Transient free convection from a vertical plate to a non-newtonian fluid in a porous medium

Transient free convection from a vertical plate to a non-newtonian fluid in a porous medium

Journal of Non-Newtonian Fluid Mechanics, 36 (1990) 395-410 Elsevier Science Publishers B.V., Amsterdam 395 TRANSIENT FREE CONVECTION FROM A VERTICA...

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Journal of Non-Newtonian Fluid Mechanics, 36 (1990) 395-410 Elsevier Science Publishers B.V., Amsterdam





Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC 2 7695- 7910 (U.S.A.) (Received


28, 1989; in revised form March 7, 1990)

Abstract An analysis of the transient, buoyancy-induced flow and heat transfer adjacent to a suddenly heated vertical wall, embedded in a porous medium saturated with a non-Newtonian fluid, is presented. It is shown that the governing equations, under boundary layer assumptions, are of a singular parabolic type and can be solved accurately in a semi-similar, finite domain using a successive relaxation method. The results show that during the initial stage, before effects of the leading edge become influential at a location, heat transfer and flow phenomena in porous media are governed by transient one-dimensional diffusion processes, for both pseudoplastic and dilatant fluids. Results are presented for the transition from this initial stage to a fully two-dimensional transient, which ultimately terminates in a steady convection. Non-Newtonian fluids which are pseudoplastic exhibit a significantly larger change in the heat transfer coefficient during the transition between the initial diffusive and final steady flow conditions, and unlike the free convection in homogeneous media neither dilatants nor pseudoplastics exhibit any undershoot in the heat transfer coefficient. Furthermore, it is shown that the time required to reach steady state increases and the heat transfer coefficient decreases with a decrease in the power law index. Keywords: free convection; non-Newtonian flow; porous media; singular parabolic; transient convection

1. Introduction Much attention has been given to the subject of free convection in a porous medium, because it has many important geophysical, environmental,

396 Natural convection from a vertical plate and engineering applications. embedded in a porous medium saturated with a Newtonian fluid has been throughly investigated in recent years [l]. Non-Newtonian flow and heat transfer in porous media also have numerous important applications. Savins [2] presented an excellent survey of the early literature describing non-Newtonian flow through porous media and considered many applications illustrating its relevance to technical areas such as oil recovery processes, naval architecture, polymeric processing, and waste disposal. Many of these have become even more important today. Sheffield and Metzner [3] presented improvements in the traditionally used capillary model (see Bird et al. [4]) to represent flow of non-linear fluids through porous media by using a sinusoidal model of pore structure. Durst et al. [5] recently explored the nature of flows through porous media using non-Newtonian fluids. Their study shows that the addition of small amounts of high molecular weight polymeric compounds to a solvent, otherwise a Newtonian fluid, causes drastic changes in pressure drop, which indicates the presence of elongation strain. Recently, Chen and Chen [6] analyzed natural convection of a non-Newtonian fluid about a horizontal impermeable surface in a porous medium under boundary layer assumptions, that is under the conditions of large Rayleigh number. More recently, free convective flow of a non-Newtonian fluid along a vertical plate embedded in a saturated porous medium was analyzed by Chen and Chen [7]. The results indicate that the heat transfer coefficient decreases with a decrease in the power law index. Both of these studies were primarily concerned with steady conditions. A great many practical applications of non-Newtonian flow in a porous medium involve conditions which are unsteady and, in particular. transient. When the restriction of steady state is removed, however, free convection in Newtonian as well as non-Newtonian fluids presents additional difficulties for both homogeneous and porous media. For example, consider the case of a semi-infinite plate impulsively heated to a constant temperature higher than that of the surroundings. During the initial phase of the flow development, at a given location above the leading edge, the temperature and the velocity fields develop as if the plate were infinite in extent, a period which is called the one-dimensional diffusion stage. This is because there is no mechanism in a boundary layer by which the presence of the leading edge can be transmitted, instantaneously. Once the signal from the leading edge reaches a given location the fluid flow and heat transfer begin a transition to a two-dimensional process, which ultimately terminates in steady convection. If the signal from the leading edge does not travel fast enough a very peculiar phenomenon may occur, as in the case of an isothermal vertical plate surrounded by a homogeneous Newtonian fluid, solved numerically by

