A Numerical Study on Cross Diffusion Cattaneo-Christov Impacts of MHD Micropolar Fluid Across a Paraboloid

Analyzing the impacts of Cattaneo-Christov flux, bioconvective Raleigh number and cross diffusion effects in electrically conducting micropolar fluid through a paraboloid revolution is assessed in this work. Non-dimensional equations are solved numerically using shooting technique with an aid of Matlab software. The impact of various parameters on velocity, temperature and concentration are discussed in detail and presented graphically. Harman number and micro rotation parameters are found and have an increasing influence on shear stress. The vertical velocity increases at free stream and the horizontal velocity increases near the surface when Grb increases, which follows the opposite trend for accumulation of Rb. The numerical results are compared with the available data indicating good agreement in a limiting case.


Introduction
The flow characteristics based on some variants such as liquid crystals, extrusion of polymer fluids, cooling of a metallic plate in a bath, exoteric lubricants, and colloidal interruption cannot be analyzed by the approach of Navier Stokes equations of Newtonian and Non-Newtonian fluids. Many researchers constructed many constitutive statements. Out of these, the micro polar fluid model was introduced by Eringen [1]. Micro polar fluid is a fluid which consists of a micro-structure that is applicable to a group of fluids along with unsymmetrical stress tensor. The analysis of micro polar fluid has attracted its diverse mechanical and manufacturing applications. Das [2] analyzed the importance of radiation specification on Magneto hydrodynamics (MHD) micropolar fluid through the reference of rotating frame. Dulal Pal and Chatterjee [3] exhibited the 2D micropolar non-Darcian flow with thermal radiation and non-uniform heat source through a permeable stretching panel with boundary layer, and employed analytically with Finite difference method (FDM). Gangadhar et al. [4] analyzed numerically the consequences of the slip effect on MHD micropolar fluid and explained the enhancement of temperature profiles with respect to the accumulation of slip factor. Ghadikolaei et al. [5] investigated the influence of TiO2 water nanoparticles on micropolar flow including dust particles.
Iraqi Journal of Science, 2021, Vol. 62, No. 4, pp: 1238-1264 1239 Gul et al. [6] presented Homotopy Analysis Method HAM to solve the heat transfer of a liquid film of a micropolar flow through a porous medium and also used ND solving technique to get numerical solutions. Impact of heat source /sink on micropolar-free convective fluid flow which passes through a shrinking panel was reported by Mishra et al. [7]. Qasim et al. [8] discussed heat transfer in a micropolar fluid over a stretching sheet with Newtonian heat source. Aurangzaib et al. [9] tested the importance of heat transfer of micropolar fluid flow over an exponentially permeable stretching sheet. In an enclosed MHD the boundary layer flow, heat and mass transmission possess an extensive attention in industrial appliances such as polymer and metal suspension, dispose of chemical material, exchangers, etc. The phenomena of transferring heat and mass play an extensive role in controlling the quality of the output in the outflows. The transformation of temperature and concentration in MHD flows also has essential applications which include magnetic substantial processing, control of the cooling rate, geophysics, etc. Kumaran et al. [10] investigated the temperature displacement on two non-Newtonian fluid flows across a paraboloid revolution and presented graphical representations of both carreau and casson fluids. Khan et al. [11] studied the bioconvection flow of a carreau fluid by a paraboloid revolution with the aid of an implicit finite difference technique (Keller-Box). The boundary layer analysis of Blasius flow of Carreau fluid on an upper horizontal surface of paraboloid of revaluation (UHSPR) was deliberated by Animasaun and Pop [12]. Abegunrin et al. [13] discussed the comparison between two non-Newtonian fluids (Casson and Williamson fluids) over an UHSPR. Animasaun [14] explored the temperature and concentration displacements about nanofluid flows across a paraboloid revolution. Rehman et al. [15] studied the physical phenomena of both the Newtonian and non-Newtonian fluid models, which constitute a set of complex mathematical equations. Vishnu Ganesh et al. [16] analyzed heat transfer and the effects on hydromagnetic nanofluid across a melting surface. Khalil et al. [17] presented the magneto hydrodynamic (MHD) Prandtl-Eyring nanofluid over a stretching surface with new mass flux conditions. Khalil et al. [18] implemented a hydromagnetic non-Newtonian fluid across a stretching cylinder. Ramzan et al. [19] investigated heat and mass transfer and, the impact on hydromagnetic Jeffery nanofluid across a stretching cylinder and also observed the exchange of heat transfer between two objects or within the object is a substantial phenomenon in the nature. Cattaneo [20] introduced the attitude of classical Fourier's law, which was examined by thermal relaxation time. Later Christov [21] implemented the upper convected Oldroyd's derivatives along with thermal relaxation time for attaining the formulation for material-invariant. Straughan [22] addressed the flow of viscous fluid in the presence of Cattaneo-Christov effect. A study of three dimensional (3D) flows of heat conduction and mass diffusion done by Hayat et al. [23]. They computed the analytical solutions of governing flow system of equations by applying OHAM. A theoretical inspection of an unsteady flow and temperature displacement of dusty nanofluid was presented by Mamatha et al. [24] and also provided the graphical representations of both dusty nanofluid and dusty fluid. Ramzan et al. [25] studied the 3D upper convected Maxwell fluid across a stretching surface and obtained a solution using MATLAB software. Ramzan et al. [26] analyzed the effect of Cattaneo-Christov heat flux with 3D non-Newtonian Maxwell fluid. Ramzan et al. [27] explored the numerical solution of two dimensional MHD stagnation point in Williamson fluid under the influence of homogeneous-heterogeneous reactions over a linearly stretched surface. Soret and Dufour effects on magneto hydrodynamic Oldroyd-B fluid across a cone/wedge were analyzed by Reddy [28] using Matlab. In the recent past, the influences of Cattaneo-Christov, thermal radiation, heat source on boundary layer flow and heat transfer were widely examined in many articles [29][30][31][32][33][34][35][36][37][38][39][40]. Many investigators concentrate on analyzing the transformation of heat and mass transfer in MHD flows. However, in the present model, the effects of MHD micropolar fluid across a paraboloid in the direction of Cattaneo-Christov heat flux along with the impacts of cross diffusion effects are evaluated. The mathematical model of the problem and the method to obtain the solution are explained in section 2. The results and discussion of the problem are analyzed in Section 3. Concluding remarks are given in section 4.

