There is considerable interest among recent researchers in the fields of flow and heat transfer in a laminar liquid film on an unsteady stretching sheet because of their numerous practical applications in many branches of manufacturing processes and technology. Some of the typical applications of such flow and heat transfer in a laminar liquid film due to an unsteady stretching sheet are food stuff processing,the process of designing various heat exchangers and chemical reaction equipments,reactor fluidization,wire and fiber coatings,transpiration cooling,reactor fluidization,polymer processing,and microchip production. The best quality of the final product in the coating process crucially depends on the rate of heat transfer at the stretching sheet. Therefore,the accumulation of knowledge of heat transfer aspects for the stretching sheet problem,especially for various thermal conditions,is required in the documentation of thermal community. Much work on the boundary-layer Newtonian fluids has been carried out both experimentally and theoretically. Crane^[1] was the first one among others who considered analytically the problem of steady flow for a Newtonian fluid driven by a stretched flat sheet moving in its own plane with a velocity varying linearly with the distance from a fixed point.Because of importance for the analysis of momentum and thermal transports within a thin liquid film over a stretching surface,a lot of papers were found. Wang^[2] was believed to be the first one who studied the flow of a Newtonian fluid in a thin liquid film over an unsteady stretching sheet where the similarity transformation was used to transform the governing partial differential equations describing the problem to a nonlinear ordinary differential equation with an unsteadiness parameter,and he solved this problem numerically and analytically. Dandapat et al.^[3] extended Wang’s problem by including the heat transfer analysis. Thin liquid film flow and heat transfer due to an unsteady stretching phenomenon with the thermocapillarity effect were discussed by Dandapat et al.^[4]. Chen^[5] investigated the heat transfer characteristics of the flow problem for a non-Newtonian power-law liquid film over an unsteady stretching surface. By the homotopy analysis method,Wang and Pop^[6] presented analytical results for the heat transfer problem for a power-law liquid film over an unsteady stretching surface. The flow of a second grade fluid liquid film over an unsteady stretching sheet was investigated by Abbas et al.^[7]. The problem of the flow and heat transfer in a thin liquid film over an unsteady elastic stretching surface with the effects of thermocapillarity and a magnetic field was analyzed by Noor and Hashim^[8]. Mahmoud and Megahed^[9] investigated the effects of variable viscosity and thermal conductivity on the flow and heat transfer of an electrically conducting power-law fluid within a thin liquid film over an unsteady stretching sheet in the presence of a transverse magnetic field. Recently,the heat transfer characteristics of a viscous liquid film flow over an unsteady stretching sheet subject to variable heat flux were investigated numerically by Liu et al.^[10].

However,the results reported in the above works are limited,since the presence of viscous dissipation has not been considered. Research into thin film flow taking into account viscous dissipation has considerable interest. The effect of viscous dissipation changes the temperature distributions by playing an important role as an energy source,which affects both the heat transfer rates and the temperature of the sheet^[11]. For instance,significant temperature rises are observed in polymer processing flows such as injection modeling or extrusion at high rates,and aerodynamic heating in the thin layer around a high speed aircraft raises the temperature of the skin. For this purpose,Chen^[12] examined the effect of viscous dissipation on the convective heat transfer of non-Newtonian power-law fluids within a thin liquid film on an unsteady stretching sheet. Abel et al.^[13] carried out a mathematical analysis of magnetohydrodynamics (MHD) flow and heat transfer to a laminar liquid film from a horizontal stretching surface in the presence of viscous dissipation.

All these above investigations assume the conventional no-slip condition at the boundary past a stretching surface. Undoubtedly,for many decades,people have conducted extensive research trying to understand and control the slip flow behaviors past a stretching surface. Wall slip readily occurs for an array of complex fluid such as emulsions,suspensions,foams,and polymer solutions. Also,the fluids that exhibit boundary slip have important technological applications such as in the polishing of artificial heart valves and internal cavities. In light of these various applications,many authors have studied and reported the results on the flow and heat transfer characteristics in the presence of slip effects. Thompson and Troian^[14] discovered that the slip velocity is related to the slip length,the shear rate at the wall,and a critical shear rate at which the slip length diverges. Megahed^[15] presented an approximate analytical solution for the unsteady slip velocity flow of a thin viscous liquid film over a heated horizontal stretching surface in the presence of variable heat flux. Turkyilmazoglu^[16] investigated the MHD slip flow of an electrically conducting viscoelastic fluid past a stretching surface. Recently,Megahed^[17] investigated the numerical solution for the slip velocity and variable viscosity effects on the flow and heat transfer of a non-Newtonian power-law fluid over a stretching surface in the presence of thermal radiation and constant heat flux. Very recently,El-Hawary et al.^[18] examined the effect of the slip velocity,concentration dependent diffusivity,thermal convective boundary conditions,and heat source/sink on the stagnation-point heat and mass transfer ofa viscoelastic fluid over a stretching sheet using the Lie group analysis. Nevertheless,to the authors’ present knowledge,the flow and heat transfer for the Casson thin film fluid in the presence of variable heat flux due to an unsteady stretching sheet with slip velocity and viscous dissipation effects are not yet studied. Thus,the main aim of this present work is to study the numerical solution for the problem of Casson thin film flow over an unsteady stretching sheet with viscous dissipation and variable heat flux involving boundary conditions of slip effect. The physical significance of the present paper is in addressing viscous dissipation and heat flux coupled with the slip velocity effect,which have an important role in governing the rate of heat transfer,which has several applications in fields of engineering and technology. To achieve this study,we use the well known numerical technique,i.e.,the shooting method.

