 2 n−1 d uO 2 d uO 2 n − V uO 2 = − + c2 = − − + c2 ; d d d
(15)
where c2 is a constant. Now, since uO 2 () → 1 and d uO 2 =d → 0 as → ∞ (see (7)), it follows from (15) that − V = c2 ;
(16)
since n ¿ 0. Further d uO 2 =d = 0 at = 1 since uO 2 has a maximum at = 1 . This gives similarly from (15) that − V uO 2 |=1 = c2 :
(17)
288
A.S. Gupta et al. / Fluid Dynamics Research 32 (2003) 283 – 294
Hence from (16) and (17), we 5nd that uO 2 |=1 = 1:
(18)
Now from continuity of velocity at = 1 , and (18) we must have uO 1 |=1 = uO 2 |=1 = 1:
(19)
This contradicts the assumption that there is a velocity overshoot in the boundary layer such that uO 1 (1 ) (=uO 2 (1 )) exceeds unity. So we conclude that there is no velocity overshoot and the velocity u() O is everywhere less than unity inside the boundary layer. This in turn implies that d u=d O ¿0 inside the boundary layer. Hence the governing equation for u() O is obtained from (9) as n
d uO d d uO −V = (20) d d d with boundary conditions u(0) O = 0;
u() O →1
as → ∞:
Integrating (20) and noting that d u=d O → 0 as → ∞, we obtain from (21) that n d uO : V (1 − u) O = d
(21)
(22)
Eq. (22) gives, on integration along with the boundary condition uO = 0 at = 0 (see (21)), the velocity distribution u() O as
(1 − n) 1=n n=(n−1) V : (23) u() O =1− 1+ n Notice that this solution is consistent with the boundary condition uO → 1 as → ∞ only when 0 ¡ n ¡ 1. We thus arrive at the novel result that steady asymptotic velocity distribution for the $ow past an in5nite porous plate subject to uniform suction exists only for a pseudoplastic (shear-thinning) $uid for which 0 ¡ n ¡ 1. Evidently for a dilatant (shear-thickening) $uid with n ¿ 1 (e.g., quicksand), no steady solution for velocity distribution exists. It is found from experiments that 23:3% Illinois clay in water (n = 0:229), 1:5% carboxymethylcellulose solution in water (n = 0:554), 33% lime in water (n = 0:171) and 4% paper pulp in water (n = 0:575) behave like pseudoplastic $uid (Bird et al., 1960). Dierenting (23) with respect to , we get
d uO (1 − n) 1=n 1=(n−1) 1=n =V V 1+ ; (24) d n which shows that d u=d O is indeed positive everywhere in any $ow for 0 ¡ n ¡ 1. Fig. 3 shows the variation of u() O with for various values of the suction parameter V when n = 0:5. It can be seen that the velocity at any point increases with increase in V . The $ow near the plate shows a boundary layer structure for large suction velocity. Fig. 4 shows the velocity distribution for several values of the power-law index n with V = 0:5. It can be seen that velocity at any point increases with increasing n, and the $ow near the plate shows a tendency towards a boundary layer structure as n increases with 0 ¡ n ¡ 1. Further the thickness of this boundary layer decreases with increase in n.
A.S. Gupta et al. / Fluid Dynamics Research 32 (2003) 283 – 294
V=2 V = 1.5 V=1 V = 0.8 V = 0.6
u(η)
1.5
1.0
0.5
0.0 0
5
10
15
η Fig. 3. Variation of velocity u() O for several values of the suction parameter V with n = 0:5.
n= n= n= n=
1.5
0.2 0.4 0.6 0.8
u(η)
1.0
0.5
0.0 0
5
10
15
η Fig. 4. Variation of u() O for several values of the power-law index n with V = 0:5.
