A study on entropy generation on thin film flow over an unsteady stretching sheet under the influence of magnetic field , thermocapillarity , thermal radiation and internal heat generation / absorption

Entropy generation is inherently affiliated with transport of thermal energy. However, not much attention has been paid to the study of entropy generation in liquid thin film flows. Hence, in this paper an analysis is carried out to study the entropy generation in a thin viscous fluid film on a stretching sheet embedded in a porous medium subject to a magnetic field and thermocapillarity force taking thermal radiation and internal heating/absorption into account. The numerical solution of the equations of momentum and energy governing the flow is obtained. Influence of various parameters emerged in the analysis on the velocity, temperature, surface drag, Nusselt number, entropy generation number and the Bejan number are graphically illustrated and discussed. Thermocapillarity number shows an enhancement of velocity at the free surface. Film thickness is found to increase with increasing thermocapillary force. Thinner films are noticed for stronger magnetic field strengths. It is found that in the absence of thermocapillary force, the effect of magnetic field on entropy generation number near the stretching surface is prominent. Thermocapillarity is seen to have a stronger effect on the entropy generation number near the stretching surface as well as at the free surface.


Introduction
Study of flow and heat transfer in a thin liquid film on a stretched surface attracted the attention of several researchers owing to its numerous engineering and industrial applications.Knowledge of flow and heat transfer within a thin film is very much helpful in manufacturing processes such as wire and fibre coating, http://www.ispacs.com/journals/cna/2017/cna-00319/International Scientific Publications and Consulting Services aerodynamic extrusion of plastic sheets, annealing and thinning of copper wires, foodstuff processing, reactor fluidization etc.In processes of extrusion of materials it is essential to maintain surface quality of the extrudate.It is necessary in all coating processes to produce a smooth glossy surface for the best appearance with low friction, good translucency and strength.Since the quality of end product in extrusion process is determined through the fluid flow and heat transfer characteristics of a thin film over a stretching sheet, it is essential to know the behavior of flow and heat transfer in these technologies.Analysis of flow in a thin Newtonian liquid film over a stretched surface was pioneered by Wang [1].Andersson et al. [2] explored the characteristics of the flow in a thin power law fluid film resting on a stretched surface.Dandapat and Ray [3] deliberated the gradual developments of a thin film on the surface of a rotating disk under the action of thermocapillarity force and Lorentz force.Chen [4] obtained the numerical solution of thermal energy transfer and velocity of flow in a thin liquid film of a power law fluid due to an unsteady stretching sheet.Chen [5] investigated the effect of Newtonian heating on the unsteady convective heat transfer in a powerlaw liquid thin film over a surface with linear stretching.Abel and co-researchers [6] explored the influence of Lorentz force and frictional heating to describe the profiles of temperature and velocity of the flow in a thin film of a liquid on an unsteady stretching surface.Santra and Dandapat [7] discussed the development of a viscous thin liquid film on a surface stretching with non-linear velocities.Dandapat et al. [8] extended this study to examine the effect of Lorentz force.Noor and Hashim [9] examined the flow and heat transfer in a liquid film over an unsteady elastic stretched surface taking the magnetic and thermocapillarity forces into consideration.Aziz et al. [10] obtained HAM solutions to examine the velocity and temperature of the flow in a thin liquid film on an unsteady stretching sheet to assess the influence of internal heating by prescribing a general temperature on the surface.Vajravelu et al. [11] examined the effect of temperature dependent thermal conductivity and viscous heating on the unsteady flow and heat transfer in a thin film of power law liquid on a permeable surface with stretching.In a later study, the investigators [12] extended this analysis to assess the effect of variable thermo-physical properties on the flow and thermal energy in a thin film over a stretching sheet.Khader and Megahed [13] obtained numerical solutions for velocity and temperature using an implicit finite difference method to examine the effect of radiative heat energy transport on the flow of a liquid thin film resting on a stretching sheet through a permeable medium.In all the above studies the traditional no-slip condition on the boundary is assumed.However, there are fluids which exhibit wall slip, for example, emulsion, suspensions, foams and polymer solutions [14].Further, fluids with boundary slip are reported to have significant technological applications such as in polishing of artificial valves pertaining to heart (see for details Mukhopadhyay [15]).In view of these various applications, several problems of flow and heat transfer with wall slip have been studied.Megahed [16] obtained numerical solutions using HPM to the problem of flow in a thin viscous liquid film over a horizontal heated stretching surface accounting for the velocity slip at the surface and heat flux as a variable.In recent years the second law of thermo dynamics has been applied to analyze thermal systems for the minimization of entropy generation.Entropy generation estimates the level of irreversibility developed during a process.Thus entropy generation can be considered as one of the criteria to assess the performance of engineering devices [17].Bejan [18,19] observed that entropy generation for forced convective heat transfer in a channel is the result of temperature gradient and viscosity effect of the fluid.In a subsequent study Bejan [20] implemented the concept of minimum entropy generation to design a counter flow heat exchanger.San [21] examined the performance of a two-dimensional fixed-bed regenerator and obtained the non-dimensional cycle time that yields the highest second law efficiency for a regenerator.Hosseinali et al. [22] presented the forced convection and entropy generation of a  2  3 water nanofluid in a micro channel subject to the position-dependent magnetic field.They observed that the Lorentz force generates vortices near the magnetic sources and the number and strength of vortices depend strongly on Hartmann number.Martin et al. [23] analyzed the impact of frictional heating on entropy generation in a MHD mixed convection flow of a nanofluid over a non-linear stretching sheet.They observed http://www.ispacs.com/journals/cna/2017/cna-00319/International Scientific Publications and Consulting Services that the surface serves as a strong source of irreversibility due to higher entropy generation number near the surface.Despite numerous investigations on thin liquid film flows it seems that not much has been done in liquid thin films dealing with entropy generation associated with stretching surfaces.Since entropy generation has a significant effect on the thermodynamic performance of a system, knowledge of entropy generation in liquid thin films enables to predict rate of transport of thermal energy.In the present investigation entropy generation in a thin liquid film on an unsteady stretching sheet embedded in a porous medium in the presence of thermocapillarity, thermal radiation and uniform magnetic field is analyzed.In section 2 we present the formulation of the problem, the governing equations and the boundary conditions.Section 3 deals with the numerical scheme adopted to solve the governing equations.In section 4 a parametric analysis is carried out on the velocity and temperature distributions, entropy generation number and Bejan number.Finally in section 5 conclusions are presented.

