Effects of Viscous Dissipation on Unsteady Free Convection in a Fluid past a Vertical Plate Immersed in a Porous Medium

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1 Transport in Porous Media (2006) 64: 1 14 Springer 2006 DOI /s Effects of Viscous Dissipation on Unsteady Free Convection in a Fluid past a Vertical Plate Immersed in a Porous Medium M. F. EL-AMIN and N. A. EBRAHIEM Mathematics Department, Aswan Faculty of Science, South Valley University, Aswan 81258, Egypt (Received: 6 September 2004; in final form: 9 July 2005) Abstract. The effects of viscous dissipation on unsteady free convection from an isothermal vertical flat plate in a fluid saturated porous medium are examined numerically. The Darcy Brinkman Forchheimer model is employed to describe the flow field. A new model of viscous dissipation is used for the Darcy Brinkman Forchheimer model of porous media. The simultaneous development of the momentum and thermal boundary layers are obtained by using a finite difference method. Boundary layer and Boussinesq approximation have been incorporated. Numerical calculations are carried out for various parameters entering into the problem. Velocity and temperature profiles as well as local friction factor and local Nusselt number are shown graphically. It is found that as time approaches infinity, the values of friction factor and heat transfer coefficient approach steady state. Key words: viscous dissipation, finite difference method, free convection, unsteady flow. Nomenclature C f c p d Ec F Gr g K Nu Pr q w t friction factor specific heat pore diameter Eckert number empirical constant Grashof number acceleration due to gravity permeability Nusselt number Prandtl number heat transfer coefficient time Author for correspondence: mfam2000@yahoo.com

2 2 M. F. El-AMIN AND N. A. EBRAHIEM t dimensionless time T temperature T dimensionless temperaturem ū, v dimensionless velocity components u, v velocity components x,ȳ space coordinates x, y dimensionless space coordinates τ w wall shear stress α thermal diffusivity β coefficient of thermal expansion ρ density of the fluid µ fluid viscosity ε porosity Subscripts w wall infinity 1. Introduction Recently, modeling of viscous dissipation due to flow in a fluid saturated porous medium was considered by several authors, and they have modeled this effect in different ways. An experimental and analytical investigation was carried by Fand et al. (1986) to study the free convection heat transfer from a horizontal cylinder embedded in a porous medium consisting of randomly packed glass spheres and the medium is saturated by water or silicon oil. They concluded that when the saturating fluid is silicon oil, the heat transfer rate changed because of viscous dissipation effects. Modeling of viscous dissipation effects on a flow of a fluid saturated porous medium which corresponding to Darcy s law was agreed by most authors. Thus, if the momentum equation is given in the form of Darcy s law (µ/k)u = p + ρg, the energy equation including viscous dissipation becomes u. T =.(α T)+ u[(µ/k)u]. (see, e.g., Bejan 1995; Magyari and Keller 2003a, b; Rees et al. 2003; El-Amin et al. 2003; and El-Amin 2003). Here, u is the Darcy velocity, K is the permeability, α is the thermal diffusivity and p is the pressure. Murthy and Singh (1997) have modeled the viscous dissipation effect on the flow of an incompressible fluid in a saturated porous medium. They applied the Forchheimer Darcy model for the momentum equation by including two terms in the energy equation to represent the viscous dissipation effect. These two terms correspond to the first and second order velocity terms in the momentum equation (Forchheimer Darcy model). Thus, if one can write the momentum equation in the form (µ/k)u+bρu 2 = p + ρg then the energy equation including viscous dissipation should written in the form u. T =.(α T)+ u[(µ/k)u + bρu 2 ]. A paradox occurring in