In this section,we investigate the physical situation for the unsteady two-dimensional laminar flow and heat transfer of a non-Newtonian Casson thin film with the uniform thickness h(t) over a horizontal elastic sheet in the presence of viscous dissipation and variable heat flux,which emerges from a narrow slot at the origin of a Cartesian coordinate system,as shown in Fig. 1. From this figure,we observe that the x-axis is chosen along the plane of the sheet,and the y-axis is taken normal to the plane. We assume that the surface starts stretching from rest with the velocity U(x,t) which can be defined as

Click Browse formula

Fig. 1 Schematic representation of problem

Figure options

where t is the time,and α and c are positive constants with the dimension reciprocal time. Here,c is the initial stretching rate. Also,we must observe that the adopted formulation of the sheet velocity in Eq. (1) is valid only for t < α⁻¹ unless α = 0. Likewise,the surface heat flux q_s(x,t) at the stretching sheet is assumed to vary with the power of distance x from the slit and with the power of time factor as

Click Browse formula

where κ is the fluid thermal conductivity,T is the temperature,T_ref is the reference temperature,d is a constant,and r and m are space and time indices,respectively. Also,the surface temperature T_s of the stretching sheet is assumed to vary with the distance x from the slit as

Click Browse formula

where T₀ is the temperature at the slit,and ν is the kinematic viscosity. Also,the reference temperature T_ref can be taken as a constant such that 0 ≤ T_ref ≤ T₀.

The rheological equation of state for an isotropic and incompressible flow of a Casson fluid is^{[19, 20]} as follows:

Here,τ_ij is the (i,j)th component of the stress tensor,ϕ = e_ije_ij,and e_ij is the (i,j)th component of the deformation rate,ϕ is the product of the deformation rate with itself,ϕ_c is a critical value of this product based on the non-Newtonian model,μB is the plastic dynamic viscosity of the non-Newtonian fluid,and P_y is the yield stress of the fluid. Therefore,when the shear stress is smaller than the yield stress P_y,the fluid exhibits no motion,i.e.,it behaves like a solid,but when the shear stress is greater than P_y,it demonstrates flow characteristics.

The governing time-dependent velocity and temperature fields of the thin film obeying such a type of flow are given by

Click Browse formula

Click Browse formula

Click Browse formula

where u and v are the velocity components along the x- and y-directions,respectively. ρ is the fluid density,β = μ_B√2ϕ_c/P_y is the Casson parameter,T is the temperature of the fluid,and c_p is the specific heat at the constant pressure.

The appropriate boundary conditions for the present problem are

Click Browse formula

Click Browse formula

Here,is the velocity slip factor which changes with the time,and N is the initial value of the velocity factor. It is noted that Eqs.(7) and (8) account for the stress-free and adiabatic free surfaces,respectively. In addition,the free surface is assumed to be uniform such that the similarity treatment is available.

The equation of continuity is satisfied if we choose a stream function ψ(x,y) such that ,and . The mathematical analysis of the problem is simplified by introducing the following dimensionless coordinates:

Click Browse formula

Click Browse formula

Equations (9)-(10) are valid only for αt << 1,where η is the similarity variable,f(η) is the dimensionless stream function,and θ(η) is the dimensionless temperature.

Using Eqs. (9)-(10),the mathematical problems defined in Eqs. (5) and (6) are then transformed to a set of ordinary differential equations and their associated boundary conditions,

Click Browse formula

Click Browse formula

Click Browse formula

Click Browse formula

Click Browse formula

where a prime denotes differentiation with respect to η, is the unsteadiness parameter, is the velocity slip parameter, is the Eckert number,and is the Prandtl number. Further,γ denotes the value of the similarity variable η at the free surface so that the first term of Eq. (9) gives

Click Browse formula

Since γ is an unknown constant,which should be determined,as a whole,from the set of the present boundary-value problem,the rate of change of the film thickness can be obtained as follows:

Click Browse formula

Thus,the kinematic constraint at y = h(t) given by Eq. (8) transforms to the free surface condition (16).