Mathematical Formulation
Consider an unsteady laminar flow and heat transfer in a thin liquid film resting on a stretching surface embedded in a porous medium exposed to a variable transverse magnetic field.The physical model and the Cartesian coordinate system are shown in Figure 1.The surface aligned with the x-axis at  = 0 is stretched to move in its own plane with a velocity [24] (, ) = where  and  are positive constants with dimension reciprocal of time t.The temperature of the stretching surface   is taken as a function of x, the distance x along the sheet from the slit and time as [25]: where  0 is the temperature of the liquid at the slit,   is the constant reference temperature such that 0 ≤   ≤  0 , ∀ < 1  .
The system is exposed to a variable magnetic field (, ) =  0 /(1 − ) where u and v are the velocity components of liquid in x and ydirections, T is the temperature, t is time,  is the kinematic viscosity,  is the density,   is the specific heat at constant pressure,  =  0 (1 − ) ( 0 is the initial permeability) is the permeability of the porous medium,  is the thermal conductivity, σ * is the electrical conductivity, Q is the heat source/sink and   is the radiative heat flux.The term Q is the heat source (> 0) or sink(< 0) per unit volume which is modeled as [27] where  * is the temperature-dependent heat source/sink and is positive in the case of source of heat and negative in the case of heat sink.The radiative heat flux   by using Rosseland approximation can be written as (Raptis [28]) where   is the Stefen-Boltzman constant and  * is the absorption coefficient,  4 may be linearly expanded in a Taylor's series about  ∞ to get and neglecting higher order terms beyond the first degree in ( −  0 ), we obtain  4 ≅ 4 0 3  − 3 0 4 .
(2.8) The boundary conditions on the velocity and temperature distribution are given by where  is the dynamic viscosity,  1 = (1 − ) 1/2 is the velocity slip factor which changes with the time, N is the initial value of the velocity factor, ℎ is the uniform thickness of the liquid film and  is the surface tension.
The surface tension that varies linearly with temperature is given by [29] (2.10) In general  decreases with temperature for most liquids so that  > 0. It is assumed that pressure in the surrounding fluid phase is uniform and the gravity gives rise to a hydrostatic variation in the liquid film.On introducing the following similarity variables: where β is the dimensionless film thickness, defined as ( [30]) (2.12) The governing partial differential equations (2.4) and (2.5) can be reduced to a set of ordinary differential equations.(, )is the stream function which automatically satisfies the equation of continuity (2.3) and hence Using equations (2.11) and (2.13), the mathematical problem stated in equations (2.4) and (2.5) can be written as ) The associated boundary conditions are ) ,  ′′ (1) =  1 (1),  ′ (1) = 0 (2.17 where primes denote differentiation with respect to ,  =   ⁄ is the dimensionless measure of unsteadiness,  = σ *  0 2  ⁄ is the magnetic field parameter,  =   0 ⁄ is the porous parameter,  =     ⁄ is the Prandtl number,  = 16   0 3 3 * ⁄ is the thermal radiation parameter,  = (  ⁄ ) 1/2 is the velocity slip parameter and  1 =  0   √ ⁄ is the thermocapillarity number.The parameters in momentum and heat transfer problems with practical significance in engineering are the skin friction coefficient   and the local Nusselt number   which characterize surface drag and heat transfer rate respectively and are defined as ) where   = / is the local Reynolds number.

