International Journal of Fluid Mechanics Research, Vol. 32, No. 3, 2005

Viscous Dissipation Effects on Natural Convection from a Vertical Plate with Uniform Surface Heat Flux Placed in a Thermally Stratified Media† M. A. Hossain Department of Mathematics, University of Dhaka Dhaka-1000, Bangladesh email: [email protected] S. C. Saha School of Computer Science, IBAIS University Dhaka, Bangladesh Rama Subba Reddy Gorla Department of Mechanical Engineering, Cleveland State University Cleveland, OH email: [email protected]

In the present study we investigate the effect of viscous dissipation on natural convection from a vertical plate placed in a thermally stratified environment. The reduced equations are integrated by employing the implicit finite difference scheme of Keller box method and obtained the effect of heat due to viscous dissipation on the local skin friction and local Nusselt number at various stratification levels, for fluids having Prandtl numbers of 10, 50, and 100. Solutions are also obtained using the perturbation technique for small values of viscous dissipation parameters ξ and compared to the finite difference solutions for 0 ≤ ξ ≤ 1. Effect of viscous dissipation and temperature stratification are also shown on the velocity and temperature distributions in the boundary layer region. * * * Nomenclature Cf Cp f g Grx

local skin-friction coefficient; specific heat of the fluid; dimensionless stream function; gravitation acceleration; Grashof number; †

Received 25.04.2005. 269

ISSN 1064-2277 c ° 2005 Begell House, Inc.

Nux Pr T T∞ T∞,0 U v S q x y

local Nusselt number; Prandtl number; temperature of the boundary layer; ambient temperature; ambient temperature at x = 0; x-component of the velocity field; y-component of the velocity field; rate of stratification; uniform heat flux; coordinate measuring distance along the plate; coordinate measuring distance normal to the plate.

Greek Symbols α β η θ ξ κ ν ψ ρ

thermal diffusivity; volumetric coefficient of thermal expansion; similarity variable; dimensionless temperature; viscous dissipation parameter; thermal conductivity; kinetic coefficient of the viscosity; stream function; density of the fluid. Introduction

Many free convection processes occur in environments with temperature stratification. Good examples are closed containers and environmental chambers with heated walls. Also of interest is the free convection flow associated with heat-rejection systems for long-duration deep ocean power modules where the ocean environment is stratified [1]. Stratification of the fluid arises due to temperature variations, concentration differences, or the presence of different fluids. Cheesewright [2] and Yang et al. [3] showed that similar solutions were not possible. This fact was supported by Eichhorn [4] and by Fujii et al. [5] and therefore developed series solutions to account for the nonzero leading-edge temperature difference. Eichhorn [4] had calculated only three terms in the series solution. On the other hand, Fujii et al. [5] gave both analytical and experimental results for a temperature stratification in which the ambient temperature distribution varies with distance. In the above investigation they also showed that the fourth term in the series solutions is necessary for comparing the experimental results. The experimental and theoretical study in which both the wall temperature and ambient temperature varied with a power of the distance along the plate was carried out by Piau [6]. His experimental temperature distributions compare well with his theoretical results; in order to make the comparison, the author had to use a nonzero starting length of the surface. Later Chen and Eichhorn [7] considered a finite isothermal vertical plate in a stable thermally stratified fluid. The experimental results of their paper have represented clear information on heat transfer to a vertical cylinder in water for both the unstratified and stratified cases. Kulkarni et al. [8] investigated the problem of natural convection from an isothermal flat plate suspended in a linearly stratified fluid medium using the von Karman – Pohlhausen integral solution method. The case of nonsimilar laminar natural convection from a vertical flat plate placed in a thermally stratified medium was studied by Venkatachala and Nath [9]. To get the desired results, they used 270

the implicit finite-difference scheme developed by Keller and Cebeci [10]. They also used the perturbation series expansion and local nonsimilarity methods. Recently, a nonsimilarity analysis has been presented by Hossain et al. [11] on the effect of uniform surface temperature and surface heat flux on the natural convection flow of a viscous incompressible fluid from a vertical cone placed in a thermally stratified medium. In this analysis, perturbation solutions are obtained appropriate to two extreme regimes of the cone and also finite difference solutions for the entire flow regime for fluid of different Prandtl number and different stratification conditions. Very recently, Saha and Hossain [12] have investigated the problem of doubly diffusive natural convection flows from a vertical plate in a stable thermally stratified medium. In all above investigations, the viscous dissipation effect had been neglected. But Gebhart [13] has shown that the viscous dissipation effect plays an important role in natural convection in various devices. Gebhart and Mollendorf [14] investigated the viscous dissipation in effects external natural convection flows. Hossain [15] studied the effect of viscous and Joule heating on the flow of an electrically conducting and viscous incompressible fluid past a semi-infinite plate of which temperature varies linearly with the distance from the leading edge and in the presence of uniform transverse magnetic field.

