Irreversibility analysis in Marangoni forced convection flow of second grade fluid

Marangoni forced convective MHD flow of second grade liquid is scrutinized. Heat source/sink, Joule heating and dissipation are addressed in energy equation. Physical aspects of entropy optimization with binary chemical reaction are addressed. Energy and entropy expressions are computed. Marangoni convection influenced on the surface pressure difference is calculated through temperature gradient, magnetic field and concentration gradient. Nonlinear PDE’s are reduced to ordinary one through suitable variables. Nonlinear system is computed for convergent solution by employing of OHAM. Characteristics of different influential parameters on entropy generation, concentration, temperature, Bejan number and velocity are graphically deliberated. Velocity enhances via Marangoni ratio parameter. Velocity and temperature have reverse effects for higher approximation of magnetic variable. For higher second grade fluid parameter the velocity is augmented. An increment occurs in temperature against higher values of Brinkman number and fluid parameter. Concentration decrease versus higher Marangoni ratio parameter. Entropy optimization upsurges for rising values of fluid parameters. Some relevant applications of Marangoni convection effect include atomic reactor, semiconductor processing, thin-film stretching, silicon wafers, soap films, material sciences, nanotechnology and applied physics etc. Entropy supports to progress the importance of numerous engineering and electronic devices development.


Introduction
Improvement of Marangoni forced convection is usually the dissipative boundary layer between two phase fluid flow like gas-liquid and liquid-liquid boundaries. Mass transportation along an interface between two liquids due to surface tension gradient is called as Gibbs-Marangoni effect (Marangoni effect). On the other hand if there is thermal dependence case, then the phenomenon called Bénard-Marangoni convection (thermocapillary convection). Marangoni convection depends upon the difference of surface pressure computed by gradient of temperature, magnetic effect and concentration gradients. Some important applications of Marangoni convection effect like atomic reactor, semiconductor processing, thin-film stretching, silicon wafers, soap films, material sciences, nanotechnology and applied physics etc. Marangoni convection is extensively used in the coloring on the ground, for instance, fine art mechanism. The most important manufacturing applications of the Marangoni convection concept are melting and welding processes. Basic concept of mass and heat transportation phenomenon in Marangoni boundary layer flow are comprehensively discussed. Impact of Brownian movement and thermophoresis effect in viscous liquid subject to Marangoni forced convection is highlighted by Sheikholeslami and Chamkha [1]. Rassol et al [2] scrutinized the Marangoni convection in MHD flow of second grade nanoliquid. Behavior of Marangoni forced convection in MHD Casson liquid flow is studied by Mahanthesh et al [3]. Hayat et al [4] scrutinized Marangoni convection impact in water based CNTs by heated impermeable stretchable surface. Characteristics of space dependent heat source/sink in MHD

Statement of problem
Marangoni forced convection magnetohydrodynamic flow of second grade liquid is addressed. Flow is generated due to concentration and temperature gradients. Marangoni influence is exploited to apprehend the flow of liquid in forward direction. Joule heating, heat generation/absorption and dissipation in energy equation are discussed. First order chemical reaction is present. Salient effects of entropy rate and binary chemical reaction are examined. Let x-direction is along surface and y-axis being normal to sheet. Magnetic field (B 0 ) is exerted normal to the sheet. Temperature and concentration both as taken as the functions of x.. The flow geometry is highlighted in figure 1.
Governing equations satisfy [1][2][3][4]: In above expression u and v represent the velocity components, m the dynamic viscosity, r the density, a the second grade liquid parameter, s 1 the electrical conductivity, s the surface tension, a the thermal conductivity, T the temperature, c p the specific heat, ¥ T the ambient temperature, L the reference length, Q 0 the heat generation/absorption coefficient, T 0 the wall temperature, D B the mass diffusivity, C the concentration, ¥ C the ambient concentration, k r the reaction rate and C 0 the wall concentration.
Let surface tension s ( ) as a linear function of temperature and concentration [1,2]: where g , T s 0 and g C show the positive constants. Considering

Entropy modeling
Entropy production is produced because of heat fluxes, friction between solid surfaces, Joule heating, diffusion, mass fluxes, dissipation and Joule-Thomson effect etc. Here entropy is produced due to heat and mass irreversibilities, liquid friction irreversibility and Joule heating irreversibility [13][14][15] i.e.

Bejan number
It is the ratio of heat and mass irreversibility to total irreversibility.

Bejan number
Entropy rate due to heat and mass transfer Total entropy ,

Definitions of operators are
The problem for mth order are

Convergence analysis
The average square residual errors as given by Liao [39][40][41][42][43][44] satisfy: in which e m t denotes total squared residual error. Total squared residual error is indicated in figure 1. Individual averaged squared residual errors versus the convergence control parameters are given in table 1.

Discussion
Here we implemented optimal homotopic analysis technique to get the convergence series solution for nonlinear system. Prominent behavior of influential parameters for velocity, Bejan number, concentration, entropy rate and temperature are analyzed.   Figure 5 displayed the effect of Marangoni ratio variable (M a ) on velocity. One can find that velocity boosts against Marangoni ratio parameter. ( ) Temperature is increased versus higher Marangoni ratio parameter. Characteristics of (b) on q h ( ) are portrayed in figure 7. Here temperature upsurges against larger (b). Impact of (M ) on q h ( ) is shown in figure 8. Clearly higher magnetic parameter yields more Lorentz forces which improve resistance to liquid flow and therefore temperature boosts. Figure 9 elucidated impact of (Q) on q h .

Temperature
( ( )) Clearly temperature augmented against (Q). Effect of (Br) on q h ( ( )) is displayed in figure 10. Clearly temperature is increased against (Br). It is because of higher Brinkman number corresponds to slow the heat produced by dissipation effect and hence temperature upsurges.

Concentration
Influence of Marangoni ratio parameter on f h ( ) is illustrated in figure 11. For higher M a ( ) the concentration decays. It is due to the surface tension created by concentration and temperature gradients. Variation of Sc ( ) on  concentration is displayed in figure 12. Here mass diffusivity decays for higher Sc and consequently reduction occurs in concentration. Figure 13 examines the behavior of (g ) on f h .
( ) Clearly for higher (g ) the liquid behaves thicker and thus concentration decays.     improves the disorderness of the thermal system and therefore entropy generation rises. Be decays versus higher Brinkman number. We clearly noticed that the viscous effect is dominant over thermal irreversibility. Influence of (b) on Be and N G is highlighted in figures 16, 17. Clearly for higher approximation of (b) N G is augmented.

Conclusions
Marangoni forced convection magnetohydrodynamic flow of second grade liquid is addressed. Flow is generated due to concentration and temperature gradients. Entropy generation is developed through second law of thermodynamic. A physical feature of irreversibility exploration is examined. Marangoni influence is   exploited to examine the flow of liquid in forward direction. Joule heating, heat generation/absorption and dissipation in energy equation are discussed. First order chemical reaction is present. The key result are given below.
• Velocity has opposite effect against M ( ) and M . a ( ) • For higher M a ( ) the temperature is enhanced.
• Similar impact of temperature is observed via b ( ) and Br . The current effort is basic and modeling about such problem can be done for third grade fluid model subject to diffusion-thermo and thermal-diffusion effects, dissipation and Joule heating effects with rheological characteristics and stretchable phenomenon with porous medium through modified Darcy's laws.