Entropy Generation
According to Bejan [31], the entropy generation per unit volume is defined as The first term in equation (2.20) is entropy generation due to heat transfer, second term is due to radiation, third term correspond to viscous dissipation and fourth term due to effect of magnetic field.The dimensionless form of the entropy generation is given by where,  0 = (  −  0 ) 2 / 2  0 2 is characteristic entropy generation rate, Ω = (  −  0 )/ 0 is dimensionless temperature ratio and  =  2 /(  −  0 ) is Brinkman number.The Bejan number (Be) is the pertinent irreversibility parameter and is defined as Entropy generation due to heat transfer Total entropy generation It is evident from (2.22) that the Bejan number takes values in the range from 0 to 1.  = 0 corresponds to the case when the irreversibility, due to fluid friction and magnetic field dominate over the heat transfer irreversibility, while  = 1 shows the domination of irreversibility due to heat transfer on entropy generation. = 1/2 amounts to case in which the irreversibility due to heat transfer is same as the sum of irreversibility due to fluid friction and magnetic fields.

Numerical Method
The model (2.14)-(2.17)consists of highly non-linear coupled ordinary differential equations.Exact solutions of the same are not possible.Literature survey suggests that such equations are solved either numerically or by using HAM.In the present case we adopt the efficient Runge -Kutta-Fehlberg method [32] along with shooting technique to obtain the numerical solutions of (2.14)-(2.17).The coupled ordinary differential equations (2.14) and (2.15) are reduced to a set of simultaneous first order equations as follows: .Hence, it is essential to iterate the value of  until it satisfies this condition with an error of tolerance of 10 −6 .
Table 1: Variation of the dimensionless film thickness and the skin friction with unsteady parameter when  =  1 =  =  =  =  * = 0 and Pr = 1.0.The accuracy of the numerical scheme employed in this analysis is ensured by comparing the present results, viz., the non dimensional thickness of the film , surface skin friction coefficient  ′′ (0) with the corresponding values evaluated by Wang [1] and Aziz et al. [10] in the absence of magnetic field parameter, thermocapillarity number, porous parameter, thermal radiation, velocity slip parameter and heat source/sink parameter for different values of unsteady parameter.These values are presented in Table 1 and it is seen that they are an excellent agreement.