1. Applications of Present Research

Natural convection induced by the simultaneous action of buoyancy forces resulting from thermal diffusion is of considerable interest in nature and in many industrial applications, such as geophysics, oceanography, drying processes, solidification of binary alloys, and electronic cooling. In many electronic cooling situations, arrays of heat-dissipating components are mounted on parallel plate channels that are open to the ambient at opposite ends. The simplest method of cooling these arrays is by circulating air vertically by natural convection. This method of cooling of electronic equipment continues to play an important role in their thermal management because it provides low noise and high system reliability. The electronic components under study are vertical walls. Important viscous dissipation effects may also be present in stronger gravitational fields and in processes in which the velocity is high. The relative magnitude of the viscous dissipation effect is given by a dissipation number. This number is an independent parameter. It has no correspondence with the Prandtl or Grashof numbers. Therefore, neither the value of the Prandtl number nor the upper limit for laminar processes prohibits important viscous dissipation effects. In none of the previous studies in the published literature has the effect of stable ambient stratification together with heat generation due to viscous dissipation on heat transfer been investigated. A stable thermal stratification in the ambient is usually present in electronic cooling applications. In this paper, we are concerned with the effect of viscous dissipation on natural convection from a vertical plate placed in a thermally stratified environment, which has not been reported in the literature. Using finite-difference techniques, the skin-friction coefficient and the rate of heat transfer are studied at various stratification levels of the ambient temperature distributions, viscous and nonviscous dissipation, three values of the Prandtl number, 10, 50, and 100. Solutions are also obtained using the perturbation technique treating the streamwise dissipation parameters as the pertinent parameters. 271

Momentum boundary layer

q

Thermal boundary layer

x

u x

v

g T∞ (x)

T∞(x)=T∞,0+Sx

T∞.0

y

T

Fig. 1. Flow configuration and coordinate system.

2. Formulation of the Problem Let us consider the two dimensional steady boundary layer flow and heat transfer of a viscous incompressible fluid along a vertical flat plate with uniform heat flux q immersed in a stable thermally stratified fluid, assuming that the ambient temperature is T∞ (x). The flow configuration and coordinate system are shown in Fig. 1. Under the usual Boussinesq approximation the governing equations of continuity, momentum, and the energy with viscous dissipation take the forms ∂u ∂v + = 0, ∂x ∂y

(1)

∂u ∂u ∂2u +v = ν 2 + gβ(T − T∞ ), ∂x ∂y ∂y µ ¶2 ∂T ∂T ∂2T ν ∂u u +v =α 2 + , ∂x ∂y ∂y Cp ∂y u

(2) (3)

where u and v are the velocity components along x- and y-directions, respectively; g is the acceleration due to gravity; T is the temperature of the fluid in the boundary layer; ν is the kinematics viscosity; Cp is the specific heat; and α is the thermal diffusivity. The temperature of the ambient fluid is considering a linear function of x, i. e., (4)

T∞ (x) = T∞,0 + Sx,

where T∞,0 is the ambient temperature at the leading edge of the plate and S is constant, which represents the rate of stratification and defined by S = ∂T∞ /∂x. The boundary conditions for the present problem are as follows: ¶ µ ∂T = −q at y = 0, u = v = 0, κ ∂y u → 0,

T → T∞ (x)

as

y → ∞,

where q is the uniform surface heat flux and κ is the thermal conductivity. 272

(5) (6)

3. Method of Solutions To integrate Eqs (1) – (3) together with the boundary conditions (5) and (6), it is necessary to reduce the equations to a convenient form. Hence, we now introduce the following group of transformations: ψ = νGr1/5 x f (ξ, η),

