1 Introduction

Fluid thin films can be seen in a variety of situations in nature and in various industrial applications, such as membrane in biophysics, tear films in eyes, coating flows, microfluidic engineering, etc. The ubiquitous presence of thin films in nature and technology has led the scientists and researchers to address the mechanism involved in the flow. Despite the diversity of phenomena and applications, the mathematical model is quite similar if the fluid is sufficiently viscous. Reynolds ^[1] was the first one who pointed out the need to model the dynamics of thin film flow, while studying the lubrication theory. Oron et al. ^[2], in their review article, presented a unified mathematical theory for the macroscopic thin fluid films and discussed linear stability of the solution, considering different aspects of the problem. O'Brien and Schwartz ^[3] found that the effects of surface tension on such flows are more pronounced for Newtonian coating when compared with shear thinning fluid. Thiele et al. ^[4] presented the analysis of the way in which thin fluid film evolves when the film was flowing down an inclined porous wall and discussed stability of the evolution equation and found that the substrate porosity tries to destabilize the flow. Siddiqui et al. ^[5] analyzed the Sisko fluid thin film flowing over a flat plate and observed that the inverse capillary number has the ability to increase the surface tension as well as the velocity of Sisko fluid.

Flow of liquid thin films over stretching surfaces has garnered a lot of attention in recent years because of their application in various industrial and engineering processes. Food stuff processing, designing of several heat exchangers, polymer processing, fiber and wire coating, large scale integration (LSI) and very large scale integration (VLSI) of microchips are some of the examples where thin liquid films over a stretching surface are apparent. Inspired by applications of such flows, a number of researchers devoted their time in analyzing the behavior of liquid thin films over a stretching surface, considering various characteristics of the problem. Some important research studies on the topic are due to Wang ^[6], Wang and Pop ^[7], Dandapat et al. ^[8], Liu and Anderson ^[9], Santra and Dandapat ^[10], Liu and Megahed ^[11] and Lin et al. ^[12].

In nuclear fusion power research, scientists use a device, known as divertor, which allows the removal of waste material from the plasma while the reactor is still operating. It has been observed that the thin liquid metal film, flowing fast enough may be a good option as a divertor surface for surface heat removal, under hydromagnetic effects. Inspired by such an application, Narula et al. ^[13] conducted an experiment to examine the effects of magnetic field and its temporal and spatial gradients on the flow of thin liquid metal film. Hayat et al. ^[14] observed in his study that the change in fluid velocity occurs with change in magnetic field at the end of thin film whereas the magnetic field has almost no effect in central region of the film. Nadeem and Awais ^[15] inspected the influence of variable viscosity on the flow of thin liquid film over a shrinking sheet under the influence of magnetic field. They pointed out that an increase in the film thickness results in the acceleration of film velocity but it causes the reduction in the film temperature. Dandapat et al. ^[16] concluded that the uniform film profile can only be obtained as a result of linear stretching, irrespective of the form of initial deposition. Das et al. ^[17] considered the thermocapillarity and found that the fluid temperature, skin friction and rate of heat transfer at sheet, all are adversely affected by thermocapillarity.

The consideration of hydrodynamic no-slip condition at the boundary, in a close neighborhood of a moving contact surface may result in a substandard model. For instance, many industrial and scientific processes involve the impinging droplets such as inkjet printing, spray cooling and fuel injection. Furthermore, the infringement of no-slip boundary condition increases the complication in the modelling of such flow processes. To alleviate this problem, the contact line sometimes is allowed to move along ^[18]. In the literature, researchers have proposed a number of ways to move the contact line. Of all the approaches, the most suitable one was proposed by Dussan ^[19]. In his study, fluid was allowed to slip along the contact line, which significantly reduced the singularity at the contact line and therefore relative velocity of fluid and solid will not be zero. Sajid et el. ^[20] in their study permitted the fluid to slip along the contact line and observed an escalation in the fluid velocity when it was tested against the slip parameter. Mukhopadhyay ^[21] found the numerical solutions for steady hydromagnetic radiative heat transfer flow past a sheet, considering velocity as well as thermal slip at the boundary. She observed that velocity slip tries to suppress the fluid velocity and thermal slip tends to reduce the fluid temperature. Megahed ^[22] observed the velocity slip effects on thin film flow of a Casson fluid past an unsteady stretching sheet and observed that velocity slip parameter tries to reduce the thickness of thin film. Ibrahim and Makinde ^[23] analyzed the hydromagnetic Casson fluid stagnation point nanofluid flow past a sheet and observed that thicknesses of velocity and thermal boundary layers increase with increasing slip parameter.

