Free convection on an inclined plate with variable viscosity and thermal diffusivity

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1 Free convection on an inclined plate with variable viscosity and thermal diffusivity G. Palani 1, J.D. Kirubavathi 1, and Kwang Yong Kim 2 1 Dr. Ambedkar Govt. Arts College, Chennai, Tamil Nadu, India 2 Inha University, Incheon, Republic of Korea gpalani32@yahoo.co.in (Received February 14, 2013) The present numerical analysis addresses free convection flow of a viscous incompressible fluid along an inclined semi-infinite flat plate considering the variation of viscosity and thermal diffusivity with temperature. The governing equations are developed with the corresponding boundary conditions are transformed to non-dimensional form using the appropriate dimensionless quantities. Due to complexity in the transformed governing equations, analytical solution will fail to produce a solution. Hence, most efficient and unconditionally stable implicit finite difference method of Crank Nicolson scheme has been used to solve the governing equations. Numerical results are obtained for different values of the viscosity, thermal conductivity, inclination angle, Grashof number, and Prandtl number. The overall investigation of the variation of velocity, temperature, shearing stress and Nusselt number are presented graphically. To examine the accuracy of the present approximate results, the present results are compared with the available results. Key words: inclined plate, variable viscosity, finite differences, Nusselt number. Introduction Natural convection flows occur due to change in the temperature differences and are of very important in a number of industrial applications. In addition, buoyancy is very important in an environment, since the temperature differences between land and air can give rise to complicated flow patterns, and in enclosures such as ventilated and heated rooms and reactor configurations. Due to enormous application in naturally occurring phenomena, the problems of twodimensional free convection flow past a semi-infinite flat plate with different conditions draw the attention of many researchers in the last few years. The problem of free convection flow past a semi-infinite plate using momentum integral method was first presented in the work [1]. Free convection flow past a semi-infinite isothermal vertical plate was first studied in the work [2] using similarity variables. Transient free convection flow past a semi-infinite vertical plate with isothermal and constant heat flux conditions was first studied in [3] using an integral method. The problem of a semi-infinite isothermal vertical plate was first studied in the work [4] by using an explicit finite-difference technique which is conditionally stable and convergent. The same problem was also studied in [5] by using the Crank Nicolson type of implicit finite difference scheme which is fast convergent, unconditionally stable, and more accurate. The authors of [6] obtained the similarity solution G. Palani, J.D. Kirubavathi, and Kwang Yong Kim,

2 G. Palani, J.D. Kirubavathi, and Kwang Yong Kim for the problem of laminar free convection in boundary layer flows of air and water in horizontal, inclined and vertical flat plates whose temperature or surface heat flux varies as the power of the axial coordinate. A finite-difference solution of unsteady natural convection boundary layer flow over an inclined plate with variable surface temperature was presented in the work [7]. In all the above papers, the fluid considered with constant properties. For the fluids, which are important in the theory of lubrication, the heat generated by the internal friction and the corresponding rise in temperature do affect the viscosity and thermal conductivity of the fluid and they can no longer be regarded as constant. The physical properties of fluids such as viscosity and thermal conductivity may change significantly with temperature (Schlichting [8]). The temperature-dependent property problem is further complicated by the fact that the properties of different fluids behave differently with temperature. The authors of the work [9] obtained the different relations between the physical properties of fluids and temperature. The temperature dependent viscosity in the algebraic or exponential form was considered in the work [10] to analyse the problem of suddenly heated or cooled channel flow of a Newtonian fluid. The effects of significant viscosity variation on convective heat transport in watersaturated porous media were studied later in [11]. Viscous incompressible fluid flow past a vertical plate along with the variation of viscosity and thermal conductivity was studied in the work [12]. MHD flow past a heated vertical plate with the variation of the viscosity and thermal conductivity with temperature was studied in the work [13]. Free convection flow past a vertical porous plate was studied numerically in the work [14] by taking into account the variable viscosity and uniform suction velocity in the presence of thermal radiation. Unsteady MHD free convection flow past a flat plate with the effect of variable viscosity and thermal radiation was studied in the work [15]. The steady locally unidirectional gravitydriven draining of a thin rivulet of Newtonian fluid with temperature-dependent viscosity down a slowly varying substrate that is either uniformly hotter or uniformly colder than the surrounding atmosphere was investigated in [16]. Radiative free convection steady laminar boundary layer flow with the effect of temperature dependent viscosity, thermal conductivity and density was studied in the work [17]. Viscous incompressible fluid flow past a continuously moving semi-infinite plate with variable viscosity and temperature was studied in [18]. The authors of the work [19] analysed the effect of thermal radiation and thermal diffusivity over a stretching surface with non-uniform heat flux. The effect of variable viscosity on radiative heat flow of an viscous incompressible electrically conducting fluid over a moving vertical plate with uniform suction and heat flux was discussed in the work [20]. Recently, the authors of [21] presented numerical solution of the effect of variable viscosity and thermal conductivity on free convection flow past a vertical plate using implicit finite difference scheme. From the literature survey, it is noticed that the effect of variation of viscosity and thermal conductivity with temperature plays an important role in fluid mechanics. Hence, the effect of variable viscosity and thermal conductivity on free convection flow past a semi-infinite isothermal inclined plate is considered in the present study. The temperature of the plate was considered higher than that of the ambient fluid temperature. Here the thermal conductivity of the fluid is considered as a linear function of the temperature, and variable viscosity is an exponential function of the temperature. In the process of solving fluid flow problems, various mathematical tools have been developed and applied, among which is the finite difference method. The non-dimensional governing equations are solved numerically using the implicit finite difference scheme of Crank Nicolson type. The effect of the viscosity and thermal conductivity variations on the transient flow variables such as velocity, temperature, shear stress and heat transfer rate is studied. 66