T − T∞ =

qx −1/5 Grx θ(ξ, η), κ

η=

y 1/5 Gr , x x

ξ=

gβx , Cp

(7)

where f is dimensionless stream function, θ is the dimensionless temperature function, η is the pseudosimilarity variable, and ξ is the streamwise distribution of the media and is termed as the viscous dissipation parameter, Cp is the specific heat due to constant pressure, and y is the stream function defined as ∂ψ ∂ψ u= and v=− . ∂y ∂x That satisfies the equation of continuity. Finally Grx is the local Grashof number and is defined by Grx = gβqx4 /κν 2 . Considering the transformations (7) into Eqs (2) and (3), we get the following nonsimilarity equations for the steady flow µ ¶ 4 3 ∂f 0 ∂f f 000 + f f 00 − f 02 + θ = ξ f 0 − f 00 , 5 5 ∂ξ ∂ξ

(8)

µ ¶ 1 00 4 0 1 0 1 0 002 0 ∂θ 0 ∂f θ + f θ − f θ − Sf + ξf = ξ f −θ . Pr 5 5 5 ∂ξ ∂ξ

(9)

The boundary conditions reduces f (ξ, 0) = f 0 (ξ, 0) = 0,

θ0 (ξ, 0) = −1, (10)

f 0 (ξ, ∞) = θ(ξ, ∞) = 0. In Eq. (9) Pr =

ν α

and

S=

κT0 1/5 GrL q

(11)

are, respectively, the Prandtl number and stratification parameter. Here we proposed to integrate the local nonsimilar partial differential equations (8) and (9) subject to the boundary conditions (10) by implicit finite-difference method. To begin with, the partial differential equations (8) and (9) are first converted into a system of first-order equations. Then these equations are expressed in finite-difference forms by approximating the functions and their derivatives in terms of the center difference. Denoting the mesh points in the ξ-η plane by ξi and ηj = sinh[(j − 1)/a], where i is the number of iterations and j = 0, 1, 2, . . . , N with N = 351 and a = 150, central difference approximations are made to get the quick convergence and thus save computational time and space. To solve resulting equations, Newton’s iteration technique together with the Keller box method is then introduced. Recently this method was discussed in more detail by Hossain [15] and Hossain et al. [16, 17] in studying the effect of oscillating surface temperature on the natural convection flow from a vertical flat plate. 273

4. Results and Discussions In Section 3, solutions are obtained for wide-range values of the pertinent parameter ξ for fluids taking values of Pr = 10, 50, and 100. In this study we discuss the effect of the viscous dissipation parameter and the stratification parameter S on the velocity and temperature profiles obtained by the method discussed above. The values of the velocity and temperature distributions have been calculated from the following relations: ux −2/5 Grx = f 0 (ξ, η), ν

(12)

(T − T∞ )κ 1/5 Grx = θ(ξ, η). qx

(13)

Velocity profiles are displayed in Fig. 2a for Pr = 10, 50, and 100 and S = 1.0, while the viscous dissipation parameter ξ = 0.0, 1.0, and 2.0. From Fig 2(a) we see that given an increase of the viscous dissipation parameter ξ, the velocity profile decreases near the surface of the plate and approaches zero at the outer region of the boundary layer. In Fig. 2b temperature profiles have been shown for the same parameter. It is seen that the values of nondimensional temperature are decreasing and becoming negative within the boundary layer. This is because for higher values of ξ the temperature in the ambient increases so rapidly with the height that the fluid coming up will be much cooler in the outer region of the boundary layer, which is often referred to as “temperature defect”. We have shown the dimensionless velocity profiles and temperature profiles for viscous dissipation parameter ξ = 0.0, 1.0, and 2.0 while Pr = 10 and S = 0.0, 1.0, and 2.0 in Figs 3a and b, respectively. We have found that the velocity and temperature profiles decrease owing to the increase of stratification parameter S. It also seen that velocity and temperature profiles decrease when the viscous dissipation parameter increases. This is an expected result since viscous dissipation is basically a thermal source in the flow. The local friction factor and the Nusselt number can be calculated by the procedure given below. One of the important physical quantities of interest is the friction factor Cf , which is defined by: Cf =

2τw , ρU02

where τw is the shearing stress at the surface and is defined µ ¶ ∂u τw = µ . ∂y y=0

(14)

(15)