289
290
A.S. Gupta et al. / Fluid Dynamics Research 32 (2003) 283 – 294
The shear stress w at the plate y = 0 is obtained from (3) as n
du ; w = k dy y=0
(25)
which gives from (10), (22) and u(0) O = 0 that w = v0 U:
(26)
It might appear paradoxical that w is independent of both the viscosity parameter k and the power-law index n. Strictly speaking, this is not a friction drag, but rather is the so-called sink drag which every body in a $ow where a certain mass is sucked in experiences. This can be easily shown from the balance of momentum. It emerges from (26) that $ow separation can be prevented by suction. Let us next consider the case of blowing v0 ¡ 0 at the plate. In this case V ¡ 0. Clearly Eq. (22) cannot then hold since the left-hand side is negative because uO ¡ 1, while the right-hand side is positive. We thus reach the conclusion that no steady asymptotic solution exists for blowing at the plate. The physical reason for this is that vorticity generated at the plate due to no-slip condition continually diuses out from the plate by viscosity and is also continually convected away from the plate by blowing. On the other hand, the steady asymptotic solution in the case of suction at the plate which we have already found stems from the fact that the vorticity generated at the plate due to no-slip condition is prevented from diusing away from the plate by convection of vorticity towards the plate by suction. To determine the temperature distribution in the above $ow, we solve the energy equation n+1 dT du d2 T − cp v0 ; (27) = 2 +k dy dy dy where cp , T and denote the speci5c heat of the $uid, temperature and thermal conductivity of the $uid, respectively. The last term in (27) represents the viscous dissipation in the $ow. Since the steady $ow solution exists only for the case of suction at the plate, it follows that a steady distribution of temperature can be found only for suction at the plate. We assume v0 ¿ 0 in Eq. (27). The boundary conditions are T = Tw
at y = 0;
T → T∞
as y → ∞;
where the wall temperature Tw is given a priori and Tw and T∞ are constants. Introducing the dimensionless temperature as T − T∞ = Tw − T ∞ and using (8) and (24) in (27), we get
d2 (1 − n) 1=n (n+1)=(n−1) d (n+1)=n 1+ + VPe ; = −EPeV V d2 d n
(28)
(29)
(30)
where a=
; cp
Pe =
UL ; a
E=
U2 : cp (Tw − T∞ )
(31)
A.S. Gupta et al. / Fluid Dynamics Research 32 (2003) 283 – 294
291
The boundary conditions for follow from (28) and (29) as =1
at = 0;
→0
as → ∞:
(32)
Integration of (30) along with the boundary conditions → 0 and d =d → 0 as → ∞ gives
1 (1 − n) 1=n 2n=(n−1) d + VPe = VEPe 1 + V : (33) d 2 n It is clear from (33) that the temperature distribution is characterized by four dimensionless parameters: the suction velocity V , the Peclet number Pe, the power-law index n and the Eckert number E. Note that the parameter E is a measure of viscous dissipation in the $ow. The rate of heat transfer at the plate is derived from (8), (29), and (31)–(33) as dT v0 U 2 − : (34) = cp v0 (Tw − T∞ ) − dy y=0 2 It is indeed remarkable that the wall heat $ux is independent of k, n and the thermal conductivity . Eq. (33) subject to the boundary condition (0) = 1 (see (32)) can be analytically integrated. The solution is
2n=(n−1) 1 1 − n () = e−VPe 1 + VEPe V 1=n " 1+ eVPe" d" : (35) 2 n 0 Fig. 5 shows the temperature distribution for various values of suction with E = 5, Pe = 20 and n = 0:5. It can be seen that at any point, temperature decreases with increase in suction parameter V . Notice that there is a temperature overshoot near the plate. This, of course, is a consequence of the fact that for E = 5, there is signi5cant generation of heat due to viscous dissipation near the plate so that the temperature in the region very close to the plate exceeds Tw . Fig. 6 shows the temperature distribution for several values of the power-law index n with V = 2; Pe = 20 and E = 1:5. It is observed that temperature at any point near the plate increases with increase in n but further away from the plate temperature decreases with increase in n. Fig. 7 displays the temperature distribution for various values of the Eckert number E with V = 2, Pe = 20 and n = 0:5. As expected, temperature at any point increases with increase in E. Notice the temperature overshoot (i.e. T ¿ Tw ) very near the plate for E ¿ 2. This is consistent with the expression for heat $ux given by (34) which shows that heat $ows from the $uid to the wall when E ¿ 2. Fig. 8 shows the variation of () with for several values of the Peclet number Pe with V = n = 0:5 and E = 1:5. It is observed that temperature at any point decreases with increase in Pe. 3. Results and discussion An investigation is made of the steady $ow of a power-law $uid past an in5nite porous $at plate subject to suction or blowing. It turns out that steady solutions for velocity distribution exist only for a pseudoplastic (shear-thinning) $uid for which the power-law index n satis5es 0 ¡ n ¡ 1 provided that there is suction at the plate. Velocity at any point increases with increase in n. No steady solution for velocity distribution exists when there is blowing at the plate. It is observed that
292
A.S. Gupta et al. / Fluid Dynamics Research 32 (2003) 283 – 294
2.0
V V V V V
θ(η)
1.5
=4 =2 =1 = 0.8 = 0.6
1.0
0.5
0.0 0
5
10
15
η Fig. 5. Variation of temperature () for several values of V with E = 5, Pe = 20 and n = 0:5.