The viscous and Joule dissipations are phenomena of practical importance in many engineering devices. The distribution of temperature is significantly altered by the effect of viscous dissipation. Therefore, the consideration of viscous dissipation effect is of utmost importance in heat transfer analysis, when the fluid is sufficiently viscous, as considered in the present paper. There is another mechanism by which the temperature distribution of an electrically conducting fluid gets altered, which may be summarized as "due to flow of an electrical current in a finitely conducting medium, electrical power is generated in the system which gets dissipated in form of heat". There are various situations where we can use this fact to our advantage such as electrical irons and immersion heaters. Nevertheless, sometimes this phenomenon becomes a burden for us, where this unnecessarily generated heat may result in the downgrading of the system. Thus viscous and Joule dissipation effects serve as a volumetric heat source. Also, because of generation of heat in the moving fluid, the analysis of heat transfer becomes important in various physical situations such as the physical problems dealing with dissociating fluids in study of chemically reacting fluids. Also, as there is a substantial difference between the fluid temperature at the sheet and ambient fluid, it becomes necessary to study the influence of temperature dependent heat source/sink. Thus, a lot of research have been carried out in the past to analyze the heat transfer process in view of the effects of viscous and Joule dissipations together with temperature dependent heat source/sink. Some of the relevant analysis on the subject are due to Yih ^[24], Abo-Eldahab and El Aziz ^[25], Alam et al. ^[26], Mahmoud ^[27], Jaber ^[28] and Rundora and Makinde ^[29].

Most of the above-mentioned studies have ignored the combined effects of viscous and Joule dissipations on the flows of magnetohydrodynamic fluid thin film. Also, such kind of flows through porous media, have not been analyzed using Darcy-Forchheimer model for flows in porous media which accounts for inertia effects caused by porous medium and pressure drop disbursed by liquid–solid contacts which dominates the viscous hindrance. Thus, the intention behind the present investigation is to understand the magnetohydrodynamic flow behavior of heat generating/absorbing Casson fluid thin film over an unsteady stretching sheet with combined viscous and Joule dissipation effects in a Darcy-Forchheimer environment, when the film velocity is assumed to slip along the boundary. Although for such flows, the richness of ideas and phenomena discussed in the proposed study can be expected to lead to highly productive interactions across disciplines.

2 Mathematical analysis of problem

Consider the hydromagnetic flow of heat generating/absorbing Casson fluid thin film through a non-Darcy porous medium, past an elastic sheet, which is being stretched unsteadily, emerging through a fine slit at the origin as depicted in Fig. 1. The sheet is lying over the x-axis and the y-axis is taken in a direction perpendicular from the sheet. The flow region is permeated by a uniform magnetic field B₀ applied, in a transverse direction (i.e., parallel to y-axis). Furthermore, the effects of viscous and Joule heating are also taken into consideration. It is presumed that the flow is prompted due to elongation of the sheet by imposing a force, applied at one of the edges of the sheet in such a way that the sheet velocity is time dependent and varies linearly with the distance. The film thickness is assumed to be h(t). The sheet starts stretching with a velocity U(x, t) given as (Mahmoud and Megahed ^[30])

(1)

where α and c are constants (positive), having dimension of t^-1. It is to be observed that the sheet velocity given in (1) is valid only when αt < 1 unless α=0. The sheet temperature is supposed to vary with x in a fashion (Mahmoud and Megahed ^[30]), such as

(2)

where T_ref is the reference temperature, T₀ is the temperature of the slit, d is the arbitrary constant, v is the kinematic coefficient of viscosity, k is the thermal conductivity of fluid, r is the space index, and m is the time index.