Using the quantity of Eq. (14) and the transformation (7) in Eq. (15), we investigate the local skin friction in terms of the dimensionless shearing stress Cf , given as 1 = f 00 (ξ, 0). Cf Gr−3/5 x 2

(16)

The exchange of heat between the fluid and a solid surface is another important quantity that needs to be discussed in the fluid dynamics of a viscous compressible or incompressible fluid. We may define a nondimensional coefficient of heat transfer, which is known as Nusselt number, as follows: Nux =

Gr1/5 h(x)x x = , κ θ(ξ, 0) 274

(17)

a)

S=1.0 Pr= 10 50 100

0.2

ux/νGrx2/5

0.15

ξ 0.0 1.0 2.0

0.1

0.05

0 0

0.5

1

1.5

2

2.5

η

b)

S=1.0 Pr=10 50 100

(T-T∞)κGrx1/5/qx

1.5

ξ 0.0 1.0 2.0

1

0.5

0 0

0.5

1

1.5

2

2.5

η Fig. 2. Dimensionless velocity profile (a) and dimensionless temperature profile (b) for viscous dissipation parameter ξ = 0.0, 1.0, and 2.0, while Pr = 10, 50, and 100 and stratification parameter S = 1.0.

275

a)

Pr=10 S=0.0 1.0 2.0

0.3

0.2

ux/νGrx2/5

ξ 0.0 1.0 2.0

0.1

0 0

1

2

3

η

b)

Pr=10

2.5

S=0.0 1.0 2.0

(T-T∞)κGrx1/5/qx

2

ξ 0.0 1.0 2.0

1.5

1

0.5

0 0

1

2

3

η Fig. 3. Dimensionless velocity profile (a) and dimensionless temperature profile (b) for viscous dissipation parameter ξ = 0.0, 1.0, and 2.0, while Pr = 10 and straification parameter S = 0.0, 1.0, and 2.0.

276

Table Comparison of the solutions obtained by different methods for skin-friction coefficients and local Nusselt numbers against the viscous dissipation parameter ξ for S = 0.0 and 1.0, while Pr = 10.

S = 0.0 ξ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2.0

00

f SS 0.5791 0.5951 0.6118 0.6292 0.6472 0.6659 0.6852 0.7053 0.7259 0.7473 0.7693

(0) FD 0.5795 0.6018 0.6188 0.6363 0.6546 0.6736 0.6936 0.7145 0.7363 0.7590 0.7829 1.0892

S = 1.0

1/θ(0) SS FD 0.94073 0.9336 0.90812 0.8933 0.87592 0.8610 0.84429 0.8298 0.81333 0.7991 0.78316 0.7689 0.75383 0.7391 0.72541 0.7099 0.69793 0.6812 0.67142 0.6531 0.64589 0.6256 0.3897

00

f (0) SS FD 0.45317 0.4550 0.46263 0.4679 0.47247 0.4778 0.48269 0.4882 0.49329 0.4990 0.50428 0.5103 0.51564 0.5220 0.52739 0.5343 0.53952 0.5472 0.55203 0.5606 0.56492 0.5747 0.7557

1/θ(0) SS FD 1.04899 1.05047 1.02555 1.02122 1.00195 0.99670 0.97828 0.97260 0.95464 0.94847 0.93110 0.92424 0.90773 0.89992 0.88458 0.87552 0.86171 0.85106 0.83916 0.82659 0.81699 0.80213 0.56599

where

q(x) , Tw − T∞ where q(x) is the quantity of heat transferred through unit area. h(x) =

(18)

Now the rate of heat transfer, in terms of the dimensionless Nusselt number, is obtained as Nux Gr−1/5 = x

1 . θ(ξ, 0)

(19)

The resulting values of the skin-friction coefficient and the local Nusselt number calculated by the aforementioned method are tabulated in the Table. Here the skin friction coefficients and local Nusselt numbers against the viscous dissipation parameter ξ for S = 0.0 and 1.0 while Pr = 10 have been displayed. It may be noted from the results that an increase of heat because of viscous dissipation causes the skin-friction coefficient to increase and the local Nusselt number to decrease. As the stratification parameter S increases, the skin-friction coefficient decreases and the local Nusselt number increases. To prove the accuracy of the result computed by the finite-difference method, the perturbation series solution method for small ξ has been employed and the results have been shown in tabular form in the Table. Solution for Small ξ (SS). Now we discuss the effect of ξ near the leading edge using the regular perturbation technique. Since near the leading edge, ξ is very small, the governing equations (8) and (9) along with the boundary conditions (10) take the following form; we assume the following series expansions for the functions f and θ: f (ξ, η) =