Results and Discussion
To analyze the effect of physical parameters, namely, magnetic parameter, film width, thermocapillarity number, unsteady parameter, porous parameter, slip parameter, Prandtl number, thermal radiation parameter and heat source/sink parameter on flow characteristics, the results presented through graphs and tables are discussed in detail.Figure2 presents influence of magnetic field (M) on horizontal velocity and temperature.Presence of magnetic field shows a reduction in velocity in the momentum boundary layer up to the special point  ≈ 0.5657 where velocity attains a minimum and thereafter starts enhancing.Further increase in the magnetic field reduces velocity up to the special point as higher values of M offer more resistance in the fluid region due to Lorentz force and after crossing the special point velocity enhances attaining its maximum at the free surface.However, enhancement of velocity in the vicinity of free surface is solely due to the thermocapillarity force.Figure 3 reveals that for a fixed value of the unsteady parameter (S), transverse velocity () enhances monotonically from the surface and increasing values of S increase the transverse velocity for  ≥ 0.2 while horizontal velocity (′) increases throughout the boundary layer with significant increase at the free surface.Figure 4 presents the effect of thermocapillarity number on horizontal velocity.It is observed that for a given value of thermocapillarity number (  1 ), velocity from its prescribed value on the surface descends along the horizontal direction from stretching surface till  approaches a specific point where  ≈ 0.6465 attaining a minimum value and thereafter it gets reversed and ultimately reaches its higher value on the free surface.For increasing values of  1 , velocity decreases attaining its minimum value at this specific point and http://www.ispacs.com/journals/cna/2017/cna-00319/International Scientific Publications and Consulting Services subsequently it increases.This may be explained as follows.Thermocapillarity number is a measure of the variation of surface tension with temperature and this surface is allowed to cool along the flow direction, and surface tension is less at the surface and consequently thermocapillarity force acts as a tangential stress on the surface of the film along the favourable flow direction.In other words, flow in the thin film develops an outward flow as the fluid away from the surface consistently gets cooled.This is evident from Figure 5 that the temperature falls throughout the thin film with significant reduction in the temperature on the free surface for an increase in  1 .Figure 6 depicts the variation of thermal radiation parameter (Nr) and heat source/sink parameter ( * ) on temperature.Larger values of Nr reduce thin film flow along the stretching boundary with reversal behaviour near the free surface.Temperature rises considerably for increasing values of Nr throughout the film.Presence of heat sink cools the fluid due to reduction in temperature.When heat sink parameter assumes the value i.e.  * = −1.0,temperature on the free surface is decreased by 23.27% than that of the temperature in the absence of any heat source/sink.This reduction is due to the absorption of energy in the thin film.In the presence of heat source ( * = 1.0) temperature is increased by 48.56% due to the release of energy in the film.In Figure 7 influence of magnetic field on entropy generation number (  ) in the absence as well as presence of thermocapillarity force is presented.It is revealed that in the absence of thermocapillarity number, the entropy generation is stronger in the vicinity of the wall at which magnetic field causes more dissipation of energy than away from the boundary.Further, increasing values of magnetic field strength increase the entropy generation.This is due to the fact that higher values of magnetic parameter amounts to stronger Lorentz force which resists fluid motion and as a result, rate of heat transfer in the boundary layer enhances.In the presence of thermocapillarity number, it is seen that the values of entropy generation number are higher than those in the absence of thermocapillarity.In fact, in the absence of thermocapillarity number when magnetic field parameter assumes values  = 0, 1, 2 and 3, the values of   increase in the range 5.652 − 7.404, while presence of thermocapillarity leads   to vary from 7.405 to 9.171.Also it is observed that in the vicinity of free surface, energy dissipation occurs due to domination of thermocapillarity which is not seen in the absence of thermocapillarity.Thus we may conclude that thermocapillarity augments the dissipation energy as a source of irreversibility.Effect of thermal radiation parameter on entropy generation number as illustrated in Figure 8 reveals that heat transfer plays a significant role in the thermodynamic irreversibility than that of viscous dissipation.As the thermal radiation parameter increases, it is noticed that entropy generation number near the stretching surface decreases.It is evident from Figure 9 that higher values of Reynolds number   increase entropy generation number significantly on the stretching surface.When   increases from 2 to 20 there is a tenfold enhancement in the entropy generation number.Effect of group parameter (/Ω ) on   is similar to that of the Reynolds number (Figure 10).However, when group parameter number /Ω = 0.1,   enhances twice to that of the corresponding case when /Ω = 0.1.Bejan number (Be), which is the ratio of heat transfer entropy generation to the overall entropy generation, is another parameter to estimate the irreversibility distribution.From Figure 11 it is observed that in the absence of magnetic field, Bejan number has higher values in the vicinity of stretching surface than those of Be in the presence of magnetic field and away from the stretching surface, we observe that the magnetic field dominates giving higher values of Bejan number.Increase in magnetic field shows a reduction of the Bejan number near the stretching surface and in a small region 0.6 ≤  ≤ 0.7, the values of Bejan number are almost same and in the rest of the region a reduction in Bejan number is noticed.Figure 12 presents plots of Bejan number for different values of Nr.It is observed that at the stretching surface, Bejan number decreases with increasing values of Nr and a little away from the surface Be increases.This shows that fluid friction reversibility dominates near the surface and away from the surface heat transfer irreversibility becomes prominent.From Figure 13 it can be concluded that, for fixed value of the group http://www.ispacs.com/journals/cna/2017/cna-00319/International Scientific Publications and Consulting Services parameter, Bejan number increases from the stretching surface and attains its maximum value away from the surface.Increasing values of group parameter decrease the Bejan number.thermal radiation and heat source parameter leads to a decrease in the local Nusselt number.This is due to the fact that thermal radiation facilitates in raising the fluid temperature at surface of the sheet and thereby reduces heat transfer at the surface.