In engineering and practical applications,our interest lies in the investigation of the important physical quantities of the flow behavior and heat transfer characteristics by analyzing the non-dimensional local skin-friction (Cf_x) or the fractional drag coefficient and the local Nusselt number (Nu_x). These non-dimensional parameters are defined as follows:

Click Browse formula

where is the local Reynolds number.

The numerical procedure is used to solve the differential system (11)-(12). This system along with the boundary conditions (13)-(15) is integrated numerically by means of the RungeKutta method with systematic estimate of f''(0) and θ(0) by the Newton-Raphson shooting technique until the outer boundary condition (14) is satisfied. The following first-order system is set:

Click Browse formula

Equations (11)-(12) with the boundary conditions (13) and (15) are then reduced to a system of first-order ordinary differential equations,i.e.,

Click Browse formula

The estimated value of γ is,therefore,systematically adjusted until Eq. (15) is satisfied within 10⁻⁷. The resulting differential equations can be integrated by the Runge-Kutta fourth-order integration scheme. The above procedure is repeated until we get the results up to the desired degree of accuracy,10⁻⁵.

The validation of the shooting method is presented in Table 1 by comparing the skinfriction coefficient −f''(0) and γ for different values of the unsteadiness parameter with the earlier problems of Wang^[2] and Abel et al.^[21]. Note that Wang^[2] has used different similarity transformations due to which the value of in his paper is the same as f''(0) of our results.

Table 1 Comparison of γ and f''(0) with published results when λ = 0 and β → ∞ for various values of S

Table options

From this comparison,we can conclude that the shooting method is valid to present the numerical results which are in excellent agreement with the earlier published papers. Also,the obtained results demonstrate reliability and efficiency of the proposed method.

In order to analyze the theoretical concept of the physical model,numerical computations are carried out for several sets of values of the physical parameters,namely,the unsteady parameter S,the Casson parameter β,the space index parameter r,the time index parameter m,the Prandtl number P r,the velocity slip parameter λ,and the Eckert number Ec. The dimensionless velocity profiles for selected values of unsteadiness parameter S are plotted in Fig. 2. From this figure,it can be seen that increasing the unsteadiness parameter leads to a decrease in the thin film thickness but it causes a rise in both the flow velocity inside the thin film layer and the free surface velocity.

Fig. 2 Behavior of velocity distribution for various values of S

Figure options

Likewise,due to the presence of heat flux along the sheet,it is found that the temperature distribution along the thin film layer and the wall temperature θ(0) decrease with an increase in the same parameter as we can see from Fig. 3,but the reverse is true for the free surface temperature θ(γ). This shows the important fact that the rate of cooling is much faster for higher values of the unsteadiness parameter whereas it may take a longer cooling time for smaller values of the unsteadiness parameter.

Fig. 3 Behavior of temperature distribution for various values of S

Figure options

Figure 4 illustrates the velocity profile with two different values of the unsteadiness parameter against the similarity variable η for various values of the Casson parameter β. It is evident from this plot that both the film thickness γ and the free surface velocity decrease monotonically when the Casson parameter β is increased for S = 0.8 and S = 1.2. In addition,this figure depicts that increasing the Casson parameter leads to an increase in the velocity distribution inside the thin layer along the sheet,but the reverse is true away from the sheet. Physically,with an increase in the non-Newtonian Casson parameter,the fluid yield stress decreases which causes a production for the resistance force which makes the fluid velocity decrease. The temperature profiles for different values of the Casson parameter with the effect of variable heat flux are presented in Fig. 5. This figure reveals that both the temperature in the thermal thin layer and the wall temperature θ(0) increase for increasing the Casson parameter. Moreover,the thermal thin layer thickness is found to decrease with increasing the same parameter.

Fig. 4 Behavior of velocity distribution for various values of β with λ = 0.2

Figure options

Fig. 5 Behavior of temperature distribution for various values of β

Figure options

To see the effects of the slip velocity parameter λ on the velocity and temperature profiles,Figs. 6-7 are drawn. Figure 6 gives the variation of λ on the velocity profile. The velocity distribution decreases as λ increases,but the free surface velocity increases by increasing the same parameter. Physically,when slip occurs,the slipping fluid decreases the surface skinfriction between the fluid and the stretching sheet. Therefore,increasing the value of λ will decrease the flow velocity in the region of the thin layer.