3 1. Mathematical problem statement A problem of transient, laminar, unsteady two-dimensional flow past a semi-infinite isothermal inclined plate is considered here. The analysis of the present paper is based on the following assumptions: the angle of the plate with the horizontal is assumed to be φ; the х axis is measured along the plate and the y axis is measured along upward normal to the plate; initially, the plate and the fluid are assumed to be at the same temperature T, the temperature of the plate is suddenly raised to temperature T w at the moment of time t > 0; the effect of viscous dissipation in the energy equation is assumed to be negligible; all the fluid physical properties are assumed to be constant except for the fluid viscosity, which varies exponentially with the fluid temperature, the thermal conductivity which varies linearly with the fluid temperature and the density variation in the body force term in the momentum equation where the Boussinesq approximation is invoked. Under these assumptions, the conservation equations for unsteady two-dimensional laminar boundary layer flow problem under consideration can be written as: u v + = 0, x y (1) u u u 1 u + u + v = gβ cosφ ( T T ) dy+ gβsin φ( T T ) + ρ, t x y x ρ y y (2) y T T T 1 T + u + v = k, t x y ρcp y y (3) where u and v are the velocity components in the x and y directions, respectively, ρ is the fluid density, T is the temperature of the fluid in the boundary layer, t is the time, T is the temperature far away from the plate, β is the volumetric coefficient of thermal expansion, C p is the specific heat, φ is the inclination angle to the horizontal, µ is the variable dynamic coefficient of viscosity, g is the acceleration due to gravity, and k is the variable thermal conductivity of the fluid. The initial and boundary conditions are t 0: u = 0, v = 0, T = T forall y, t > 0 : u = 0, v = 0, T = T at y = 0, (4) u = 0, T = T at x = 0, u 0, T T as y. On introducing the following non-dimensional quantities: x y ul vl t X =, Y = Gr, U = Gr, V = Gr, t = Gr, L L ν ν L 1/4 1/2 1/4 ν 1/2 2 67

4 G. Palani, J.D. Kirubavathi, and Kwang Yong Kim T T gβ L ( Tw T ) T =, Gr =, T 2 w T ν µ 0C p µ 0 Pr =, ν =, κ ρ 0 3 (5) where L is the length of the plate, ν is the kinematic viscosity, Gr is the Grashof number, Pr is the Prandtl number, µ 0 and к 0 are, respectively, the viscosity and thermal conductivity at temperature T w and having the variations of viscosity and thermal conductivity with T, as proposed by the authors of the works [10, 12, 16, 19, 22]: µ /µ 0 = e λt, (6) k/k 0 = 1 + γ T, (7) where λ and γ denote the viscosity and thermal conductivity variation parameters, respectively, depending on the nature of the fluid. Equations (1) (3) are reduced to the following dimensionless form: 2 U V + = 0, X Y U U U 1/4 + U + V = Gr cosφ TdY + Tsinφ+ t X Y X λt U XT T U + e λe, 2 Y Y Y T T T 1+ γ T U V T γ T + + = +. t X Y Pr 2 Y Pr Y 2 Y 2 (8) (9) (10) The corresponding initial and boundary conditions in a dimensionless form are as follows: t 0: U = 0, V = 0, T = 0 for all y, t > 0: U = 0, V = 0, T = 1 at Y = 0, U = 0, T = 0 at X = 0, (11) 68 U 0, T 0 at Y. Equations(8) (10) along with the initial and boundary conditions (11) describe the unsteady laminar boundary layer free convective flow past a semi-infinite isotheral inclined plate with variable viscosity and thermal conductivity. 2. Numerical techniques The two-dimensional, non-linear, unsteady, coupled and integro partial differential equations (8) (10) under the initial and boundary conditions in equation (11) are replaced by finite difference quotients and then solved using Crank Nicolson implicit finite difference scheme which is unconditionally stable, more accurate and fast convergent. The finite difference equations corresponding to Equations (8) (10) are given by:

5 k+ 1 k+ 1 k k k+ 1 k+ 1 k k Ui, j Ui 1, j+ Ui, j Ui 1, j+ Ui, j 1 Ui 1, j 1 + Ui, j 1 U i 1, j X k+ 1 k+ 1 k k Vi, j Vi, j 1 + Vi, j Vi, j 1 + = 0, 2 Y (12) k+ 1 k k 1 k 1 k k k 1 k 1 k k Ui, j U + + i, j Ui, j Ui 1, j Ui, j U + + i 1, j U k k i, j 1 Ui, j 1 Ui, j 1 U i, j 1 + U i, j + V i, j = t 2 X 4 Y 1/4 1 k 1 k φ TdY + Ti, j Ti, j X 2 Y = Gr cos + + sinφ+ k+ 1 k+ 1 k+ 1 k k k k+ 1 k λ ( T,, ) 2, 1 2,, 1, 1 2 i j + T U i j i j Ui j + Ui j+ + Ui j Ui, j+ U i, j+ 1 + e 2 2( Y ) k+ 1 k+ 1 k k k+ 1 k+ 1 k k k+ 1 k λ ( Ti, j + T, ) 2 T i j i, j+ 1 Ti, j 1 + Ti, j+ 1 T i, j 1 Ui, j+ 1 Ui, j 1 + Ui, j+ 1 U i, j 1 λe, 4 Y 4 Y (13) k+ 1 k k+ 1 k+ 1 k k k+ 1 k+ 1 k k Ti, j T i, j Ti, j T k i 1, j+ Ti, j T i 1, j T k i, j+ 1 Ti, j 1 + Ti, j+ 1 T i, j 1 + U i, j + V i, j = t 2 X 4 Y k k+ 1 k+ 1 k+ 1 k k k 1+ γt T, i, j 1 2 i, j i, j 1 i, j 1 2 i j T + T + + T Ti, j+ T i, j+ 1 = + Pr 2 2( Y ) γ T T T T + Pr 4 Y k+ 1 k+ 1 k k i, j+ 1 i, j 1 + i, j+ 1 i, j 1 The region of integration is considered as a rectangle with sides X max = 1 and Y max = 16, where Y max corresponds to Y =, which lies very well outside the momentum and energy boundary layers. The maximum of Y was chosen as 16 after some preliminary investigations so that the last two of the boundary conditions (11) are satisfied. Here, the subscript i designates the grid point along the X-direction, j along the Y-direction, and the superscript k along the t-direction. During any one time step, the coefficients U and V k, appearing in the difference equations are treated as constants. The values of U, V, and T are known at all grid points at t = 0, from the initial conditions. The computations of U, V, and T at time level (k + 1) using the values at previous time level k are carried out as follows: The finite difference equation (14) at every internal nodal point on a particular i-level constitute a tridiagonal system of equations. Such a system of equations is solved by Thomas algorithm [23]. Thus, the values of T are found at every nodal point for a particular i at (k + 1)th time level. Using these values in equation (13), the values of U at (k + 1)th time level are found in a similar manner. Thus, the values of T and U are known on a particular i-level. Finally, the values of V are calculated explicitly using equation (12) at every nodal point on a particular i-level at (k + 1)th time level. This process is repeated for various i-levels. Thus, the values of T, U, and V are known at all grid points in the rectangular region at (k + 1)th time level. 2. i k, j i j (14) 69