3 EFFECTS OF VISCOUS DISSIPATION ON UNSTEADY FREE CONVECTION 3 the non-darcy term in the approach of Murthy and Singh (1997) has been resolved by Nield (2000). However, the Forchheimer drag does indeed contribute to the viscous dissipation. Nield (2000) suggested a modified formula to model viscous dissipation in non-darcy porous media. He concluded that modeling of viscous dissipation is related with the local drag modeling. Thus, if the local drag is correctly modeled, for example by the Brinkman Darcy model (µ/k)v µ eff 2 V, where µ eff is the effective viscosity, the viscous dissipation will be equal to the power of the drag, i.e. (µ/k)v.v µ eff V. 2 V (for more details and comprehensive discussion see Nield, 2000). Pop and Herwig (1990) studied the transient development of the concentration boundary layer with impulsive mass diffusion from a heated vertical surface. The problem of unsteady free convection adjacent to an impulsively heated horizontal circular cylinder in porous media was investigated by Bradean et al. (1997). Numerical study of double-diffusive free convection from a vertical surface is presented by Srinivasan and Angirasa (1988). Angirasa et al. (1997) investigated combined heat and mass transfer by natural convection with opposing buoyancy effects in a fluid saturated porous medium. In this paper, the problem of modeling of viscous dissipation effects on unsteady free convection from a vertical flat plate to fluid saturated porous medium is studied. The Darcy Brinkman Forchheimer model which includes the effects of boundary and inertia forces is employed. The dimensionless non-linear partial differential equations are solved numerically using an explicit finite-difference scheme. The values of friction factor and heat transfer coefficient are determined for steady and unsteady free convection. 2. Analysis In the case of modeling the momentum equation according to the Darcy Brinkman Forchheimer model the viscous dissipation modeling in the energy equation should due to contribution of the terms which content the viscosity in the momentum equation as well as Forchheimer drag (see Nield, 2000). Thus, when the local drag is modeled by (µ/k)v + (F/K 1/2 )V.V µ eff 2 V,F is an empirical constant, then the viscous dissipation should equal the power of the drag (µ/k)v.v +(F/K 1/2 )V.(V.V ) µ eff V. 2 V. Consider the unsteady, laminar boundary layer in a two-dimensional free convective flow of a fluid over a vertical flat plate embedded in a porous medium. At time t = 0, the temperature of the surface immersed in a fluid is raised suddenly from that of surrounding fluid T, up to a higher and constant value T w and kept at this value thereafter. Physical model

4 4 M. F. El-AMIN AND N. A. EBRAHIEM Figure 1. Physical model and coordinate system. and coordinate system are shown in Figure 1. The Brinkman Forchheimer model is used to describe the flow in porous media with large porosity and permeability. It is assumed that there is a local thermal equilibrium, i.e. the temperature of the fluid equals the temperature of the solid. Under the Boussinesq and boundary layer approximations, the governing mass, momentum and energy conservation equations become ū x + v ȳ = 0, (1) ū t +ū ū ū + v x ȳ = gβ( T T ) + µ 2 ū ρ ȳ µε 2 ρkū Fε2 ū ū, (2) 1/2 T t +ū T x + v T ȳ = α 2 T ȳ 2 + ū c p [ K µε ρk ū + Fε2 K ū ū µ 1/2 ρ 2 ū ȳ 2 ]. (3) In the previous equations, ū and v are the velocity components along the x and ȳ axes. T w is the surface temperature, while T is the temperature inside the boundary layer. ρ, α, β, ε, F, µ, K, c p and g are the density, the thermal diffusivity, the volumetric coefficient of thermal expansion, the porosity, the empirical constant, the fluid viscosity, the permeability, the fluid specific heat and the acceleration due to gravity, respectively. Further discussion on the validation of this model of momentum equation may be found in Nield and Bejan (1999). One may note that the differ-

5 EFFECTS OF VISCOUS DISSIPATION ON UNSTEADY FREE CONVECTION 5 ence between Equation (1.18) in their book and Equation (2) is that here we use the velocity denoted by ū while in the equation of Nield and Bejan (1999) they used the Darcy velocity (let s say u d ), and the relation between them is u d = ε.ū. This may be interprets appearing the porosity ε in both terms of the porous media in the momentum equation. The viscous dissipation terms appear in the energy equation (the final three terms on the right-hand side of Equation (3)) are modeled according to Nield (2000) and the above introduction and argument. The viscous dissipation represents the conversion (the rate of conversion per unit volume) of mechanical energy into thermal energy. Nield (2004) concluded in his comments on the paper by Al-Hadhrami et al. (2003) that modeling of viscous dissipation on the Darcy Brinkman Forchheimer by just a power-of-drag term, or a Darcy power term plus a term involving velocity derivatives, is still a moot point. However, it is clear that one should not use just the term involving velocity derivatives, as some authors have done in the past. The initial and boundary conditions are t = 0:ū = v = 0, T = T for all x and ȳ, ū = v = 0; T = T at x = 0. t>0: ū = v = 0; T = T w at ȳ = 0, x>0, ū = 0, T = T at ȳ, x>0. (4) We introduce the following dimensionless variables: x = x/l, y =ȳ/l, u=ū/u, v = v/u, T = ( T T )/( T w T ) and t = U t/l, (5) where, U is a reference velocity and L is a suitable length scale. Introducing expressions (5) into Equations (1) (3) we have the transformed equations in the following form: u x + v y = 0, (6) u t + u u x + v u y = GrT + 2 u y 1 2 Da u Fr u u, (7) Da T t + u T x + v T y = 1 2 [ T 1 P r.re y + Ec.u 2 Da u + Fr ] Da u u 2 u, (8) y 2 where, Gr = Lgβ( T w T )/U 2 is the Grashof number, Da = K/L 2 ε the Darcy number, Fr= FεK 1/2 /L the inertia coefficient, Pr= µ/ρα the Prandtl number, Re= ρlu/µ the Reynolds number and Ec= U 2 /c p ( T w T ) is the Eckert number.