1.0
θ(η)
0.8
n= n= n= n=
0.6
0.8 0.6 0.4 0.2
0.4
0.2
0.0 0
1
2
3
4
η Fig. 6. Variation of () for several values of n with E = 1:5, Pe = 20 and V = 2.
A.S. Gupta et al. / Fluid Dynamics Research 32 (2003) 283 – 294
293
3.0 E E E E E
2.5
=1 =2 =4 =6 =8
θ(η)
2.0
1.5
1.0
0.5
0.0 0.0
0.5
1.0
1.5
2.0
2.5
η Fig. 7. Variation of () for several values of Eckert number E with n = 0:5, Pe = 20 and V = 2.
1.0
θ(η)
0.8 Pe = 5 Pe = 10 Pe = 15 Pe = 20
0.6
0.4
0.2
0.0 0.0
0.5
1.0
1.5
2.0
2.5
η Fig. 8. Variation of () for several values of Peclet number Pe with n = 0:5, E = 1:5 and V = 0:5.
294
A.S. Gupta et al. / Fluid Dynamics Research 32 (2003) 283 – 294
temperature at any point near the plate increases with n but further away from the plate, temperature decreases with increase in n. When the plate temperature Tw exceeds the free stream temperature T∞ , heat $ows from the plate to the $uid if the Eckert number E (characterizing viscous dissipation) is less than 2. But if E ¿ 2, heat $ows from the $uid to the plate. Acknowledgements One of the authors (A.S.G) acknowledges the 5nancial support of Indian National Science Academy for carrying out this work. We thank the referees for their comments which enabled us to make an improved presentation of the paper. References Acrivos, A., Shah, M.I., Petersen, E.E., 1960. Momentum and heat transfer in laminar boundary-layer $ows of non-Newtonian $uids past external surfaces. A.I.Ch.E. J. 6, 312–317. Acrivos, A., Shah, M.I., Petersen, E.E., 1965. On the solution of the two-dimensional boundary-layer $ow equations for a non-Newtonian power-law $uid. Chem. Eng. Sci 20, 301–305. Bird, R.B., Stewart, W.E., Lightfoot, E.N., 1960. Transport Phenomena. Wiley, New York. Chang, P.K., 1976. Control of Flow Separation. Hemisphere Publ. Corp., Washington, DC. Gersten, H., KKorner, H., 1968. WKarmeKubergang unter BerKucksichtigung der ReibungswKarme bei laminaren KeilstrKomungen mit verKanderlicher Temperatur und Normalgeschwindigkeit entlang der Wand. Int. J. Heat Mass Transfer 11, 655–673. GriJth, A.A., Meredith, F.W., 1936. The possible improvement in aircraft performance due to the use of boundary layer suction. Report No. 2315, Aeronautical Research Council, London. Lachmann, G.V. (Ed.), 1961. Boundary layer and $ow control, Vols. I and II. Pergamon Press, London. Lyche, B.C., Bird, R.B., 1956. The Graetz–Nusselt problem for a power-law non-Newtonian $uid. Chem. Eng. Sci. 6, 35–41. Metzner, A.B., 1965. Heat transfer in non-Newtonian $uids. In: Advances in Heat Transfer, Vol. 2. Academic Press, New York. Mishiyoshi, M.R., 1964. Heat transfer in slurry and $ow with internal heat generation. Bull. JSME 7, 376–384. Schlichting, H., 1978. Boundary Layer Theory. McGraw-Hill, New York. Schowalter, W.R., 1960. The application of boundary layer theory to power-law pseudoplastic $uids: similar solutions. A.I.Ch.E. J. 6, 24–28. Shul’man, Z.P., 1980. Heat transfer to non-Newtonian boundary layers. Heat Transfer-Sov. Res. 12, 17–30.