397 Hellums and Churchill ]8]. During the initial one-dimensional conductive transient, the boundary layer becomes thicker than the steady state value and a local maximum in the temperature and velocity fields occurs. For such a case the heat transfer coefficient exhibits a significant undershoot and the approach to the steady state value occurs from below. If the signal from the leading edge travels rapidly, as in the case of a porous medium saturated with a Newtonian fluid, then such a behavior does not occur [9,10]. In general, equations governing the momentum and thermal boundary layers must be solved to determine the actual behavior of the transient development of fluid flow and heat transfer in a particular situation, and herein lies the problem. Boundary layer formulations of problems involving transient free convection of Newtonian fluids have been shown [9-131 to exhibit an essential singularity. The existence of a transient pattern consisting of a distinct one-dimensional conduction or diffusive regime followed by a distinct two-dimensional convective approach to steady flow is associated with this essential singularity, which occurs at the juncture of these flow regimes. A great deal of analytical and numerical difficulty has been reported to be associated with this singularity. Reviews of the literature on this topic are presented by Williams et al. [13], Telionis [14], and Haq et al. To our knowledge no study has yet been published to describe the behavior of the transient free convection of a non-Newtonian fluid in a saturated porous medium, and the singular behavior of the boundary layer equations for such a case. Hence, in this paper the transient case of a power-law fluid in a porous medium about an isothermal semi-infinite vertical plate is analyzed. In this study it is assumed that the Rayleigh number is such that the flow is always macroscopically laminar and stable, and that the mean pore size and permeability of the medium are sufficiently small to discount a presence of the type of superimposed wave motion which has been observed in beds of large diameter particles, large void spaces, and very non-uniform boundary regions. It is also assumed that the thermal properties of the solid and fluid phases, and the microscopic interphase heat transfer coefficient, are such that thermal equilibrium can be assumed. This should be the case in porous media involving relatively fine beds of materials encountered in most applications. Hence, the characteristic response time at the microscopic level is assumed to be small compared with that at the macroscopic level. Additionally, it is assumed that the properties of the bed and the interstitial fluid are such that, even in the relatively rapid transient conditions, variations of the boundary layer characteristics occur over distances which are large compared with the pore size of the bed. We also assumed the bed to be spatially homogeneous. Dilatant and pseudoplastic non-Newtonian fluids were of primary interest

398 in the work reported here. These are inelastic power-law fluids and display both shear thickening and shear thinning when contained in a porous medium. Such fluids are not able to deform radially and axially sufficiently well to conform to irregular flow paths in a porous medium, and hence display a “short circuit” flow path and a reduced tortousity [2]. Dharmadhikari and Kale [16] have shown empirically how these effects must be incorporated into the traditional power law relationships between stress and strain rate. Chen and Chen (6,7] also used this formulation in their analysis of steady natural convection. 2. Analysis Consider a Cartesion coordinate system with the origin fixed at the leading edge, the x-axis is directed upwards along the wall, and the y-axis is normal to the wall, into the porous medium. The governing equations for free convection of a power-law fluid under boundary layer assumptions are given as [7,16] u, + u,.= 0, d = - (L’~)(dp,‘dx 07; + UT, i- VT), = aT,,, ,

(W - PR),

W 04

where u is the heat capacity ratio of the saturated porous medium, which is the ratio of a volume-weighted capacity (PC,) of the porous medium to that of fluid, and (Y is the thermal diffusivity of the porous medium defined as k/( ,oc~)~“~~, k being the volume-weighted conductivity. The heat capacity ratio, u, arises because the convective enthalpy transport involves only the fluid phase heat capacity while the conductive energy transport, and the energy storage, involve both solid and fluid phases. The axial conduction term in the energy equation is neglected under boundary layer assumptions, which are valid for free convection involving large Rayleigh numbers. The momentum boundary term (i.e. the gradient of the macroscopic shear stress), the convective terms, and the time derivative of the velocity have been shown to be negligible in the case of Newtonian fluid in a porous medium of low permeability or small Darcy number [9]. A similar behavior is expected in the case of the non-Newtonian flow and has been assumed in the present model. p in eqn. (lb) is the fluid density and K,, M, and n are the modified permeability of the porous medium, the modified power law constant, and the modified power law index respectively. These constants are modified to incorporate effects of the porous medium, or flow geometry, on the properties of non-Newtonian fluids. Such effects have been shown to exist, in contrast to the flow of Newtonian fluids for which the fluid properties are