6 G. Palani, J.D. Kirubavathi, and Kwang Yong Kim After experimenting with few sets of mesh sizes, they have been fixed at the level X = 0.05, Y = 0.25, and the time step t = In this case, spatial mesh size are reduced by 50% in one-direction then in both directions, and the results are compared. It is observed that, when mesh size is reduced by 50 % in X-direction and Y-direction the results differ in fourth decimal place. Hence the above mentioned sizes have been considered as appropriate mesh size for calculations. Computations are carried out until the steady-state is reached. The steady-state solution is assumed to have been reached, when the absolute difference between the values of U as well as temperature T at two consecutive time steps are less than 10 5 at all grid points. The local truncation error is O( t 2 + Y 2 + X) and it tends to zero as t, X, and Y tend to zero, which shows that the scheme is compatible. Also the Crank Nicolson type of implicit finite difference scheme is proved to be unconditionally stable for a natural convective flow in which there is always a nonnegative value of velocity U and a nonpositive value of V (see [7]). Thus, compatibility and stability ensure the implicit finite difference scheme convergence. 3. Discussion of results The ranges for λ, γ, and Pr can be taken as follows in the present study [8,12]: for air: 0.7 λ 0 γ 6, Pr = 0.733; for water: 0 λ 0.6, 0 γ 0.12, 2 Pr 6; for lubrication oils: 0 λ 3, 0.1 γ 0, 500 Pr From equation (6), we observe that for λ < 0, the viscosity of the fluid increases with an increase in the temperature, and this is the case for fluids such as air. For λ > 0, the viscosity of the fluid decreases with an increase in the temperature and this is the case for the fluids such as water and lubrication oils. From Equation (7), we observe that γ > 0 implies that the thermal conductivity increases with an increase in the temperature, and this is the case for fluids such as water and air. While for γ < 0, the thermal conductivity decreases with an increase in the temperature, and this is the case for fluids such as lubrication oils. To examine the accuracy of the present computed results, we compare our results with the results of the work [12] (in the case of vertical plate with variable viscosity and thermal diffusivity) at the steady state level for velocity U and temperature T for φ = 90, λ = 0.4 with different values of γ for air (Pr = 0.733). The results of the present work, as shown in Fig. 1, agree very close with the results of the work [12]. In addition, the numerical values of velocity U are compared with the theoretical results of the work [6] for φ = 57.65, Gr = 10 6, Pr = 0.7, λ = 0, γ = 0 (corresponding to ζ = 16 of [6]) in Fig. 2 and are in good agreement. 70 Fig. 1. Comparison of the velocity and temperature profiles at Х = 1 for λ = 0.4 and Pr = φ = 90 ; 1 the results of the present work, 2 the results of the work [12].

7 Fig. 2. Comparison of steady velocity profiles at Х = 1. φ = 57.65, Gr = 10 6, Pr = 0.7, λ = 0, γ = 0; 1 the results of the present work, 2 the results of the work [6]. In the transient free convection problem, it was seen that the time is an important factor to study this transport phenomenon. The variation of velocity and temperature profiles with time was shown on graphs, and it was predicted that after certain lapse of time, the velocity and temperature reach steady state. Temporal maximum in velocity and temperature profiles is also observed. This type of phenomena was observed by several investigators for the problem of transient free convection on vertical or inclined flat plate. Now, we have to predict the effects of the inclination of the plate on the time to reach the steady state. When the plate makes an inclination angle φ = 60 with the horizontal, the velocity increases with time and reaches a temporal maximum at Y = 1.75 for t = 2.58 and subsequently it reaches steady state when t=4.95 this is in the case of Gr = 10 5, whereas in the case of Gr = 10 7, the velocity attains maximum value of at Y = 1.5 for t = 2.61 and subsequently it reaches the steady state. The steady state velocity is found to decrease with the increasing value of Grashof number. From the numerical results, we observed that the difference between the temporal maximum and steady state decreases with the increasing value of Grashof number. The variations of velocity and temperature are shown in Figs. 3 and 4, respectively, for Pr = (air), γ = 1, λ = 0.1, Gr = 10 5 and 10 7, and φ = 30 and 60. When the inclination angle φ increases, the normal component of the buoyancy force decreases near the leading edge, which causes an impulsive driving force to fluid motion along Fig. 3. Variation of the velocity profiles at different Gr and ϕ. * steady-state solution; λ = 0.1, γ = 1, Pr =

8 G. Palani, J.D. Kirubavathi, and Kwang Yong Kim Fig. 4. Variation of temperature profiles at different Gr and ϕ for Х = 1. * steady-state solution; λ = 0.1, γ = 1, Pr = the plate, that is, the impulsive force along the plate decreases with increasing φ. The time required to reach the steady state is less with an increase in the inclination angle φ. Since the tangential buoyancy force dominates in the downstream and also increases with φ, the velocity increases with φ, but it is observed to be in the other way in the case of temperature distribution. From the numerical results, it is observed that there is no appreciable change in the temperature distribution due to the change in Gr. Figures 5 8 demonstrate the variation of velocity and temperature for different viscosity and thermal conductivity in air (Pr = 0.733) at their transient, temporal maximum, and steady state for X = 1. The fluid velocity increases and reaches its maximum value near the wall (i.e., 0 Y 2) and then decreases monotonically to zero as Y becomes large for all times t. It is also noticed that the velocity and temperature increases with time t, it reaches a temporal maximum thereafter a reasonable decrease is observed and, consequently, it reaches the steady state. Figures 5 and 6 show the variation of transient velocity and temperature profiles with λ for a fixed value of γ = 2 in air (Pr = 0.733). The velocity of the fluid increases with time until a temporal maximum is reached and thereafter a moderate reduction is observed until the final steady state is reached. Time taken to reach the steady state decreases slightly with an increase 72 Fig. 5. Velocity profiles at different λ for γ = 2. * steady-state solution; Pr = 0.733, Gr = 10 6, φ = 60.

