ELECTRO-THERMO-CONVECTION FLOW OF CONDUCTING NON-NEWTONIAN FLUID OVER UPRIGHT WAVY CONE ALONGSIDE IRREGULAR ELECTRICAL CONDUCTIVITY

This work aims to investigate the numerical analysis of the electrohydrodynamic (EHD) effect using a Casson fluid model. The EHD flow for the laminar forced convection and heat transfer on a vertical wavy cone with variable electric conductivity in the existence of internal heat generation/absorption and Ohmic dissipation was simulated. The problem model was characterized by highly partial differential equations that were concluded via the approximation of the boundary layer. The governing model was then turned into ordinary paired differential equations that applied convenient similarity transformations. Thus, the resulted equations were numerically investigated using the Keller-box method. The impact of the different physical parameters on the velocity profiles and temperature distributions, and also on the shear stress and heat transfer rate, were presented. It was found that a decrease in the electric field parameter leads to heighten the velocity, local skin friction coefficient, and local rate of heat transfer whereas the temperature reduces. Further, rising the Eckert number was deduced to decrease the local rate of heat transfer whilst the velocity, temperature, and local skin friction coefficient were enhanced.


INTRODUCTION
In many engineering and manufacturing applications, roughened surfaces are commonly present. It is oftentimes developed to enhance heat transfer in heat transfer devices. The natural laminar convection flow over rough surfaces can be utilized for heat transfer in countless thermal transfer devices, for instance, flat plate condensers in refrigerators, heat exchanger, flat plate solar collectors, industrial heat radiators, geothermal reservoirs, and functional clothing design, etc. It is therefore that the investigation of the flow over the rough surface has interested many researchers in different physical situations [1][2][3][4][5][6][7][8][9][10].
It is well known that Electrohydrodynamics (EHD) is a cross-disciplinary field with important applications in industrial processes addressing the interaction among fluids, electric charges, and electrical fields [11,12]. The effects of an electromagnetic fields on convective heat transfer are of great interest since they induce currents in the fluid. the hydrodynamic and Maxwell's equations of electromagnetism are the equations that describe this category of flow which is of considerable prominence in the industrial, technological, and geothermal applications. Further research on this type of electromagnetic fluid flow numerically can be followed through the references [13] to [20]. Many engineering applications are available for the magnetic field, which affect the heat generation/absorption processes in the conduction of electrical fluid flows [21][22][23][24][25][26][27][28][29].
Casson fluid is a type of visco-plastic fluids that reveals yield stress and whose constitutive equation was described by Casson [30]. Insomuch as the shear stress is weaker than the yield stress, Casson 's liquid functions like a solid and behaves like a fluid when the shear stress is larger than the yield stress. It has considerable usages in polymer treatment industries and biomechanics. Casson fluid model examples jelly, juices, honey, tomato sauce, silicon suspension, human blood, polymeric liquids, and many others. Some relevant studies about this fluid model can be seen in references [31] up to recently [36].
To the authors' knowledge, the heat transfer problems of viscous flows alongside roughened surfaces have been extensively discussed by many authors. However, solutions are not yet presented for the Casson fluid flow in the existence of an electric field together with variable electric conductivity. Therefore, The goal of the current seeking is to investigate the problem of effects of electromagnetic field on forced convective heat transfer over a vertical wavy cone retained at an unvarying surface temperature flooded with a Casson fluid beside an irregular electric conductivity and take into consideration the internal heat generation/absorption and Ohmic heating (Joule electromagnetic dissipation) to explore the impact of the germane parameters, such as amplitude wavelength ratio, Casson fluid parameter, electrical field parameter, magnetic field parameter, Prandtl number, Eckert number and heat generation/absorption parameter on fluid velocity, temperature, local coefficient of skin friction, and local heat transfer rate. The governing nonlinear equations are constricted to ordinary differential equations via the fitting similarity transformations and then they are numerically solved applying the Keller-box implicit finite difference method.