∞ X

ξ i fi (η),

θ(ξ, η) =

i=0

∞ X

ξ i θi (η),

(20)

i=0

where fn (η) and θn (η) are the arbitrary functions depending on η. Now substituting the expression (40) into Eqs. (5) – (7) and then equating powers of ξ from O(0) to O(2), the three equations 277

are obtained as follows:

4 3 f0000 + f0 f000 − f 02 + θ0 = 0; 5 5

(21a)

1 00 4 1 S θ0 + f0 θ00 − f00 θ0 − f 0 = 0; Pr 5 5 5

(21b)

f0 (0) = f00 (0) = 0,

θ00 (0) = −1,

f00 (∞) = θ0 (∞) = 0;

(21c)

4 9 6 f1000 + f0 f100 + f000 f1 − f00 f10 + θ1 − f00 f10 = 0; 5 5 5

(21d)

9 6 1 1 00 4 θ1 + f0 θ10 + f1 θ00 − f00 θ1 − (f10 θ0 + Sf10 ) + f000 2 = 0; Pr 5 5 5 5

(21e)

f1 (0) = f10 (0) = θ10 (0) = 0,

f10 (∞) = θ1 (∞) = 0;

(21f)

9 14 16 8 4 f2000 + f0 f200 + f100 f + f000 f2 − f00 f20 + θ2 − f10 2 = 0; 5 5 5 5 5

(21g)

1 00 4 9 14 11 6 θ2 + f0 θ20 + f1 θ10 + f2 θ00 − f00 θ2 − f10 θ1 Pr 5 5 5 5 5 1 0 − (f2 θ0 + Sf20 ) + 2f100 f100 = 0; 5

(21h)

f2 (0) = f20 (0) = θ20 (0) = 0,

(21i)

f20 (∞) = θ2 (∞) = 0.

Now we find the local skin-friction coefficient and local Nusselt number for small ξ as 1 Cf Gr−3/5 = f000 (0) + ξf100 (0) + ξ 2 f200 (0) x 2

(22)

and

1 . θ0 (0) + ξθ1 (0) + ξ 2 θ2 (0) Three terms of the above series have been taken to attain the good agreement. NuGr−1/5 = x

(23)

For example, taking Prandtl number Pr = 10 and S = 1.0, the above series (22) and (23) can be written as 1 Cf Gr−3/5 = 1.5378 − 0.3398ξ + 0.1781ξ 2 + . . . x 2 and 1 . NuGr−1/5 = x 0.4398 − 0.3936ξ + 0.0509ξ 2 + . . . Substituting the values of the viscous dissipation parameter ξ in the above series expansion, we can obtain the numerical values of the skin-friction coefficient and the local Nusselt number from for the different values and Nux Gr−1/5 the Table. Similarly, we obtain the values of 0.5Cf Gr−3/5 x x of viscous dissipation parameter ξ and the stratification parameter S by using above expansion. , for different values of Prandtl number and the stratification parameter The result of 0.5Cf Gr−3/5 x S have been compared to those found by other methods in the Table. The results obtained by the series solutions approach are in good agreement with those obtained from the finite-difference method. 278

