Hall Current and Joule Heating Effects on Flow of Couple Stress Fluid with Entropy Generation

In this work, an analytical study of the effects of Hall current and Joule heating on the entropy generation rate of couple stress fluid is performed. It is assumed that the applied pressure gradient induces fluid motion. At constant velocity, hot fluid is injected at the lower wall and sucked off at the upper wall. The obtained equations governing the flow are transformed to dimensionless form and the resulting nonlinear coupled boundary value problems for velocity and temperature profiles are solved by Adomian decomposition method. Analytical expressions for fluid velocity and temperature are used to obtain the entropy generation and the irreversibility ratio. The effects of Hall current, Joule heating, suction/injection and magnetic field parameters are presented and discussed through graphs. It is found that Hall current enhances both primary and secondary velocities and entropy generation. It is also interesting that Joule heating raises fluid temperature and encourages entropy production. On the other hand Hartman number inhibits fluid motion while increase in suction/injection parameter leads to a shift in flow symmetry. Keywords-Hall current; Joule heating; entropy generation; couple stress fluid; Adomian decomposition method


INTRODUCTION
The study of hydromagnetic flow has been extensively investigated in the past years due to its applications in MHD generators, flow control, shock damping in car absorbers, nuclear reactors, plasma studies, purifications of metal from non-metal enclosures, geothermal energy extractions, polymer technology and metallurgy.Relevant investigation was pioneered in the first half of the twentieth century.Thereafter several studies have been conducted.Author in [1] investigated the spontaneous magnetic field in a conducting liquid in turbulent motion.In [2], magnetohydrodynamics at high Hartmann numbers were investigated.Author in [3] considered the effect of a uniform magnetic field on the Eckman layer over an infinite horizontal plate at rest relative to an electrically conducting liquid rotating with uniform angular velocity about a vertical axis while authors in [4] analyzed the combined effect of free and forced convection on MHD flow in a rotating porous channel, authors in [5] investigated the radiation effect of magnetohydrodynamics flow of gas between concentric spheres.In [6], authors considered the radiative effect on velocity, magnetic and temperature fields of a magnetohydrodynamic oscillatory flow past a limiting surface with variable suction In [7], author considered the hydromagnetic natural convection flow between vertical parallel plates with time-periodic boundary conditions, Authors in [8] studied convection heat and mass transfer in a hydromagnetic flow of second grade fluid in the presence of thermal radiation and thermal diffusion.In the studies above where the effect of magnetic field is reported, small and moderate values of the magnetic field are assumed.However, the current trend of research is geared toward a strong magnetic field and a low density gas due to its numerous applications such as in space flight, nuclear fusion research, magnetohydrodynamic generators, refrigeration coils, electric transformers, Hall accelerators and biomedical engineering (e.g.cardiac MRI and ECG).Hall current occurs when the applied magnetic field is very strong or gas is ionized with low density leading to a reduction in conductivity normal to the magnetic field, as a result of the free spiraling of electrons and ions around the magnetic lines of force before collisions.This then induces a current in the direction of both electric and magnetic fields.This is referred to as Hall effect, the induced current is called Hall current [9][10][11][12][13][14][15].
In this study, the Hall current and Joule heating effects on the flow of couple stress fluid with entropy generation is considered.The study is essential due to the fact that entropy generation occurs in moving fluid with high temperature which can lead to loss of resources and effort if the inherent irreversibility in the fluid flow is not well addressed.Authors in  [16] submitted that all processes that produce, convert and consume energy must be re-examined very carefully and all available-work destruction mechanisms must be removed.To the best of our knowledge similar study has not yet been reported in literature, although various factors responsible for the entropy production have been studied, for instance in [17] authors considered effects of velocity slip and temperature jump on the entropy generation in MHD flow over a porous rotating disk.Also, authors in [18] presented entropy generation on MHD nanofluid blood flow due to peristaltic waves.Recently, authors in [19] studied the thermodynamic analysis of hydromagnetic third grade fluid flow through a channel filled with porous medium.In [20], the effect of thermal radiation on the entropy generation of hydromagnetic flow through porous channel was investigated.More studies on the factors responsible for entropy production are reported in [21][22][23][24][25][26].Several techniques such as Homotopy perturbation [27], differential transform method [28][29], variational iteration technique [30], finite difference technique [31] etc. are available in literature.However, the Adomian decomposition method is applied in this work due to its simplicity in application and rapid convergence, (see Tables I and II).Furthermore, the method has been used to analyze various linear and nonlinear problems such as the fractional-order differential equations [32], the time dependent Edem-Fowler type equation [33], the Navier-Stokes equations [34], the evolution model [35], the Flierl-Petviashivili equation [36], the fourth-order wave equation [37], the peristaltic transport model [38], the Fokker-Planck equation [39] and the Bratu's problem [40].

II. PROBLEM FORMULATION
Fully developed, steady, incompressible and electrically conducting couple stress fluid between two parallel plates of distance h apart has been investigated.The coordinate system is taken such that the x-axis is along the lower plate in the flow direction, the y-axis is normal to the xy-plane while z-axis is made perpendicular to the plates.A constant pressure gradient is induced along the x-direction.Hot fluid is injected into the channel wall at the lower plate and sucked off at the upper plate with the same velocity.Following [41], the generalized Ohm's law with Hall current is It is further assumed that if (j x , j y , j z ) are the components of the current density J, the equations of conservation of electric charge 0 J   shows that j z is constant which is assumed to be zero because j z =0 at the plates which are electrically nonconducting.It then implies that j z =0, everywhere in the flow.Furthermore, the electrical field E=0 [42].Following the given assumptions, (1) becomes: Note that e e m    represents the Hall parameter.Solving (2) and (3) for j x and j y gives: The governing equations for the flow following [43][44] are: Momentum equation along axis x: Momentum equation along axis y: Energy equation: Introducing the following dimensionless variables, Equations ( 6), (7) yield the following dimensionless form; (0) 0, (1) 1. where , .
III. ADOMIAN DECOMPOSITION   The Adomian decomposition method is applied by writing (10), (11) and (12) in integral form as: and   Using the partial sum in (17) and the Adomian polynomials for the non-linear terms in ( 18), ( 14)-( 16) yield ( 19)- (21).The final stage for the implementation of ADM is the coding of equations ( 19)-( 21) in symbolic Mathematica software, which yields a very large symbolic solution.
   

A. Results Verification
The approximate solution obtained by ADM can be verified by comparing our result and the one in [44] as displayed in Table I.

B. Entropy Generation Analysis
The local entropy generation expression [35] for the flow is given as Using (9) Entropy generation within the flow can be analysed by letting   If Bejan number (Be) is less than one-half, irreversibility due to viscous dissipation dominates entropy generation and when (Be) is greater than one-half irreversibility due to heat transfer dominates the fluid flow.Bejan number (Be) equal to one-half indicates that both contribute equally to entropy generation.The Bejan number can be written as: Fig Br (J) vel y i the fro inc eff ma dem tre suc shi m=0.5, Jh=2, nt (m), Joule he (M) paramete city u and w a a) and 1(b) illu ocity.It is app anced as hall c current reduc osed by the ap res 2

Figure 5
Figure 5 dem rease in the J ect of varying mperature.It i gnetic field in ating which e gures 7-9 rep ating and su neration.Entro ll current and gures 7-8.Joul d Hartman num rresponding ris suction/injecti ile opposite enomenon is orted in Figur Fig.

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