6 6 M. F. El-AMIN AND N. A. EBRAHIEM The initial and boundary conditions are now given by t = 0:u = v = T = 0 for all x and y, u = v = T = 0 at x = 0, t>0: u = v = 0,T = 1 at y = 0, x>0, u = 0,T = 0 at y, x>0. (9) In technological applications, the wall shear stress and the local Nusselt number are of primary interest. The wall shear stress may be written as τ w = µ ū ȳ=0 = µu u ȳ L y. (10) y=0 Therefore the local friction factor is given by C f = 2τ w ρu = 2 u 2 y. (11) y=0 From the definition of the local surface heat flux T q w = k e ȳ=0 = k e( T w T ) T ȳ L y, (12) y=0 where, k e is the effective thermal conductivity of the saturated porous medium, together with the definition of the local Nusselt number Nu= q w L = T T w T k e y. (13) y=0 3. Method of Solution The numerical integration was carried out using the time dependent form of the nonlinear partial differential equations (6) (8), with initial and boundary conditions (9) by the explicit finite-difference method, as explained by Carnahan et al. (1969). The spatial domain under investigation was restricted to finite dimensions, such that the length of the plate x max was assumed to be 100 and the boundary layer thickness y max was taken as 60 (this value verifies the boundary conditions u( ), T ( ) which converge asymptotically as will shown in Figures 2 and 3 below). If u,v and T denote the values of u, v and T at the end of a time-step. Then, the set of approximate finite difference equations corresponding to Equations (6) (8) are

7 EFFECTS OF VISCOUS DISSIPATION ON UNSTEADY FREE CONVECTION 7 Figure 2. Unsteady velocity and temperature profiles for various times at Da = 10, Ec= 0.01, Fr= 0.5, Gr = 1, Pr= Re= 10.0 and x = 50. Figure 3. Steady velocity profiles as a function of y for different position x with Gr = 1, Fr= 0.5, Da = 10, Pr= Re= 10 and Ec= 0.01.

8 8 M. F. El-AMIN AND N. A. EBRAHIEM (u i,j u i 1,j )/ x + (v i,j v i,j 1 )/ y = 0, (14) (u i,j u i,j)/ t + u i,j (u i,j u i 1,j )/ x + v i,j (u i,j+1 u i,j )/ y = GrT + (u i,j+1 2u i,j + u i,j 1 )/( y) 2 (1/Da)u i,j (Fr/Da) ui,j u i,j, (15) (T i,j T i,j)/ t + u i,j (T i,j T i 1,j )/ x + v i,j (T i,j+1 T i,j )/ y = (T i,j+1 2T i,j + T i,j 1 )/P r.re.( y) 2 + Ec.u i,j {(1/Da)u i,j + +(Fr/Da) u i,j u i,j (u i,j+1 2u i,j + u i,j 1 )/( y) 2 }, (16) where primed variables indicate the values of the variables at a new time and (i, j) represent grid points. Successive steps in time may be regarded as successive approximations towards the final steady state solution, for which both u/ t and T / t are zero. The coefficients u i,j and v i,j in Equations (15) and (16) are treated as constants, during any one time-step. Then, at the end of any time-step t, the new velocity components u and v, and the new temperature T at all interior grid points may be obtained by successive applications of (14) (16), respectively. This process is repeated in time and, provided the time-step is sufficiently small, u, v and T should eventually converge to values which approximate the steady state solution of Equations (14) (16). A selected set of results has been obtained covering the ranges 0.1 Da 10, 0.0 Ec 0.2, 0.0 Gr 5, Re = 10, Pr= 10 and 0.0 Fr 2.0. The system of equations was solved for the dependent variables u, v and T as functions of x, y and t. The velocity and temperature fields were calculated for various time steps t = 0.01 for a grid i.e. x = 1, y = 0.5. The space and time steps x, y, t are determined such that the solutions are not dependent on it (i.e. numerical solution is not affected by grid size). An examination of complete results for t = 1, 10, 20,...,250, revealed little or no change in u, v and T after t =100 for all computations. Thus the results for t = 100 are essentially the steady-state values. 4. Results and Discussion Figure 2 shows the development of the velocity and the temperature profiles with time t. This figure indicates that the development of the velocity and the temperature distributions with time approaches steady-state conditions after a certain time (t = 100). From the same figure we note that the momentum boundary layer thickness (δ u 60) is thicker than the thermal boundary layer thickness (δ T 18). Both the momentum boundary layer thickness and the thermal boundary layer thickness increase with time.