Conclusions In this paper, effect of viscous dissipation on natural convection from a vertical plate placed in a thermally stratified media has been investigated. Two solution methodologies, namely, the implicit finite-difference method and the perturbation series solution method, are employed to integrate the equations governing the flow. The numerical computations are carried out for the fluid having Prandtl number Pr = 10, 50, and 100 against the value of the local viscous dissipation parameter ranged to ξ ∈ [0, 2]. The following conclusions may be drawn: • Values of the skin-friction coefficient increase and the rate of heat transfer decreases for increasing values of the viscous dissipation parameter ξ. • Increasing value of the viscous dissipation parameter leads to a decrease in the velocity profile near the surface of the plate. The temperature of the fluid decreases within the boundary layer region as the viscous dissipation parameter increases. • Values of the velocity and temperature profiles decrease due to an increase in the values of stratification parameter S. REFERENCES 1. Yang, K. T., Szewczyk, A. A., and Novotny, J. L., Problems in Free Convection of HeatRejection Systems, In: Proc. Sympos. Deep Submergence Propulsion and Marine Systems, AIAA Chicago Section, 1966. 2. Cheesewright, R., Natural Convection from a Plane Vertical Surface in Non-Isothermal Surroundings, Int. J. Heat Mass Transfer, 1967, 10, pp. 1847–1859. 3. Yang, K. T., Novotny, J. L., and Cheng, Y. S., Laminar Free Convection from a Non-Isothermal Plate Immersed in a Temperature Stratified Medium, Int. J. Heat Mass Transfer, 1972, 15, pp. 1097–1109. 4. Eichhorn, R., Natural Convection in a Thermally Stratified Fluid, In: Progress in Heat and Mass Transfer, Vol. 2, Pergamon, New York, 1969, pp. 41–53. 5. Fujii, T., Takeuchi, M., and Morioka, I., Laminar Boundary Layer of Free Convection in a Temperature Stratified Environment, In: Proc. 5th Int. Heat Transfer Conf., Tokyo, NC2, 1974, pp. 44–48. 6. Piau, J. M., Influence des Variations des Properties Physiques et de la Stratification en Convection Naturelle, Int. J. Heat Mass Transfer, 1974, 17, pp. 465–476. 7. Chen, C. C. and Eichhorn, R., Natural Convection from a Vertical Surface to a Thermally Stratified Fluid, ASME J. Heat Transfer, 1976, 98, pp. 446–451. 8. Kulkarni, A. K., Jacobs, H. R., and Hwang, J. J., Similarity Solutions for Natural Convection Flow Over an Isothermal Vertical Wall Immersed in Thermally Stratified Medium, Int. J. Heat Mass Transfer, 1986, 30, pp. 691–698. 9. Venkatachala, B. J. and Nath, G., Non-Similar Laminar Natural Convection in a Thermal Stratified Fluid, Int. J. Heat Mass Transfer, 1981, 24, No. 11, pp. 1848–1850. 10. Keller, H. B. and Cebeci, T., Accurate Numerical Methods for Boundary Layer Flows, Part II. Two Dimensional Turbulent Flow, AIAA J., 1972, 10, No. 9, pp. 1193–1199. 11. Hossain, M. A., Paul, S. C., and Mandal, A. C., Natural Convection Flow along a Vertical Circular Cone with Uniform Surface Temperature and Surface Heat Flux in a Thermally Stratified Medium, Int. J. Numer. Methods Heat Fluid Flow, 2002, 12, pp. 290–306. 279

12. Saha, S. C. and Hossain, M. A., Natural Convection Flow with Combined Buoyancy Effects Due to Thermal and Mass Diffusion in Thermally Stratified Media, Nonlinear Anal. Mod. Cont., 2004, 9, pp. 89–102. 13. Gebhart, B., Effects of Viscous Dissipation in Natural Convection, J. Fluid Mech., 1962, 14, pp. 225–232. 14. Gebhart, B. and Mollendorf, J., Viscous Dissipation in External Natural Convection Flows, J. Fluid Mech., 1969, 38, No. 1, pp. 97–107. 15. Hossain, M. A., Viscous and Joule Heating Effects on MHD-free Convection Flow with Variable Plate Temperature, Int. J. Heat Mass Transfer, 1992, 35, No. 12, pp. 3485–3487. 16. Hossain, M. A., Banu, N., and Nakayama, A., Non-Darcy Forced Convection Boundary Layer Flow Over a Wedge Embedded in a Saturated Porous Medium, Numer. Heat Transfer, A, 1994, 26, pp. 399–414. 17. Hossain, M. A., Pop, I., and Vafai, K., Combined Free Convection Heat and Mass Transfer Above a Near-Horizontal Surface in Porous Media, Hybrid Methods Engng, 1999, 1, pp. 87– 102. 18. Hung, C. I. and Chen, C. B., Non-Darcy Free Convection in Thermally Stratified Medium Along a Vertical Plate with Variable Heat Flux, Heat Mass Transfer, 1997, 33, pp. 101–107. 19. Akhter, C. and Hossain, M. A., Effect of Transpiration and Heat Source on Steady Natural Convection Flow from a Plate Immersed in Non-Isothermal Surroundings, M. Sci. Thesis, Univ. Dhaka, Bangladesh, 2000.

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