Research Article | Open Access
Flow and Heat Transfer in a Liquid Film over a Permeable Stretching Sheet
An analysis has been carried out to study the flow and heat transfer in a liquid film over a permeable stretching sheet. Using similarity transformations, the time-dependent boundary layer equations are reduced to a set of nonlinear ordinary differential equations. The resulting parameter problem and velocity as well as temperature fields are solved using the homotopy analysis method (HAM). Analytic series solutions are given, and numerical results for velocity and the temperature profiles are presented through graphs of different values for pertinent parameter. The effects of unsteadiness parameter and permeability parameter on the velocity and temperature profiles are explored for different values of blowing or suction parameter.
The study of flow and heat transfer in a thin liquid film is often encountered in industrial and engineering applications. Such processes are wire and fibre coating, cooling of metallic plates, drawing of a polymer sheet, aerodynamic extrusion of plastic sheets, and thinning of copper wires. In polymer processing application, the flow over a permeable stretching sheet has received considerable attention from researchers because of its widespread applications. Therefore, the analysis of momentum and thermal transport within thin liquid film on a continuously stretching surface is important for gaining some fundamental understanding of such processes. Crane  studied the Newtonian fluid flow induced by the stretching of an elastic sheet. The boundary layer equation is considered, and the boundary conditions are prescribed at the sheet and on the fluid at infinity. Nonetheless, in real physical application involving coating processes, the consideration of the fluid adhering to the finite liquid film is important. The hydrodynamics of a flow in a thin liquid film driven by an unsteady stretching surface was first investigated by Wang . Andersson et al.  explored the heat transfer characteristics of the hydrodynamical problem. Wang  investigated the same problem of Andersson et al. , presenting analytic solutions. Several researchers extended Wang’s  classical problem in non-Newtonian problems [5–8], thermocapillarity effects [9–12], and magnetic effects [13, 14].
Wang  was probably the first to analyze the flow and heat transfer in a thin liquid film on an unsteady stretching surface using homotopy analysis method (HAM). The HAM is a general analytic method to obtain series solutions for various types of nonlinear equations [15, 16]. Several studies have successfully applied HAM to various nonlinear problems in science and engineering such as in heat transfer , upper-convected Maxwell fluid , Kawahara equation , diffusion and reaction in porous catalyst , micropolar fluid flow [21, 22], and MHD flow [23, 24].
The purpose of the present study is to give a numerical analysis of the flow and heat transfer in a thin liquid film over a porous stretching sheet, and the effects of unsteadiness parameter on the film thickness are explored for different values of blowing or suction parameter. Here the momentum and energy equations are highly nonlinear; hence a similarity transformation is used to transform the nonlinear partial differential equations into nonlinear ordinary differential equations. The resulting parameter problem and velocity as well as temperature fields are solved using the homotopy analysis method (HAM). The analytical solutions reached using HAM are presented and implications discussed.
2. Mathematical Model
Consider that the Newtonian fluid flow in a thin liquid film of uniform thickness on a horizontal thin elastic sheet emerges from a narrow slot at the origin of a Cartesian coordinate system. The fluid motion and heat transfer are due to the stretching of a permeable elastic sheet parallel to the -axis at . Two equal and opposite forces are applied along the -axis, keeping the origin fixed. Under these assumptions, the governing conservation equations of mass, momentum, and energy at unsteady state can be expressed as where is the velocity component of the fluid in the direction is defined as , is the mass transfer velocity at the surface of the sheet with for blowing (injection) and for suction, and corresponding to an impermeable sheet. Further, is the temperature, is the kinematic viscosity, is the density, is the specific heat at constant pressure, and is the thermal diffusivity. It is assumed that the surface of the planar liquid film is smooth and free of surface waves, and the viscous shear stress as well as the heat flux vanishes at the adiabatic free surface. Thus the boundary conditions are The elastic sheet’s temperature is assumed to vary both along the sheet and with time in accordance with where is the temperature at the slit, is the constant reference temperature for all , and is the positive constant of proportionality with dimension ( ).
Consider employing the similarity transformations which are given as where is the dimensionless film thickness and is the stream function defined by where a prime denotes differentiation with respect to . Apparently, (7) has already satisfied the continuity equation, (1). Consequently, (1)–(4) are transformed to the following nonlinear boundary-value problem: subject to where is a dimensionless measure of the unsteadiness, is an unknown constant which must be determined as a part of the boundary value problem, is Prandtl number, and is permeability parameter. It is worth mentioning that the momentum boundary layer problem defined by the ODE (8) subject to the relevant boundary conditions (10) is decoupled from the thermal boundary layer problem, while the temperature field is on the other hand coupled to the velocity field. The most important characteristics of flow and heat transfer are the shear stress and the heat flux on the stretching sheet that are defined as where is the fluid dynamic viscosity. The local skin-friction coefficient and the local Nusselt number can be defined as Thus, the skin friction and the rate of heat transfer for fluid flow in a thin film can be expressed as where is the local Reynolds number.