Also, the heat flux at the surface is assumed to be varying with respect to distance and time, given by the following expression:

(3)

For an incompressible and isotropic Casson fluid, the rheological equation of state is given as (Makinde and Eegunjobi ^[31] and Benazir et al. ^[32])

(4)

where e_ij is the (i, j) th component of the stress tensor, φ = e_ije_ij is the product of deformation rate with itself, φ_c is the critical value of the φ based on the non-Newtonian model, μ_B is the plastic dynamic viscosity of non-Newtonian fluid and P_y is the yield stress of the fluid.

The fluid thin film is considered as non-volatile. Therefore, the influence of latent heat due to evaporation is ignored. The governing equations for fluid velocity and temperature field of the thin film, following the above mentioned behavior, are given as (see Liu et al. ^[33])

(5)

(6)

(7)

where is the Casson parameter, u, v, T, σ, ρ, c_p, K and C_b, are respectively, the fluid velocity in the x-direction, fluid velocity in the y-direction, fluid temperature, electrical conductivity, fluid density, specific heat at constant pressure, dynamic permeability of porous medium and the quadratic drag coefficient. In the energy equation (7), the term Q denotes the heat generation (for positive values) or absorption (for negative values), which is given as

(8)

where A^* is the temperature dependent heat generation/absorption coefficient.

It is worthwhile to observe that while writing the governing equations (5) to (7), induced magnetic field is ignored under the consideration of small magnetic Reynolds number. Since we are dealing with flow of a thin film over a sheet and thickness of the film, which in this case is characteristic length, is small and fluid velocity u, which evolve from the elongation of the sheet, is also small, thus the supposition of small magnetic Reynolds number is justified (Davidson ^[34]).

The conditions at the boundary are as follows:

(9)

(10)

where N₁= N(1 -αt)^1/2 is time dependent velocity slip factor with N being the initial velocity slip factor.

The Mathematical analysis of the flow of thin film can be made much simpler if we can convert the governing partial differential equations given by equations (5) to (7) along with the boundary conditions (9) and (10) into ordinary differential equations. To do so, following similarity transformations are introduced:

(11)

These similarity transforms are valid only when αt≪1.

Making use of (11), equations (6) and (7) get converted into following system of ordinary differential equations:

(12)

(13)

Transformed boundary conditions are obtained as follows:

(14)

(15)

(16)

where is local inertia parameter, M=σB₀²(1 -αt)/(ρc) is magnetic parameter, K₁=v²Re_x/(KU²) is permeability parameter, Re_x = Ux/v is local Reynolds number, Pr = ρvc_p/k is Prandtl number, S = α/c is unsteadiness parameter, λ = N(c/v)^1/2 is velocity slip parameter and E_c = U²/(c_p(T_s-T₀)) is the Eckert number. Here, γ denotes the value of similarity variable η at the free surface. Thus the boundary condition (10) gives

(17)

2.1 Wall velocity gradient and wall temperature gradient

For engineering and practical purposes, our main concern is the exploration of physical entities of the flow and heat transfer attributes by investigating the non-dimensional skin friction coefficient and local Nusselt number.

These dimensionless quantities are presented as follows:

(18)

(19)

2.2 Numerical implementation

Differential equations (12) and (13), along with boundary conditions (14), (15) and (16) are solved numerically by a shooting method along with 4th order Runge-Kutta technique. Differential equations (12) and (13) are converted into a set of 5 first order linear differential equations. For an initial guess of γ, this set is solved, subject to conditions at the boundary (14) and (15) until the outer boundary condition (16) is satisfied.

2.3 Validation of numerical solution

To judge the precision of numerical method, we have realized an assessment for the numerical values of skin friction coefficient -f^"(0) and γ with λ = M = E_c = F =K₁ = 0 and β→∞, against S with those of Abel et al. ^[35]. An excellent agreement is found between our result and that of Abel et al. ^[35], which is shown in Table 1. This guarantees that our numerical scheme is in confidence and it can be used for further computation of results.