Conclusions
An analysis of boundary layer behaviour in a thin liquid film on an unsteady stretching sheet embedded in a porous medium is carried out.Results of the present work are validated with those available in the literature and are found to be in good agreement.It is observed that the magnetic field suppresses the horizontal velocity.Thermocapillarity number has a dual effect on the horizontal velocity by reducing it in a region enclosing the stretching surface and enhancing it in the other region enclosing free surface.Unsteady parameter and magnetic field parameter are observed to decrease the film thickness while thermal radiation parameter and thermocapillarity number enhance the film thickness.Free surface temperature is decreased by thermocapillarity number, heat sink parameter and Prandtl number while an enhancement is noticed with thermal radiation parameter and heat source parameter.Lorentz force has a stronger influence on entropy generation near the stretching surface.Thermocapillarity number increases entropy generation significantly at the stretching surface while at the free surface it is nominal.Bejan number is decreased near the surface with increasing values of magnetic field parameter.

Figure 11 :Figure 12 :
Figure 11: Variation of M on Be

Table 2
enables us to understand the behaviour of film thickness, free surface temperature and quantities of physical interest such as skin friction coefficient and local Nusselt number for different variations in the unsteadiness parameter (S), magnetic field parameter (M), porous parameter (D), Prandtl number (Pr), wall slip parameter (), thermal radiation parameter (Nr), heat source parameter ( * > 0) and the thermocapillarity number ( 1 ).It can be seen that the local skin friction coefficient increases by increasing unsteadiness parameter, magnetic field parameter and porous parameter whereas film thickness decreases with increasing values of the same parameters.It is noticed that both local skin friction coefficient and thin film thickness show an increasing trend with increasing slip parameter.The local skin friction coefficient is found to be a decreasing function of thermocapillarity number while film thickness is an increasing function of the same parameter.It is observed that an increase in Prandtl number results in an increase in the local Nusselt number while the free surface temperature decreases.As expected, an increase in the value of http://www.ispacs.com/journals/cna/2017/cna-00319/International Scientific Publications and Consulting Services [26]pplied in a direction normal to the stretching surface.The effect of internal heating and thermal radiation is taken into account.The boundary layer equations governing the flow and heat transfer are[26] International Scientific Publications and Consulting Services

Table 2 :
Values of the dimensionless film thickness , skin friction  ′′ (0), Free surface temperature (1) and the wall-temperature gradient − ′ (0) for different values of the physical parameters