9 EFFECTS OF VISCOUS DISSIPATION ON UNSTEADY FREE CONVECTION 9 Although the momentum boundary layer thickness δ u 60, the maximum velocity is close to the wall y 5. Steady state velocity profiles as a function of y for different positions of x are plotted in Figure 3. This figure shows that both the thickness of the momentum boundary layer and the maximum velocity depends on the position of x. Thus, the momentum boundary layer thickness and the maximum velocity increase as x increases. Figure 4 displays the results for steady state local friction factors and local Nusselt numbers, as a function of x for various values of Da. Itis worth mentioning that the Darcy results holds if the Darcy number Da < 10 3 while for Da>10 the clear fluid limit (Nield and Bejan, 1999). Hence, the range 0.1 Da 10 was used. From Figure 4a, it is observed that the Darcy number enhances the local friction factor. It is interesting to note that for the small values of Da, the effect of the position x on the local friction factor will be small. The opposite is true when Da has large values, the local friction factor increases as x increases. Figure 4b indicates that the Darcy number enhances the heat transfer rate. At small Darcy number a slight effect on the heat transfer rate along x is observed, while when Da has large values, the heat transfer rate decreases as x increases. Steady state local friction factors and local Nusselt numbers are plotted as a function of x for various values of Ec in Figure 5. Figure 5a shows that the local friction factor has a high gradient along x at Ec=0.1, while it has a very small gradient along x for small value of the Eckert number (= 0.01) and when Ec = 0.0 (without viscous dissipation). Thus, when viscous dissipation is taken into account it has an evident effect on the local friction factor. The Eckert number represents the ratio of square velocity to the buoyancy, noting that for incompressible fluids the Eckert number should be in the range 0 <Ec<1. However, increasing the Eckert number meant increasing the inertial force with respect to the buoyancy force which causes the viscous dissipation. It is obvious that as the Eckert number increases the heat transfer rate in terms of local Nusselt number decreases as illustrated in Figure 5b. In Figure 6, steady state local friction factors and local Nusselt numbers are plotted as a function of x for various values of the inertia coefficient Fr. The inertia coefficient Fr reduces the local friction factor while it has a slight effect on the local Nusselt number (large values of Fr has an essential effect on the Nusselt number (see Nield and Bejan, 1999, p. 80). The effects of the Grashof number on the local friction factor and the local Nusselt number are shown in Figures 7a and b, respectively. The Grashof number represents the ratio of the buoyancy force to the velocity square. Thus, when the Grashof number increases (the buoyancy force dominants the velocity) the local friction factor increases due to the

10 10 M. F. El-AMIN AND N. A. EBRAHIEM Figure 4. Steady (a) local friction factor (b) local Nusselt number, as a function of x for various values of Da with Fr= 0.5, Gr = 1, Pr= Re= 10 and Ec= increase in the velocity gradient on the wall ( u/ y) y=0. On the other hand, when the velocity dominants the buoyancy force the velocity gradient on the wall decreases then the local friction factor is reduced. It is interesting to note, from Figure 7a, that the local friction factor is vanished at Gr = 0.0. Figure 7b indicates that the absolute values of the heat transfer rate are increased with increasing the Grashof number (when buoyancy force dominants the velocity) because the temperature gradient increases.