399 independent of the flow geometry (see Savins [2]). These parameters defined as [16] M = ($/p)[5n’/(2


+ 3n’)] (150/32)““,

K, = (2/~)[ Dc2/(8 - gc)] n+*, n = n’ + 0.3(1 - n’),

n” = 3n’/(2

+ n’),

where p’ is the power law constant and is equal to the viscosity of the fluid if the power law index n’ is equal to 1. D is the mean diameter of solid particles in the porous medium of porosity 6. The initial and boundary conditions are T= T,,

for t IO at x 2 0, y 2 0,

T= T,,

for t>Oat




T= T,, v=O,

for t > 0 at x 2 0, y = 0,

T= T,,

for t > 0 at x = 0, y 2 0.



x= x/x0,

Y =y/yO,

U = u/u’, V = u/u’,

variables are now defined as 7 =


I!I= AT/AT,,


where x0 = (K2gB AT,/J,c~~)~‘“,

u” = x’(M/K)“~,

YO= /m,

u” =Y’(M/K)“~,

to = cr( K/M)““,


A non-dimensional



heat transfer group is also defined as

(3) where Ra: = xKgp AT,/Md. Using the Boussinesq approximation and introducing non-dimensional variables yields general equations in the format ux+









400 B=O,

for 710at




for 7>0at



(5a) co,




for 7>0

at X20,









Equations (4) are a set of coupled, second-order, differential equations in three independent variables. highly non-linear for dilatants and pseudoplastics for different from 1 and it is linear for Newtonian fluids. reduction in the number of independent variables, behavior of these equations, we introduce semi-similar 5 = I-

exp( -T/X),

stream function

non-linear partial The U-8 coupling is the value of M very In order to obtain a and to evaluate the coordinates:

n = Y/dm,

I/ = [d(Xt) ]f([,q).


Hence the infinite range of X and 7 in the physical problem is collapsed to a finite range in the transformed problem and the number of the independent variables is reduced from three to two. Equations (4) and (5) then become

U” = q‘t, rl),


where U( t’, v) = f,,

C(t, 77)= O/N5 + 0 - 0 141 - t)] f+ ((1 - 5) 141 - C>V/X +(1 -Eb?P. For{=0

For c=l










0777-t (f/2)0,=0, ,lJ=o

l9= 1,














The energy equation written in this form clearly exhibits the singular parabolic nature of the problem. A change of the sign of the factor 1 + ln(1 - <)U in eqn. (7) from positive to negative causes the equation to become singular. This factor can be written in terms of non-dimensional variables as y( X, Y, T) = 1 + ln(1 - c)U = 1 - 7U/X.



0.6 q=o, e=1


Fig. 1. Physical and semi-similar domains. in 4-77 space represent for different values of n, where the domain of inflience of eqn. (7) changes

the critical direction.


The value of the coefficient y at a location X0 above the leading edge of the plate decreases from 1.0 and ultimately becomes negative as time 7 increases from zero. A change of sign of this coefficient alters the direction in which the information is being transmitted by the PDE equation (7), which is of a parabolic type. As shown in Fig. 1, we designate by 5, the value of < at which the coefficient changes sign. For 5 I EC, the domain of influence of the PDE is 0 I < I 6, and eqn. (9a) is considered as the initial condition, which describes one-dimensional conduction heat transfer and can be represented analytically in terms of complementary error functions. However, for E 2 & the domain of influence of eqn. (7) is 5, I 5 I 1 and, with the change of sign of y, eqn. (9b) becomes the initial condition. Equation (9b) describes steady state convective heat transfer. This is a second-order ordinary differential equation and can be solved numerically by using a fourth-order Runge-Kutta scheme. The leading edge effect propagates with the velocity field which results during one-dimensional conduction, as though the plate were infinite and conducting into a semi-infinite medium. At the instant the leading edge effect reaches a location X0, the heat transfer process changes from that of

402 one-dimensional conduction to a two-dimensional transient which finally approaches a steady state. The distance of penetration of this effect can be given as [17] Xp =



7) d7.