3 Results and discussion

Extensive numerical computations are performed for velocity field and temperature distribution within the film boundary layer together with wall velocity gradient and wall temperature gradient, to get insight of the physics involved in the flow regime for several values of flow parameters which characterize the features of the flow. Numerical findings are well demonstrated in Figs. 2-16 along with Table 2. The default values of pertinent flow parameters are taken as

until otherwise specified particularly.

Figures 2 to 7 portray the behavior of fluid velocity f^' under the impact of unsteadiness parameter S, velocity slip parameter λ, magnetic field parameter M, permeability parameter K₁, local inertia parameter (Forchheimer number) F and Casson parameter β, respectively. We perceive from Fig. 2 that a rise in the value of unsteadiness parameter S leads to a decrement in the film thickness, but simultaneously, the velocity inside the thin film as well as the surface velocity increases on increasing S. Since S can be increased by increasing α, an increase in α results in an increase in the stretching velocity U(x, t). Thus on increasing S, U(x, t) is getting increased. Since the flow is solely induced due to elongation of the sheet, therefore, an increase in unsteadiness parameter results in increased fluid velocity. Figure 3 indicates that free surface velocity escalates on increasing λ whereas we observe an adverse effect of λ on fluid velocity inside the thin film and film thickness. Since an increase in slip parameter means an upsurge in the slip velocity, and an increase in slip velocity will obviously result in reduction of fluid velocity inside the thin film. We observe from Fig. 4 that an increase in M causes fluid velocity inside the thin film as well as the film thickness to decrease significantly. The reason behind such effect of magnetic field may be owed to the induction of a retarding body force, referred to as Lorentz force, due to the existence of magnetic field in an electrically conducting thin film. This force acts in a direction perpendicular to both fields. Since M suggests the ratio of hydromagnetic body force and viscous force, greater value of M indicates a stronger hydromagnetic body force which has a tendency to decelerate the fluid flow. Figure 5 reveals that, for an increase in the parameter K₁, the velocity is observed to decrease. As we can see from the expression of

a rise in the value of K₁ means a decrease in the permeability of porous medium. Consequently, less space is available for fluid to flow and hence, we observe a decrease in the velocity of thin film. Figure 6 elucidates that, on increasing local inertia parameter F, fluid velocity inside the thin film decreases whereas there is almost no effect of F on film thickness. There is barely an effect of F on free surface velocity, which is clear from Fig. 6. In case of porous spaces with bigger pore sizes, Forchheimer number accounts for the inertia effects due to porous medium and pressure drop disbursed by fluid-solid interaction which dominates the viscous hindrance. Thus, an increase in local inertia parameter F means a greater resistance to the flow, therefore, the fluid velocity is getting decreased. Figure 7 is plotted to analyze the effect of Casson parameter β on fluid film velocity. It is apparent from this figure that thickness of the film and free surface velocity increase on increasing Casson parameter β whereas fluid velocity inside the thin film decreases on increasing Casson parameter β in the region away from the sheet. The Casson parameter is expressed as

(20)

where μ_B is plastic dynamic viscosity, φ_c is the critical value of φ which is product of the deformation rate with itself and P_y is the yield stress of the fluid. For such fluid, once the initial shear threshold is reached, the fluid shows a linear shear stress behavior with respect to shear rate. It is clear from the expression that Casson parameter is directly proportional to plastic dynamic viscosity. Therefore, an increase in β means an increase in plastic dynamic viscosity μ_B which means a greater resistance to the flow of thin film and as a result, a downfall is observed in the fluid velocity on increasing the Casson parameter.