9 Fig. 6. Temperature profiles at different λ for γ = 2. * steady-state solution; Pr = 0.733, Gr = 10 6, φ = 60. in the viscosity variation parameter. From Fig. 5, it is clear that velocity U near the wall increases as λ increases (the viscosity of air decreases). But, an opposite trend is noticed at a certain distance from the wall. From Fig. 6, it is observed that the temperature of the fluid decreases as λ increases (the viscosity of air decreases). Fig. 7. Velocity profiles at different γ for λ = 0.2. * steady-state solution; Pr = 0.733, Gr = 10 6, φ = 60. Fig. 8. Temperature profiles at different γ for λ = 0.2. * steady-state solution; Pr = 0.733, Gr = 10 6, φ =

10 G. Palani, J.D. Kirubavathi, and Kwang Yong Kim The numerical values of the variation of transient velocity and temperature profiles with γ for a fixed value of λ = 0.2 in air (Pr = 0.733) are depicted graphically in Figs. 7 and 8. When γ = 5 the velocity attains its maximum value at Y =1.75 for t = 2.25 and moderate reduction is observed then it reaches steady state for t = For fixed value of λ, the velocity and temperature distribution in the fluid increases as γ increases(thermal conductivity of air increases). It can also be observed that with an increase in γ, the rise in the magnitude of the velocity and temperature is significant, which implies that the volume flow rate increases with an increase in γ. The effect of variation of thermal conductivity on velocity and temperature is more significant even in the initial transient period. Time to reach the temporal maximum and steady state decreases with increasing γ. The difference between the temporal maximum and steady state decreases as γ decreases. Figures 9 16 show the variation of velocity and temperature at their transient, temporal maximum and steady state against the coordinate Y at the leading edge of the plate viz., X = 1.0 for different Grashof numbers, inclination angle, viscosity, thermal conductivity and Prandtl numbers in water. The variation of velocity and temperature for different values of Grashof number Gr and inclination angle φ in water are calculated numerically and represented in the graphical form in Figs. 9 and 10. Due to rise in the inclination angle with the horizontal, the velocity is found to increase. Also we observed that the velocity increases as the Grashof number decreases. Fig. 9. Velocity profiles at different Gr and ϕ. * steady-state solution; Pr = 3, λ = 0.2, γ = Fig. 10. Temperature profiles at different Gr and φ. * steady-state solution; Pr = 3, λ = 0.2, γ = 0.02.

11 Fig. 11. Velocity profiles at different λ for γ = * steady-state solution; Pr = 3, Gr = 10 5, φ = 45. The numerical values of variation of the velocity and temperature calculated from equations (13) and (14) are depicted in the graphical form in Figs. 11 and 12 for various values of λ for fixed value of γ = 0.02 in water (Pr = 3). We observed that the time taken to reach the temporal maximum and steady state decreases with an increase in the viscosity variation parameter λ. It can be seen from Fig. 11 that an increase in the viscosity variation parameter λ increases the velocity of the flow near the wall, because the viscosity of water decreases with an increase of the viscosity variation parameter λ, as seen in equation (6). In addition, the maximum velocity gets very closer to the wall (Y = 1.25) for higher values of λ. This qualitative effect arises because, for the case of variable viscosity (λ > 0), the fluid is able to move more easily in a region close to the heated surface in association with the fact that the viscosity of the fluid with λ > 0 is lower relative to the fluid with constant viscosity. This results in thinner velocity and thermal boundary layers. It is observed that as λ increases (the viscosity of water decreases), the velocity of the fluid particle increases only in the region 0 Y 1.5. Figure 12 shows the variation of temperature profiles with increasing λ. This is in association with the fact that an increase in λ yields an increase in the peak velocity as shown in Fig. 11. However, two opposing force effects, due to an increase in λ, on the fluid can be considered. The first effect increases the velocity of the fluid particle, due to the decrease in the viscosity, and the second effect decreases the velocity of the fluid particle, due to Fig. 12. Temperature profiles at different λ for γ = * steady-state solution; Pr = 3, Gr = 10 5, φ =

12 G. Palani, J.D. Kirubavathi, and Kwang Yong Kim Fig. 13. Velocity profiles at different γ for λ = 0.3. * steady-state solution; Gr = 10 6, φ = 45, Pr = 3. the decrease in the temperature. In the region (0 Y 1.5), the temperature Т is high, consequently, the first force will be dominant and the velocity U increases as λ increases (Fig. 11). On the other hand, as the temperature T is low far away from the wall, the second effect will be dominant and the velocity decreases as λ increases (Fig. 11). From the discussion, we notice that neglecting the variation of fluid viscosity and thermal conductivity will introduce a substantial error. Figures 13 and 14 show the variation of velocity and temperature for various values of γ for fixed value of λ =0.3 in water (Pr = 3). We observe from these curves that the time taken to reach the steady state decreases with the increasing value of γ. It is also observed that the velocity of water decreases as γ decreases and the reverse trend is noticed as Y increases. It is also observed that the temperature distribution of the fluid increases with the increasing value of γ. The variation of transient velocity and temperature with Prandtl numbers for fixed values of other parameters is shown in Figs. 15 and 16. Higher value of Prandtl number required more time to reach the temporal maximum and steady state in comparison with the lower values of Prandtl number. From Fig. 15, we notice that the velocity profile decreases with the increasing value of Prandtl number. Larger Prandtl number values give rise to thinner temperature profiles, because a larger Prandtl number value means that the thermal diffusion from the wall is not prevailing, whereas the velocity diffusion extends far from the wall. 76 Fig. 14. Temperature profiles at different γ for λ = 0.3. * steady-state solution; Gr = 10 6, φ = 45, Pr = 3.