The governing


(11) for the one-dimensional


stage are

0, = o,,,




where for 7 = 0 or Y = 0, B = 1 and as Y 4 cc, 13---f0. The solution (12) is then *

of eqn.

8 = erfc( Y/2&F),




(14) The penetration distance X, at any time can be determined by using eqns. (13) and (14) in eqn. (11) and integrating numerically. Equation (13) implies that 6 has a maximum value of unity at Y = 0, and hence for all values of n the maximum value of U also exists at the wall and is equal to unity. This means that the maximum value of X,/T is ( Xp/7)crit = I!.&,, = 1, and the factor 7U/X of eqn. (10) at the critical point becomes equal to one. Therefore, at the critical point the value of y is equal to 0 at Y = 0. Hence. y = 0 corresponds to the first arrival of a leading edge effect. Thus we conclude that the solution of the initial transient problem given by eqns. (13) and (14) is valid only for the initial domain given as 0 5 r 5 X or 0 I 5 5 1 - exp( - 1) = 0.6321. The temperature and velocity field given during transformed semi-similar domain, are


the initial


in a


8 = erfc((q/2)/[


- <>I ) for 0 2 E I S,,it?

u = (I’/,*


and the critical value of the heat transfer Nu,//R


group is

= 0.564 at c = tcrit = 0.632.

Analytical solutions are difficult to obtain for the time 7crit < 7 < ?sS, and no accurate prediction is available to date for either this regime or the time required to reach steady state. 3. Numerical solutions The semi-similar transformations defined by eqns. (6) map the domain of the governing equations from 0 I 7 < cc, 0 I X s cz3, and 0 I Y < 00 to

403 0 I ,$I 1 and 0 I n < cc. Accordingly, the parabolic-type initial conditions and boundary conditions are mapped to four conditions which are similar to those encountered with elliptic equations and are therefore referred to here as “boundary” conditions. The mapping is shown in Fig. 1. Convection phenomena in (5, n) are then mathematically described by eqns. (7)-(9), and even though the equations remain parabolic their representation is elliptic. These equations can now be solved numerically by a successive relaxation (SOR) method, generally used in solving elliptic-type PDEs. The detailed analysis of this method as applied in this context was carried out by Wang [18], and the technique has been verified for the solution of singular parabolic Darcy flow of a Newtonian fluid in a porous medium by Haq [9]. Equation (7) is discretized by a second-order central difference, which yields e(i,






-e(i U(i, j) =e(i,




- 1, j)]/2

j+l) j)][e(i+l,



j) (18) (19)

where A(i) = [l -t(i)]

ln[l -t(i)],

C(i, j) = (1/2)[5(i)+A(i)lf(i, j> + [1-Wl71W/2 +


c(i) = (i - 1) A< and n(j)

+ 1, j) -f(i - 1, j>]/2 At,

= (j - 1) All for i, j = 1, 2, 3, . . . .

At E = 0 eqn. (9a) gives the boundary condition for eqn. (7). Since we can find an exact analytical solution for the domain 0 I 5 I 5,,it, the computational time can be significantly reduced if the solution at 5 = t,,+ = 0.6321 is used as a boundary condition rather than the solution at 5 = 0. The analytical solution at the critical line tcrit is determined using eqns. (16)-(17). At 5 = 1 eqn. (9b) gives another boundary condition for eqn. (7). This is a second-order ordinary differential equation which is solved numerically by using a fourth-order Runge-Kutta scheme. With all four “boundary” conditions known the finite difference equation (18) is solved using an SOR method for the two-dimensional transient region. If m denotes the iteration

404 level and p denotes the relaxation (18)-(20) is given as



= (1 -p)B(i,









j+l)“‘-B(i, +&)u(i,

- 1, j) m+‘]/2

the procedure

l)‘“+O(i, j-l)“‘+‘]/2



for solving



l)‘+‘]/(An) AIJ

1, .j)“I (21)

where U(i,


= #/n(i,



An initial guess is obtained for all i, j by linear interpolation between the at 6 = tcrit and 5 = 1. Then J’(i, j) is solutions for the “boundaries” computed before each iteration using a corrected trapezoidal rule and the iterations are terminated after a convergence criterion defined as max l0(i,j)m'+' - B(i,j)"'I< To1