FLOW ANALYSIS
The physical problem deals with the steady-state of an incompressible electrically conducive Casson fluid boundary layer flow near a wave cone whose transverse vertical sinusoidal surface.
The wavy surface cone is illustrated by where a is the wavy surface amplitude and 2L is the characteristic length associated with the uneven surface. The coordinate system is defined in such a way that x-axis runs from the apex to the cone's flat surface, and y-axis is perpendicularly measured outward as seen in Fig. 1. The surface of the wave cone is assumed to be at a uniform temperature T w and the regular ambient equations of the problem are now built based on previous assumptions as following (see [8]) where ρ is the fluid density, (u, v) are the components of the velocity in (x, y) respectively, is the local radius of the cone's flat surface, γ is the half-angle of the cone's flat surface, p is the pressure of fluid, µ is the viscosity of dynamic, α is the parameter of Casson fluid, 2 is the Laplacian operator, β is the volumetric expansion coefficient, g is the gravitational acceleration, T is the temperature, C p is the specific heat at invariable pressure, k is the fluid thermal conductivity and Q 0 is the dimensional heat generation (Q 0 > 0) or heat absorption (Q 0 < 0). Here σ is presumed to rely on the fluid velocity and to have the format The suitable boundary conditions can often be modeled as In order to dimensionless the system, we introduce the following transformations [34] where a, Gr are the amplitude wavelength ratio and Grashof number in term of the temperature, respectively. Now, on substituting the transformations (7) into Eqs. (2) to (5) and assuming that Gr −→ ∞ [3], the equations system can be converted as follows in equations of boundary layers and the boundary conditions (6) turned out to are the magnetic field parameter, electric field parameter, heat generation/absorption parameter, Prandtl number, and Eckert number, respectively.
From Eq. (10), it can be noticed that along y-direction the pressure gradient will be O(Gr −1/4 ), which means that an inviscid flow solution should be used to evaluate the lowermost order of pressure gradient along x-direction, which leads to ∂ p/∂ x = 0, for details see [6]. Excluding the terms of the pressure gradient in both Eqs. (9) and (10), thus it is reduced to a single equation Now, to constringe governing Eqs. (11) and (13) to an opportune form, the following transformations are announced herein ψ is the stream function while η is the pseudo-similarity variable that fulfills Eq. (8) and is realized by On the substitution Eq. (14) in Eqs. (11) and (13), we have In addition, the boundary conditions in Eq. (12) take the the next formula The important physical quantities are the local coefficient of skin friction, C f x , and also the local heat transfer rate in form of the local Nusselt number, Nu x , and they are defined respectively as (19) C in which U = µGr 1/2 /ρL is characteristic velocity, τ w and q w are the shear stress and heat flux at the wall which they are provided respectively by (20) .
Applying the non-dimensional quantities (7) and the non-similar transformation (14) on Eq.
(20), the local coefficient of skin friction, C f x , and also the local heat transfer rate, Nu x , can be evidenced by ,

NUMERICAL PROCEDURE
Local, not similar, partial differential Eqs. (16) and (17) are integrated along with the boundary conditions (18) by combining the implicit finite difference approximation upon Keller-box method described by Keller [38,39], Keller and Cebeci [40], Cebeci and Bradshaw [41], and Vajravelu and Prasad [42]. This method is widely used in solving the physical problems, see for example the work of Swalmeh et al. [43] and Ahmad et al. [44].
First, Eqs. (16), (17) and (18) are mutated to a set of a five-order system of equations accordingly these equations are formulated in finite-difference procedures with a second-Newton 's approach is used to linearize nonlinear algebraic equations resulting and has been represented in matrix-vector notation. Resolution of these equations and their boundary conditions through the Thomas algorithm for the block tridiagonal systems. The solution starts at x = 0, with the right step size ∆η, including interval 0 ≤ η ≤ η ∞ by iteration, and continues to an iteration of x > 0 position with the appropriate step size ∆x. The step sizes are ∆η = 0.02, and ∆x = 0.04, further, the boundary thickness is η ∞ = 5. The iterative process was stopped when the error in computing the wall shear stress parameter f ηη (x, 0) in the next procedure convergent criteria 10 −5 . Furthermore, the technique followed here is unconditionally stable besides realizes extraordinary precision, converges rapidly, and awards stable numerical meshing functionality.
MATLAB software (R2018b) provides computations and graphical analyses.