13 Fig. 15. Velocity profiles at different Pr. * steady-state solution; Gr = 10 6, φ = 45, λ = 0.2, γ = 0.02; Pr = 2 (1), 4 (2), 6 (3). Figures show the variation of velocity and temperature at their transient, temporal maximum and steady state against the coordinate Y at the leading edge of the plate viz., X = 1.0 for different Grashof numbers, inclination angle, viscosity, thermal conductivity, and Prandtl numbers in lubricating oils. Fig. 16. Temperature profiles at different Pr. * steady-state solution; Gr = 10 6, φ = 45, λ = 0.2, γ = 0,02; Pr = 2 (1), 4 (2), 6 (3). Fig. 17. Velocity profiles at different Gr and φ. * steady-state solution; Pr = 500, λ = 1, γ =

14 G. Palani, J.D. Kirubavathi, and Kwang Yong Kim Fig. 18. Temperature profiles at different Gr and φ. * steady-state solution; λ = 0.2, γ = The variations of velocity and temperature of lubricating oil for different values of Grashof number and inclination angle are calculated numerically and presented in a graphical form in Figs. 17 and 18, respectively. Velocity increases drastically near the wall and then decreases far away from the wall. The effect of Gr on velocity is more as φ decreases. The velocity attains a temporal maximum value at Y = 0.5, t = and moderate reduction is observed then it reaches a steady state at t = (Gr = 105, φ = 30 ). Velocity decreases as Gr increases, the same trend was noticed in the case of air and water (Figs. 3 and 9). In addition, the velocity is found to increase due to rise of inclination angle with the horizontal. The numerical values of variation of velocity and temperature profiles with λ for a fixed value of γ = 0.05 for lubrication oil (Pr = 500) are shown graphically in Figs. 19 and 20. From these figures, it is observed that time taken to reach the steady state is more when λ decreases. It is also noticed that the velocity increases as λ increases (the viscosity of oil decreases). The temperature of the fluid decreases as λ increases. The numerical values of the variation of transient velocity and temperature profiles with γ for a fixed values of λ = 1.0 for lubrication oil (Pr = 500) are presented in the graphical form in Figs. 21 and 22. It is noticed that the velocity and temperature distribution in the fluid increases as γ increases. Fig. 19. Velocity profiles at different λ for γ = * steady-state solution; Pr = 500, Gr = 10 6, φ = 30 ; λ = 1 (1), 2 (2), 3 (3).

15 Fig. 20. Temperature profiles at different λ for γ = * steady-state solution; Pr = 500, Gr = 10 6, φ = 30 ; λ = 2 (1), 3 (2). Fig. 21. Velocity profiles at different γ for λ = 1. * steady-state solution; Pr = 500, Gr = 10 6, φ = 30 ; γ = 0.05 (1), 0.1 (2). Fig. 22. Temperature profiles at different γ for λ = 1. * steady-state solution; Pr = 500, Gr = 10 6, φ = 30 ; γ = 0.05 (1), 0.1 (2). 79

16 G. Palani, J.D. Kirubavathi, and Kwang Yong Kim Fig. 23. Velocity profiles at different Pr. * steady-state solution; Gr = 10 6, φ = 30, λ = 1, γ = 0.1; Pr = 500 (1), 750 (2), 1000 (3). Knowing the velocity and temperature field, it is quite interesting to study from the practical point of view, the most important characteristics of the flow are the shearing stress and the rate of heat transfer at the plate. The local shear stress at the plate is defined by u τ x = µ y y= 0. (15) By introducing the non-dimensional quantities given in equations (5) and (6) in (15), we get non-dimensional form of local skin friction, and it is given by τ X λ 34 U = e Gr. Y The integration of equation (16) from X = 0 to X = 1 gives the average skin friction, and it is given by 1 λ 34 U τ = e Gr dx. Y (17) Y = 0 0 Y = 0 (16) 80 Fig. 24. Temperature profiles at different Pr. * steady-state solution; Gr = 10 6, φ = 30, λ = 1, γ = 0.1; Pr = 500 (1), 750 (2), 1000 (3).