is satisfied 4. Results The results presented are based on a mesh size of AT = 0.1, for the criterion of domain 0 < 17-C 11 for all values of n, and a convergence To1= 0.00001. The number of meshes used in [ coordinates was 37 (A< = 0.01) for 0.632 -C$ < 1.0. Numerical experiments indicated that p = 0.6 (underrelaxation) minimized the number of iterations for the successive relaxation scheme and ensured the stability and convergence of the solutions. Further mesh refinement did not alter the results significantly. While the number of iterations required was found to be dependent on the initial guess, the final converged solution was independent of the initial guess and the mesh size. The results are presented in Figs. 2-8. Figure 2 presents temperature profiles against 5 showing the two-dimensional domain between the initial transient 6 < 0.632 and steady state < = 1. Temperature profiles are plotted against n for different values of < in Figs. 3-7 for II = 2. 1.5. 1.0. 0.8, and 0.5 respectively. The temperature profile in Fig. 2 goes through a local maximum for 0 < tcrit -C 1 for larger values of the power law index n. As the value of 17 decreases the maximum in the temperature field moves toward < = 1 and



0.0 0.5




Fig. 2. Transient different n.





exp (-T/X)



field obtained

in semi-similar



Fig. 3. Transient dimensionless temperature field in semi-similar domain showing an overshoot near the wall and monotonic behavior far away from the wall for n = 2.0.

Fig. 4. Transient dimensionless temperature field in semi-similar domain showing an overshoot near the wall and monotonic behavior far away from the wall for n =1.5.


Fig. 5. Transient dimensionless temperature shoot near the wall and monotonic behavior

field in semi-similar domain showing far away from the wall for n = 1.0.

an over-

Fig. 6. Transient dimensionless temperature shoot near the wall and monotonic behavior

field in semi-similar domain showing far away from the wall for n = 0.8.

an over-

Fig. 7. Transient dimensionless temperature field in semi-similar behavior throughout the region for n = 0.5.





Fig. 8. Transient heat transfer group in physical domain obtained by transformation solution in (5, q) showing a common initial transient period for different values of n.


finally disappears. The temperature profiles plotted in Figs. 3-6 vs. 17 indicate an increase in boundary layer thickness with a decrease in the value of n. Figures 3-5 indicate a local maximum in the slope of the temperature field at the wall (i.e. dV at T,J= 0), and hence an overshoot after passing through a local minimum is present in the value of - 19, at 17= 0, whereas Fig. 7 for n = 0.5 indicates a monotonic behavior of the slope of the temperature field near the wall. Such results are associated with an overshoot in boundary layer thickness for the classical problems of free convection in homogeneous Newtonian fluids with low to intermediate Prandtl numbers, such as air and water. In the present study the boundary layer thickness, however, is not affected by this behavior of the temperature field near the wall and increases monotonically between < = 0 and 6 = 1 as shown in Figs. 3-6. The results presented in Fig. 8 indicate a monotonic transition of the heat transfer group from one-dimensional conduction, through the two-dimensional domain of essential singularity, to steady state, without a local minimum or an undershoot. This result is in contrast with the classical natural convection problem, where the essential singularity is associated with an overshoot in the heat transfer group [13]. The value of the heat transfer group at the end of the one-dimensional transient is 0.564. For T/X> 5, the value of the heat transfer group found in this study is in agreement to three decimal digits with the published steady state results of Ref. 7. The results further indicate that the dilatants, with increasing value of power law index, rapidly deviate from the one-dimensional conduction solution at an earlier time as compared with the pseudoplastics. The time required to reach steady state is higher for the pseudoplastics and increases with decreasing values of n.