RESULTS AND DISCUSSION
In the existence of internal heat generation/ absorption, and Ohmic dissipation, the influence respectively, for distinctive values of the electric field parameter E. It is noticed from Fig. 3(a) that u dwindles with a growth in E which is consequence of the momentum boundary layer thickness reduces. Truth be told that the Lorentz force grows with increased electrical strength, which makes the fluid flow resistant and thus shrinkages the velocity. Also, from Fig. 3(b), it is depicted that the temperature distributions enhance for expansion for E due to the thickness of thermal boundary heightens by raising of Lorentz force.    Fig. 4(b), it is evident that buildups α, the thermal thickness of the boundary layer shrinks and therefore the temperature inside the boundary layer. growing the value of heat generation/absorption parameter Q. We can observe from Fig. 5(a) that u decays rapidly to its asymptotic estimates and oncoming to zero in the stream region.
Moreover, the velocity achieved to its extreme value away from the wavy surface when Q growths. It is also evident that the thickness of momentum increases and the velocity. Fig.   5(b) reveals that the temperature distribution slightly increase versus the variable values of Q.  Fig. 6(a) that growing value of Ec rises u. This is because the growing value of Ec elevates thermal energy inside the boundary layer owing to fluid friction which visibly increases convection and eventually increase velocity.
The effectiveness of separate Ec on θ is indicated in Fig. 6(b). Likewise contrasted with the instance no viscous dissipation, it comprehends that the temperature may be enhanced for enhancing those Eckert number to every one sorts of fluids. Because of viscous heating, those enhanced in the temperature will be improved and calculable for higher value of Ec. In different words, enhancing those Eckert number prompts a cooling of the surface. Moreover, a transfer of heat into the fluid happens which causes an ascent in the temperature. transfer Nu x for variation in the Casson fluid parameter α. It is detected from Fig. 9(a) that the growth in α produces a decline in C f x . This is forasmuch the enlargement in α causes an intension in the fluid motion. Fig. 9(b) exhibits that the local rate of heat transfer Nu x rises inasmuch an increase in α. It is because the thickness of thermal boundary layer reduces with α causing the temperature gradient at the cone surface to increase. The influence of the heat    accelerates the fluid flow and enhances the temperature so from those figures it is detected that Ec, C f x enhances on the upstream orientation of the surface to increase Ec and reduces the local heat transfer rate.

CONCLUSIONS
In the existent study, concerted influences of electrohydrodynamic and magnetohydrodynamic on Casson fluid and heat transfer characters in a variable electric conductivity with the internal heat generation/absorption besides Ohmic dissipation are scrutinized. The governing nonlinear partial differential equations to this problem were simplified via the similarity transformations so they could be solved numerically via Keller-box finite difference method. The graphs are structured to analyze the impact of the relevant flow parameters, overall the magnetic field parameter, electric field parameter, Casson fluid parameter, heat generation/absorption parameter, and also Eckert number. The main results have indicated that • Increasing the electric field parameter and magnetic field parameter, shrink the velocity, local coefficient of skin friction, and local heat transfer rate, further, it elevates the temperature.
• Growing the Casson fluid parameter affords more significantly physically affected, it strongly raises the velocity and local heat transfer rate near the surface but decelerates the temperature and local coefficient of skin friction far away the surface.
• Rising the Eckert number and also the heat generation/absorption parameter enhances the velocity, temperature and local coefficient of skin friction, however, they reduce the local heat transfer rate.
• The velocity is lower when the fluid flows over the rough surface (a = 0) compared to the fluid flow over the flat surface (a = 0) while this behavior is reflected with the temperature.