Figures 8 to 16 are plotted to analyze the behavior of fluid temperature θ against the parameters β, M, K₁, Pr, E_c, λ, S, r and m respectively. It is witnessed from Figs. 8 to 10 that with an increase in β, M and K₁, temperature of thin film increases throughout the film region. Since the action of Casson parameter, magnetic field and permeability parameter has led to the decrement in the velocity of the fluid in thin film region, therefore, extra work done in dragging the fluid against these three physical entities dissipates in the form of energy and hence increased fluid temperature is observed in the thermal boundary layer. It is evident from Fig. 11 that an increase in Pr compels the fluid temperature in the thin film region to decrease. Also, wall temperature is much higher for smaller values of Pr, i.e., thermal boundary layer thickness decreases on increasing the value of Prandtl number. This result confirms the actual nature of Prandtl number, because Prandtl number is a measure of relative importance of momentum and thermal diffusivities, smaller thermal diffusion corresponds to larger Prandtl number. Hence the results are in perfect agreement with physical meaning of Prandtl number. Figure 12 elucidates that fluid temperature is getting increased on increasing the Eckert number E_c which in fact supports the physics. Since Eckert number is the ratio of kinetic energy to enthalpy, an increase in E_c means the dissipated heat is stored in the liquid, which increases the fluid temperature. Figure 13 is drawn to understand the effect of slip parameter λ on fluid temperature. It is interesting to see here that an increase in the value of λ causes the fluid temperature to fall significantly whereas the thermal boundary layer is getting thicker with an increase in it. Figure 14 demonstrates the dependence of fluid temperature θ (η) on S. One can observe that an increase in unsteadiness in the stretching results in the decreased fluid temperature inside the thin film as well as free surface temperature whereas we obtain a thickened thermal boundary layer on increasing the value of S. The value of similarity variable at free surface, γ, given by (17) depends on α in such a way that an increase in α results in the increment in the value of γ. And an increase in the value of α results in increment in unsteadiness parameter. Thus an increase in unsteadiness parameter S thickens the thermal boundary layer. Figures 15 and 16 present the influences of space index r and time index m and the effects of these two indices are to decrease the fluid temperature.

In order to analyze the behavior of physical quantities of interest viz. local skin friction co-efficient and local Nusselt number Nu_xRe_x^-1/2, effects of pertinent flow parameters such as S, β, F, λ and E_c on these two quantities are computed and are presented in Table 2. One can observe from Table 2 that skin friction co-efficient is getting enhanced on increasing either of unsteadiness parameter, Casson parameter and Forchheimer parameter. On the other hand, local Nusselt number is getting enhanced on increasing only unsteadiness parameter and Casson parameter and decreases on increasing either of Forchheimer parameter, velocity slip parameter and Eckert number.

4 Conclusions

A mathematical model is established for magnetohydrodynamic non-Darcy flow of an incompressible, electrically conducting, viscous and heat absorbing/generating thin fluid film past a horizontal unsteady stretching sheet with non-uniform heat flux, taking Navier's velocity slip and Joule heating into consideration. Noteworthy results are summarized as follows:

(ⅰ) The Forchheimer parameter, which is responsible for inertial drag, reduces the fluid velocity inside the thin film. An intensification in magnetic field leads to a significant fall in fluid velocity and reduces the film thickness as well. An increase in the unsteadiness in the stretching of sheet causes fluid velocity inside thin film to increase. An enhancement in velocity slip tends to decrease the fluid velocity inside the film for obvious reasons. A greater resistance to the flow is offered on increasing the plastic dynamic viscosity by virtue of increasing the Casson parameter.

(ⅱ) As a result of increased resistance to the fluid movement due to increase in Casson and magnetic parameters, the fluid temperature is getting increased. Due to dissipation of heat, temperature is getting increased on increasing Eckert number. Fluid temperature is observed to be decreasing due to increase in Prandtl number, i.e., on decreasing thermal diffusivity. A decrease in fluid temperature is also witnessed on increasing either of space index, time index, slip parameter and unsteadiness parameter.

(ⅲ) Coefficient of skin friction is getting boosted on increasing either of unsteadiness parameter, Casson parameter or Forchheimer parameter whereas the velocity slip parameter does the vice-versa. On the other hand, local Nusselt number is perceived to rise on increasing unsteadiness and Casson parameters while the rest of the parameters, namely, Forchheimer parameter, velocity slip parameter and heat dissipation parameter have adverse effect on local Nusselt number.

Acknowledgements One of the authors, Mr. R. TRIPATHI is thankful to University Grants Commission of New Delhi in India, for providing funds to accomplish this research work.