17 Fig. 25. Local skin friction distribution. Gr = 10 6, φ = 30, Pr = 0.733; 1 λ = 0.2, γ = 2; 2 λ = 0.4, γ = 4; 3 λ = 0.6, γ = 3; 4 λ = 0.4, γ = 3. The local Nusselt number is defined by L( k( T y) ) y= 0 Nu x =. k ( T T ) 0 w (18) Using the non-dimensional variables defined in equations (5) and (7) in (18), the nondimensional form of local Nusselt number is given by Nu X 14 T = (1 + γ )Gr. Y The integration of equation (19) from X = 0 to X = 1 gives the average Nusselt number, and it is given by 1 14 T τ = (1 + γ)gr dx. Y (20) The derivatives involved in equations (16), (17), (19), and (20) are evaluated by using a fivepoint approximation formula, and then the integrals are evaluated by Newton Cotes closed integration formula. Figs. 25 and 26 reveal the local variation of shearing stress with different values of variation parameter λ and γ for air and water, respectively. The local shearing stress increases 0 Y = 0 Y = 0 (19) Fig. 26. Local skin friction distribution. Gr = 10 6, φ = 30, Pr = 5; 1 λ = 0.2, γ = 0.08, 2 λ = 0.2, γ = 0.06, 3 λ = 0.4, γ =

18 G. Palani, J.D. Kirubavathi, and Kwang Yong Kim Fig. 27. Distribution of the local Nu number. Gr = 10 6, φ = 30, Pr = 0.733; 1 λ = 0.2, γ = 2; 2 λ = 0.2, γ = 4; 3 λ = 0.6, γ = 3; 4 λ = 0.4, γ = 3. gradually as X increases. From the numerical results, we noticed that the local shearing stress increases with the decreasing value of the viscous parameter λ. In addition, we observed that local skin friction increases with the increasing value of thermal conductivity parameter γ. The local heat transfer distribution is presented in Figs. 27 and 28 for air and water, respectively, with different values of variation parameter λ and γ. The local heat transfer is found to increase as both the parameters λ and γ increase. The average values of skin friction are given in Figs. 29 and 30. The average skin friction increases with time and asymptotically reaches a constant value. It is observed that average skin friction increases with the decreasing value of λ. But the reverse trend is noticed with respect to γ. Figures 31 and 32 show the average values of Nusselt number for air and water, respectively. It is observed that average Nusselt number increases as both the viscous parameter λ and thermal conductivity parameter γ increases. 82 Fig. 28. Distribution of the local Nu number. Gr = 10 6, φ = 30, Pr = 5; 1 λ = 0.2, γ = 0.04; 2 λ = 0.2, γ = 0.06; 3 λ = 0.4, γ = 0.06.

19 Fig. 29. Average skin friction vs. time. Gr = 10 6, φ = 30, Pr = 0.733; 1 λ = 0.2, γ = 2; 2 λ = 0.2, γ = 4; 3 λ = 0.6, γ = 3; 4 λ = 0.4, γ = 3. Fig. 30. Average skin friction vs. time. Gr = 10 6, φ = 30, Pr = 5; 1 λ = 0.2, γ = 0.04; 2 λ = 0.2, γ = 0.06; 3 λ = 0.4, γ = Fig. 31. Average Nu number vs. time. Gr = 10 6, φ = 30, Pr = 0.733; 1 λ = 0.2, γ = 2; 2 λ = 0.6, γ = 4; 3 λ = 0.6, γ = 3; 4 λ = 0.4, γ = 3. 83

20 G. Palani, J.D. Kirubavathi, and Kwang Yong Kim Fig. 32. Average Nu number vs. time. Gr = 10 6, φ = 30, Pr = 5; 1 λ = 0.2, γ = 0.04, 2 λ = 0.2, γ = 0.06, 3 λ = 0.4, γ = Conclusions A numerical study has been performed for the flow of a fluid along a semi-infinite isothermal inclined plate taking into account the effect of variable viscosity and thermal conductivity. The fluid viscosity is assumed to vary as an exponential function and the thermal conductivity as a linear function of the temperature. A family of governing partial differential equations is solved by an implicit finite difference scheme of Crank Nicolson type. The effect of variable viscosity and thermal conductivity on velocity, temperature, shearing stress and heat transfer is studied in detail. A comparison between the present numerical results and previously published works is also made. The agreement between the two results is found to be excellent. Conclusion of this study is as follows: 1. The frictional coefficient increases with the decrease of λ (the viscosity of water decreases), but increases with the increase of γ (the thermal diffusivity of water increases). 2. The viscosity, thermal conductivity, and Prandtl number of a working fluid have turned out to be sensitive to the variation of the temperature in a natural convection problem. Hence, the effect of variable viscosity, thermal conductivity, and Prandtl number has to be taken into consideration to accurately predict the skin-friction coefficient and heat transfer rate. 3. The results pertaining to the fluid with variable viscosity and thermal conductivity differ significantly from those of the fluid with constant properties. 4. It is observed that, neglecting the viscosity and thermal conductivity variation will give substantial errors. Therefore, we conclude that, in order to predict more accurate results, the effects of the variable viscosity and thermal conductivity have to be considered. References 1. E. Pohlhausen, Der Wärmeaustausch zwischen festen körpern und Flüssigkeiten mit kleiner Reibung und kleiner Wärmeleitung, Z. Angew. Math. Mech., 1921, Vol. 1, P S. Ostrach, An analysis of laminar free convection flow and heat transfer about a flat plate parallel to the direction of the generating body force, NACA Report, 1953, No. 1111, P R. Seigel, Transient free convection from a vertical flat plate, Trans. Amer. Soc. Mech. Engng., 1958, Vol. 80, P J.D. Hellums and S.W. Churchill, Transient and steady state free and natural convection, numerical solution, Рart 1. The isothermal vertical plate, AIChE J., 1962, Vol. 8, P