3. Analytical Approach
In this section we apply HAM to solve systems (8)–(11). We assume that the solutions of and can be expressed by a set of base functions with the following forms: where and are constants. Under the rule of solution expression given by (8) and (9), subject to the boundary conditions (10) and (11), it is straightforward to choose as the initial guesses of and . The auxiliary linear operators and are chosen with the following properties: where , , , , and are constants of integration. From (8) and (9), the nonlinear operators are defined as where and are both unknown functions of and while is a function dependent on the embedding parameter . Let and denote the non-zero auxiliary parameters whereas and denote nonzero auxiliary functions, respectively. The zero-order deformation equation can be constructed as subject to the boundary conditions From (17) and (18), it is straightforward to show that when , the solutions of (21)-(22) are Since and , when , (21)-(22) are equivalent to (8)–(11), respectively, provided that Thus, as increases from to , and vary from the initial guesses and to the solutions and in (8)–(11). In addition, also varies from the initial guess to . Applying the Taylor series to (23) and (25), , , and can be expanded as series of where Thus, using (24), we have By differentiating (21) times with respect to , then setting , and finally dividing by , the th-order deformation equations are obtained: subject to the boundary conditions for , where Let and denote the particular solutions of (29) as The general solutions for (19) are Hence, the th-order approximations of , , and are given, respectively, by
4. Results and Discussion
We solve (29)-(30) with the aid of Maple, a symbolic computation software. The auxiliary functions and in (29) were set to be equal to 1 in all calculations reported in this paper. The solution of (29)-(30) can be expressed as for , where and are the coefficients, which can be obtained recursively for using given by (16) and (17). When , we have the analytic solutions as follows: where where . Note that (37) will be simplified to the expression of Wang  when .
We noted that the HAM analytic solutions contain two nonzero auxiliary parameters and that can be used to adjust and control the convergence of the series solutions. Liao [15, 25] indicates that the analytic solutions given by HAM contain an auxiliary parameter which could be chosen by means of what he calls the -curve.
Figures 1(a) and 1(b) show the variation of with using 10th-order HAM approximation when for suction and for blowing, respectively. For suction case, it is shown that the convergence region can be obtained by choosing a value of in the ranges for , for , for , for , and for .
For blowing case, the appropriate ranges of the auxiliary parameter to maintain the convergent rate and region of are for , for , for , for , and for .
The curves of are graphed in Figures 2(a) and 2(b) for suction and blowing, respectively. For suction case, it is shown that the convergence region can be obtained by choosing a value of in the ranges for , for , for , for , and for .
For blowing case, the appropriate ranges of the auxiliary parameter to maintain the convergent rate and region of are for , for , for , for , and for .
In order to retain the convergent rate and region of , Figures 3(a) and 3(b) show the appropriate ranges of for both suction and blowing. For suction case, the appropriate value of is in the ranges for , for , for , and for .
For blowing case, the appropriate value of is in the ranges for , for , for , for , and for .
To demonstrate the accuracy of the present findings, results for dimensionless film thickness and the skin friction are compared with the available results obtained by Wang . It is found that all the numerical results in Table 1 agree with the values of and in Wang . As the unsteadiness parameter increases, the skin friction increases and thereby the film thickness reduces.
The effects of permeability parameter on the velocity distribution are presented in Figures 4(a) and 4(b) for suction and blowing, respectively. It is found that, as permeability parameter increases, the flow velocity of the thin film increases for suction case, while for blowing case, the flow velocity decreases.
The temperature profiles for different values of suction and blowing are demonstrated in Figures 6(a) and 6(b). It is seen that the temperature of the fluid flow increases with the increase in suction whereas blowing () has quite the opposite effect on the temperature of the flow.
The effects of the unsteadiness parameter on the temperature profile are shown graphically in Figures 7(a) and 7(b). It is observed that by increasing the value of unsteadiness parameter, the temperature profiles decrease for both cases of suction and blowing.
5. Concluding Remarks
In this paper an analysis has been carried out to study the effects of porous stretching sheet for both cases of suction and blowing. Analytical and numerical solutions are obtained for momentum and heat transfer over a permeable stretching sheet. The effect of permeability parameter is to increase the flow velocity for suction case and to decrease the flow velocity for the blowing case. Besides, we found that the permeability parameter has significant effects on the temperature distribution. When increases, the temperature of the fluid flow increases for suction case and decreases for blowing case. When the unsteadiness parameter increases, the velocity of thin film flow increases for both cases of suction () and blowing (); thereby the film thickness reduces.
The authors would like to acknowledge the financial support received from the Grant UKM-ST-07-FRGS0028-2009.
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