11 EFFECTS OF VISCOUS DISSIPATION ON UNSTEADY FREE CONVECTION 11 Figure 5. Steady (a) local friction factor (b) local Nusselt number, as a function of x for various values of Ec with Fr= 0.5, Gr = 1, Pr= Re= 10 and Da = 1. The opposite is true with decreasing the Grashof number the velocity dominants the buoyancy force. 5. Conclusions In this problem, the effects of viscous dissipation on unsteady free convection from a vertical flat plate to a fluid saturated porous medium are

12 12 M. F. El-AMIN AND N. A. EBRAHIEM Figure 6. Steady (a) local friction factor (b) local Nusselt number, as a function of x for various values of Fr with Da = Gr = 1, Pr= Re= 10 and Ec= studied using the Darcy Brinkman Forchheimer model. The problem is treated as unsteady-state problem and after a certain time the steady-state is approached. The inertia effect reduces the local friction factor while it has a slight effect on the heat transfer rate. The buoyancy enhances the local friction factor and reduces the heat transfer rate. The permeability

13 EFFECTS OF VISCOUS DISSIPATION ON UNSTEADY FREE CONVECTION 13 Figure 7. Steady (a) local friction factor (b) local Nusselt number, as a function of x for various values of Gr with Fr= 0.5, Da = 1,Pr= Re= 10 and Ec= enhances both the heat transfer rate and the local friction factor. Viscous dissipation enhances the local friction factor as well as the absolute values of the heat transfer rate. Acknowledgement The authors would like to express their gratitude to Professor D. A. Nield, for modification of the viscous dissipation model of this problem.

14 14 M. F. El-AMIN AND N. A. EBRAHIEM References Al-Hadhrami, A. K., Elliott, L. and Ingham, D. B.: 2003, A new model for viscous dissipation in a porous medium across a range of permeability values, Transport in Porous Media 53, Angirasa D., Peterson G. P. and Pop I.: 1997, Combined heat and mass transfer by natural convection with opposing buoyancy effects in a fluid saturated porous medium, Int. J. Heat Mass Transfer 40, Bejan, A.: 1995, Convection Heat Transfer, 2nd edn., Wiley, New York, pp Bradean R., Ingham D. B., Heggs P. J. and Pop I.: 1997, Unsteady free convection adjacent to an impulsively heated horizontal circular cylinder in porous media, Num. Heat Transfer Part A 31, Carnahan B., Luther H. A. and Wilkes J. O.: 1969, Applied Numerical Methods, Wiley, New York, pp El-Amin, M. F., El-Hakiem, M. A. and Mansour, M. A.: 2003, Effects of viscous dissipation on a power-law fluid over plate embedded in a porous medium, Heat Mass Transfer 39, El-Amin, M. F.: 2003, Combined effect of magnetic field and viscous dissipation on a power-law fluid over plate with variable surface heat flux embedded in a porous medium, J. Magnetism Magnetic Materials 261, Fand, R. M., Steinberger, T. E. and Cheng, P.: 1986, Natural convection heat transfer from a horizontal cylinder embedded in a porous medium, Int. J. Heat Mass Transfer 29, Magyari, E. and Keller, B.: 2003a, The opposing effect of viscous dissipation allows for a parallel free convection boundary-layer flow along a cold vertical flat plat, Transport in Porous Media 51, Magyari, E. and Keller, B.: 2003b, Buoyancy sustained by viscous dissipation, Transport in Porous Media 53, Murthy, P. V. S. N. and Singh, P.: 1997, Effects of viscous dissipation on a non-darcy natural convection regime, Int. J. Heat Mass Transfer 40, Nield, D. A. and Bejan, A.: 1999, Convection in Porous Media, 2nd edn., Springer-Verlag, New York. Nield, D. A.: 2000, Resolution of a paradox involving viscous dissipation and non-linear drag in a porous medium, Transport in Porous Media 41, Nield, D. A.: 2004, Comments on A new model for viscous dissipation in a porous medium across a range of permeability values in Transport in Porous Media 53, , 2003, Transport in Porous Media 55, Pop I. and Herwig H.: 1990, Transient mass transfer from an isothermal vertical flat embedded in a porous medium, Int. Comm. Heat Mass Transfer 17, Rees, D. A. S., Magyari, E. and Keller, B.: 2003, The development of the asymptotic viscous dissipation profile in a vertical free convective boundary layer flow in a porous medium, Transport in Porous Media 53, Srinivasan J. and Angirasa D.: 1988, Numerical study of double-diffusive free convection from a vertical surface, Int. J. Heat Mass Transfer 31,