Mathematical formulation
Consider the two-dimensional (2D) micropolar mixed convection flow over a paraboloid is the region of flow (here A and b are positive and 1 m  ) and is the stretched parallel with the velocity of the imminent fluid layers on the surface. Here the horizontal surface is placed along X-axis, while Y-axis is perpendicular to it. The suitable governing equations to explore the micropolar fluid of a boundary layer flow passing through paraboloid revolution are given by Kalyani et al.
where ' U and ' V are expressed as the velocities along the directions X and Y ; k ,  and  are known as the vortex, dynamic and kinematic viscosities respectively;  and g are the density of the fluid and the gravitational acceleration;  and *  are the thermal and solutal expansion coefficients; is the spin gradient viscosity; The similarity transformations are as follows Using the transform (7), the equations (2)-(6) are transformed as follows: together with the boundary conditions: are used to alter the system as follows: is the local Reynolds number.

Results and Discussion
In the present model, a powerful numerical mechanism of R-K fourth order inclusive of shooting      and therefore no changes in shear stress occur at the free stream. Figure-2 Figure-2(e).
The influence of thermal Grashof number ) ( b Gr is presented in Figure- . We noticed that ) ( h scales down near the surface and that ) ( is increased near USHPR, although the response is reversed beyond the surface for accumulating values of K. The influence of different values of Nr on thermal profile is shown in Figures 6(a)-6(b). They show that the thermal boundary layer thickness and temperature are improved for increasing values of Nr . Higher values of Nr afford more temperature to the fluid flow which generates an increment in temperature and thermal boundary layer thickness. We identify that the thermal gradient enlarges at USHPR for values from 0 2.5   and the effect is in opposite far from the plate for higher values of Nr (Figure-6b). The impact of Pr on thermal profile is shown in Figure-7(a). It is illustrated that the thermal profile decreases upon the increase of Pr . Figure-7(b) describes the thermal gradient for various values of Pr . It is observed that the thermal gradient depreciates near UHSPR, but the impact is opposite far off from the plate when 1. 5 5  , while the thermal gradient difference is not considered thereafter up to the free stream.
The influence of thermal distribution for distinct values of thermal relaxation specification (  ) is illustrated in Figure-8(a). It is noticed that the thermal profile is depreciating upon rising  . In addition, the thickness of the thermal boundary layer was generally declined when  was increased.
Particles of the object need a long extent to shift temperature to their neighboring particles. Especially, we can clarify that for large values of  material that represent a non-conducting attitude which is important in the minimization of thermal profile. Analysis of  on thermal gradient profile is presented in Figure-8(b). The thermal gradient decreases near USHPR and the result is reversal away from the plate.
The influence of the concentration profile for various traits of Schmidt number (Sc) is shown in Figures-9(a) and 9(b). It is indicted from Figure-9(a) that the concentration profile drops rapidly when Sc increased from 1.0 to 1.5, hence the thickness of the solutal boundary layer decreased. This explains that large diffusing species have more decelerating effects on the concentration profile. Figure-9 Figures-10(b) and 10(d). The accumulation of thermal gradient and concentration gradient near USHPR are observed, while the effect is reversed away from the plate.
The impact of Soret effect on the concentration profile is shown in Figure-11(a). It is observed that the profile of concentration is increased with the increase of Soret effect as well as the thickness of the boundary layer Figure-11(b) depicts the concentration gradient for a distinct value of Sr. It is revealed that the concentration gradient increases near USHPR, but the effect is reversed from 1.8 6   and not considerable change observed till free stream. Table 2

Concluding remarks
The effects of Cattaneo-Christov heat flux, cross diffusion effects, thermal radiation and chemical reaction on MHD micropolar fluid flow through a paraboloid are examined in the present paper. The fourth order R-K method as well as the shooting technique was applied to solve nonlinear ODE's of the problem. The consequences of few flow parameters on the skin friction coefficient to the surface are discussed and the profiles of momentum, heat and mass are graphically illustrated for different flow parameters. The main findings from our analysis involve a significant increase in the perpendicular velocity at the free stream with a parallel velocity expand near the surface when b Gr rises, followed by the reverse trend upon the accumulation of b R . Temperature gradient is an increasing function for the raising values of Nr and Du, while it is a decreasing function with higher values of Pr and  at an adjacent to the plate, but the effect is the opposite at the point far away from the plate. Concentration profile is increased with increase of Soret effect and reduced when Schmidt number is increased. As a last observation Hartman number and micro rotation parameter have an increasing influence on shear stress.