21 5. V.M. Soundalgekar and P. Ganesan, Finite difference analysis of transient free convection on an isothermal flat plate, Reg. J. Energy Heat Mass Transfer, 1981, Vol. 3, P T.S. Chen, H.C. Tien, and B.F. Armaly, Natural convection on horizontal, inclined and vertical plates with variable surface temperature or heat flux, Inter. J. Heat Mass Transfer, 1986, Vol. 29, P K. Ekambavannan and P. Ganesan, Finite difference solution of unsteady natural convection boundary layer flow over an inclined plate with variable surface temperature, Wärme- und Stoffübertragung, 1994, Vol. 30, P H. Schlichting, Boundary Layer Theory, McGraw Hill, New York, W.M. Kays and M.E. Crawford, Convective Heat and Mass Transfer, McGraw Hill, New York, H. Ockendon and J.R. Ockendon, Variable-viscosity flows in heated and cooled channels, J. Fluid Mech., 1977, Vol. 83, No. 1, P J. Gary, D.R. Kassory, H. Tadjeran, and A. Zebib, Effect of significant viscosity variation on convective heat transport in water saturated porous media, J. Fluid Mech., 1982, Vol. 117, P E.M.A. Elbashbeshy and F.N. Ibrahim, Steady free convection flow with variable viscosity and thermal diffusivity along a vertical plate, J. Physics D: Applied Physics, 1993, Vol. 26, No. 12, P E.M.A. Elbashbeshy, Free convection flow with variable viscosity and thermal diffusivity along a vertical plate in the presence of the magnetic field, Int. J. Engng. Sci., 2000, Vol. 38, P M.A. Hossain, K. Khanafer, and K. Vafai, The effect of radiation on free convection flow of fluid with variable viscosity from a porous vertical plate, Int. J. Therm. Sci., 2001, Vol. 40, P M.A. Seddeek, Effect of variable viscosity on a MHD free convection flow past a semi-infinite flat plate with an aligned magnetic field in the case of unsteady flow, Int. J. Heat Mass Transfer, 2002, Vol. 45, No. 4, P S.K. Wilson and B.R. Duffy, Strong temperature-dependent-viscosity effects on a rivulet draining down a uniformly heated or cooled slowly varying substrate, Phys. Fluids, 2003, Vol. 15, No. 4, P E.M. Abo-eldahab, The effects of temperature-dependent fluid properties on free convective flow along a semiinfinite vertical plate by the presence of radiation, Heat Mass Transfer, 2004, Vol. 41, No. 2, P V.M. Soundalgekar, H.S. Takhar, U.N. Das, R.K. Deka, and A. Sarmah, Effect of variable viscosity on boundary layer flow along a continuously moving plate with variable surface temperature, Heat Mass Transfer, 2004, Vol. 40, No. 5, P M.A. Seddeek and M.S. Abdelmeguid, Effects of radiation and thermal diffusivity on heat transfer over a stretching surface with variable heat flux, Phys. Letters A, 2006, Vol. 348, Nos. 3 6, P A.A. Mahmoud Mostafa, Variable viscosity effects on hydromagnetic boundary layer flow along a continuously moving vertical plate in the presence of radiation, Appl. Math. Sci., 2007, Vol. 1, No. 17, P G. Palani and Kwang-Yong Kim, Numerical study on a vertical plate with variable viscosity and thermal conductivity, Arch. Appl. Mech., 2010, Vol. 80, No. 7, P J.C. Slattery, Momentum, Energy and Mass Transfer in Continua, McGraw Hill, New York, B. Carnahan, H.A. Luther, and J.O. Wilkes, Applied Numerical Methods, John Wiley and Sons, New York,

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