Fig. 6 Behavior of velocity distribution for various values of λ

Figure options

Fig. 7 Behavior of temperature distribution for various values of λ

Figure options

The dimensionless temperature distribution within the thin layer region for various values of the slip velocity parameter is illustrated in Fig. 7. From this figure,we can observe that an increase in the slip velocity parameter may result in an enhancement in the wall temperature θ(0),the fluid temperature distribution,and the free surface temperature θ(γ) in the thin layer.

The effect of the Prandtl number Pr on the temperature distribution is demonstrated in Fig. 8 for S = 0.8 and S = 1.2. It can be found that the temperature profiles decrease for increasing values of the Prandtl number. Also,the temperature profiles for small Prandtl numbers show a stronger wall temperature θ(0) and a stronger free surface temperature θ(γ) than those for large Prandtl numbers. Physically,this is due to the fact that an increase in the Prandtl number decreases the temperature distribution along the thin layer because higher values of the Prandtl number correspond to the weaker thermal diffusivity.

Fig. 8 Behavior of temperature distribution for various values of Pr

Figure options

Figure 9 is obtained by plotting the temperature distribution against the variable η for different values of the Eckert number Ec. From this graph,it is clear that the temperature distribution increases with the increase in the value of the Eckert number. Physically,this behavior is observed because in the presence of viscous dissipation,heat energy is stored in the fluid,and there is more significant generation of heat along the sheet. Also,it is evident from this figure that the effect of the viscous dissipation parameter is to enhance both the wall temperature θ(0) and the free surface temperature θ(γ) in the thermal thin layer. Thus,viscous dissipation in a flow due to a stretching sheet is beneficial for gaining temperature.

Fig. 9 Behavior of temperature distribution for various values of Ec

Figure options

Figures 10-11 illustrate that how profiles of temperature distribution are affected by the variations in the space index parameter r or the time index parameter m when other parameters remain fixed. These figures indicate that the dimensionless temperature profile turns depressed for the increasing values of the space index parameter or the time index parameter. Likewise,it is shown that the effect of these index parameters causes a drop in both the wall temperature θ(0) and the free surface temperature θ(γ) for the thin film flow.

Fig. 10 Behavior of temperature distribution for various values of r

Figure options

Fig. 11 Behavior of temperature distribution for various values of m

Figure options

At this stage,we present Table 2 in order to more fully characterize the behavior of the quantities of relevant physical interest like the local skin-friction coefficient and the local Nusselt number with changes in the unsteady parameter S,the Casson parameter β,the space index parameter r,the time index parameter m,the Prandtl number P r,the velocity slip parameter λ,and the Eckert number Ec. One can then see from Table 2 that the local skin-friction coefficient decreases by increasing both the unsteadiness parameter and the Casson parameter,whereas the local Nusselt number increases with the increasing values of the same parameters. As it is observed that both the local skin-friction coefficient and the local Nusselt number decrease with increasing the slip parameter λ. Likewise,as expected,an increase in the Prandtl number causes an increase in the local Nusselt number. This is because that the fluid with a higher value of the Prandtl number possesses a larger heat capacity and hence intensifies heat transfer. Moreover,it is noticed that the increase in the value of the Eckert number leads to a decrease in the local Nusselt number. This is because the viscous dissipation is a source for gaining the temperature at the surface of sheet,thereby decreasing the heat transfer at the sheet. Finally,the local Nusselt number increases as both the space index parameter and the time index parameter increase with all other parameters fixed.

Table 2 Variation of f''(0) and for various values of S, β, λ, P r, Ec, r, and m

Table options

The present study provides the numerical solutions for the thin film flow and heat transfer of a Casson fluid over an unsteady stretching sheet with slip effect,viscous dissipation,and variable heat flux. The governing partial differential equations with the boundary conditions are reduced to ordinary differential equations by a suitable similarity transformation with appropriate boundary conditions. These equations are solved numerically via the shooting technique involving the Runge-Kutta integration scheme together with the Newton-Raphson method. The accurate numerical results are obtained and compared with the previously published papers,and the results are found to be in good agreement. It is found that the effect of increasing values of the unsteadiness parameter or the Casson parameter causes a decrease in both the film thickness and the skin-friction coefficient. However,the local Nusselt number is enhanced by increasing the same parameters. Moreover,it is obtained that the thin film thickness,the skin-friction coefficient,and the local Nusselt number decrease as the slip velocity parameter increases. Finally,it is interesting to find that as the space index parameter and the time index parameter increase in magnitude,the local Nusselt number increases.

Acknowledgements The author wishes to express his sincere thanks to the editor and referees for their valuable comments and suggestions,which lead to a significant improvement of the paper.