408 5. Conclusions Transient, buoyancy-induced flow of a non-Newtonian power-law fluid in a fluid-saturated porous medium adjacent to an impulsively heated semi-infinite vertical wall has been analysed and the heat transfer has been shown to be initially governed by one-dimensional conduction. After a signal from the leading edge arrives at a location and before steady state is reached, the heat transfer transient is shown to be singular parabolic. An accurate solution valid for the entire time domain has been obtained numerically. The results show that the Nusselt number decreases continuously with time and monotonically approaches the steady state value for all values of power law index representing the non-Newtonian effects. Overshoot of the boundary layer thickness at the end of the transient one-dimensional period, which occurs in classical transient-free convection in homogenous boundary layers, does not exist for dilatant, pseudoplastic, or Newtonian fluids in a porous medium. The time required to reach steady state and the fractional change in the Nusselt number, between the critical point and steady state, increases with a decreasing value of the power law index. References 1 P. Cheng, Natural convection in porous medium: external flows, in: S. Kakac, W. Aung and R. Viskanta (Eds.), Natural Convection, Hemisphere Publishing Co., New York, 1985. 2 J.G. Savins, Non-Newtonian flow through porous media, Ind. Eng. Chem., 61 (10) (1969) 18-47. 3 R.E. Sheffield and A.B. Metzner, Flows of nonlinear fluids through porous media, AIChE J., 22 (4) (1976) 736-741. 4 R.B. Bird, W.E. Stewart and E.N. Lightfoot, Transport Phenomena, Wiley, New York, 1960. 5 F. Durst, R. Haas and W. Interthal, The nature of flows through porous media, J. Non-Newtonian Fluid Mech., 22 (1987) 169-189. 6 H.T. Chen and C.K. Chen, Natural convection of non-Newtonian fluids about a horizontal surface in a porous medium, J. Energy Resource Technol.. 109 (1987) 119-123. 7 H.T. Chen and C.K. Chen, Free convection flow of non-Newtonian fluids along a vertical plate in a porous medium, J. Heat Transfer, 110 (1988) 257-260. 8 J.D. Hellums and S.W. Churchill, Transient and steady state, free and natural convection. numerical solutions: part 1. The isothermal, vertical plate, AIChe J., 8 (5) (1962) 690-692. 9 S. Haq, Transient free convection in a fluid saturated porous medium. Ph.D. Dissertation, North Carolina State University at Raleigh, NC, 1989. 10 D.B. Ingham and S.N. Brown, Flow past a suddenly heated vertical plate in porous medium, Proc. R. Sot. London, Ser. A., 403 (1986) 51-80. 11 K. Nanbu, Limit of pure conduction for unsteady free convection on a vertical plate, Int. J. Heat Mass Transfer, 14 (1971) 1531-1534. 12 D.B. Ingham, Singular parabolic partial differential equations that arise in impulsive motion problems, J. Appl. Mech., 44 (1977) 396-400.

409 13 J.C. Williams, J.C. Mulligan and T.B. Rhyne, Semisimilar solutions for unsteady free convection boundary layer flow on a vertical flat plate, J. Fluid Mech., 175 (1987) 309-322. 14 D. Telionis, Unsteady Viscous Flows, Springer, Verlag, New York, 1981. 15 S. Haq, C. Kleinstreuer and J.C. Mulligan, Transient free convection of a non-Newtonian fluid along a vertical wall, J. Heat Transfer, 110 (1988) 604-607. 16 R.V. Dharmadhikari and D.D. Kale, Flow of non-Newtonian fluids through porous media, Chem. Eng. Sci., 40 (3) (1985) 527-529. 17 R.J. Goldstein and D.G. Briggs, Transient free convection about vertical plate and circular cylinder, J. Heat Transfer, 86 (1964) 490-500. 18 J.C.T. Wang, On the numerical methods for the singular parabolic equations in fluid dynamics, J. Comput. Phys., 52 (1983) 465-479.

Appendix A: Nomenclature D g i, j k


%I n


Nux M P Ra,* t

T u

u v

V x


mean diameter of solid particles in porous media acceleration due to gravity grid points volume-weighted averaged conductivity of the porous medium permeability of the porous medium modified permeability of the porous medium modified power law index power law index local Nusselt number modified power law constant relaxation factor modified Rayleigh number for Darcian porous media time temperature velocity along x-coordinate (along the wall) dimensionless u velocity velocity along y-coordinate (normal to the wall) dimensionless u velocity distance along wall dimensionless distance along wall distance normal to the wall dimensionless distance normal to the wall

Greek symbols

; f A

thermal diffusivity coefficient of thermal expansion porosity of the medium difference


similarity variable dimensionless temperature ratio power law constant similarity variable fluid density heat capacity ratio of saturated porous media dimensionless time stream function Subscripts C

crit ss ; 00

critical critical point steady state value wall value fixed value bulk values far away from wall

Superscripts 0


fixed value iteration level