Dynamics of thermal radiation and heat generationabsorption on a viscous MHD micropolar fluid over a

Abstract

Keywords: Galerkin Weighted Residual; Heat Generation/Absorption; Micropolar Fluid; Thermal Radiation; Viscous Dissipation

Introduction

The study of micropolar fluid has been a popular research area for many years. Goud [1] explained that it is an area of interest to many researchers because of the extensive relevance of micropolar fluids to technology. The solution of the governing equation responsible for the flow and heat transfer are solved by different methods in the literature among which are Shooting Method, Weighted Residual and Adomial Decomposition, among others. Micropolar fluids represent fluids consisting of rigid randomly oriented particles suspended in a viscous medium,

where the deformation of particles is ignored. It is a subclass of microfluids as simplified and defined by Erigen [2, 3]. Erigen’s theory can be used to study how liquid crystals, polymeric fluids and exotic lubricants behave. There are lots of industrial processes that require fluid as a working medium. This prompted Ariman et al. [4] to do a review of micropolar fluids and discussed how it can be applied to technology. Nadeem et al. [5] studied how an unsteady biconvection nanomaterial Micropolar fluid behaves over an exponentially stretching surface when subject to multi slip conditions. The results show that the fluid temperature

*Corresponding author. *E-mail address: akinbomaths@gmail.com This paper was recommended for publication in revised form by Editor-in-Chief Ahmet Selim Dalkılıç Published by Yıldız Technical University Press, İstanbul, Turkey Copyright 2021, Yıldız Technical University. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

gained strength with the smaller values of the Prandt than the higher values. Damseh et al. [6] examined natural convection heat and mass transfer adjacent to a continuously moving vertical permeable infinitely long surface for incompressible micropolar fluid in the presence of heat generation/absorption with the first-order chemical reaction. It was reported that heat and mass transfer coefficients are enhanced with a rise in the heat generation/ absorption and the chemical reaction parameters. Damseh [7] researched unsteady natural convection heat and mass transport of micropolar fluid over a vertical surface with constant heat flux where the enhancement in the vortex viscosity was viewed to magnify the rotation within the boundary layer which has a propensity to strengthen the coefficient of friction. Baharifard et al. [8] studied the flow of MHD micropolar over a moving plate with suction and injection boundary conditions. The impacts of radiation, heat generation, and viscous dissipation play significant roles in both free and forced convection flow and have also caught the attention of researchers. For example, engineering and physics both benefit from the use of radiation. In space technology and high-temperature operations, the effects of thermal radiation heat transfer on various flows are crucial while significant of heat generation/absorption as well as the viscous dissipation which describes how kinetic energy is being transformed into internal energy by work done against viscous stresses, making the process irreversible has been studied in the literature. In research into the unstable case of free convection across an infinite vertical plate in the presence of thermal radiation, Das et al. [9] found that the plate shear stress is marginally larger for uniform heat flux (UHF)than uniform wall temperature (UWT). Koriko et al. [10] reported that the flow near a free stream’s temperature distribution is unaffected by thermal stratification while investigating the effects of thermal stratification as well as nonlinear thermal radiation on the behaviour of micropolar fluid along a vertical surface. Magnetic force’s effectiveness on ferrofluid’s thermal performance in a screw tube was examined by Barzegar Gerdroodbary et al. [11]. Mamun et al. [12] observed that variation in heat generation accelerates the mobility of fluid molecules while investigating the impact of heat generation and viscous dissipation on magnetohydrodynamics natural convection The presence of permeability in the flow of heat and mass transport cannot be overemphasized as its dynamics play a significant role in Science and Engineering field among which is filtration, as a result of its various importance, a good number of authors have worked on permeability effect in the literature. Postelnicu [13] examined the impact of the magnetic field on the transfer of heat and mass numerically by means of natural convection on vertical surfaces that are porous in nature. The study remarked that the thickness of the hydrodynamics, thermal and concentration layers improve over the strength of magnetic intensity. Haque [14] studied the effects of an induced magnetic field

on the transient heat and mass transfer magneto micropolar fluid flow past a semi-infinite vertical plate surrounded by a porous medium in the presence of a constant heat sink. Using the homotopy analysis method, Akinbo and Olajuwon [15, 16] worked on various forms of heat and mass transports in the flow of Walters’ B fluid through a porous medium where the random movement of the fluid molecules was found to be resisted by the porosity impact. Reddy et al. [17] investigated mass transfer and radiation effects of unsteady MHD-free convection fluid flow via in porous medium with heat generation/ absorption. Mamta and Krishna [18] studied the effects of radiation absorption, mass diffusion, chemical reaction and heat source of heat generating fluid past a vertical porous plate subjected to variable suction with the assumption that the plate is embedded in a uniform porous medium and moves with a constant velocity in the flow direction in the presence of a transverse magnetic field. Other areas of fluid dynamics that are also essential for the current trend of scientific development aside from the Micropolar fluid have also been considered in the literature. Joshi et al. [19-22] investigated via horizontal pipeline the Slurry flow characteristics at various Prandtl numbers as well as other investigations involving transportation in a straight pipe where the Maximum specific energy consumption for each particle and efflux is shown by the fluid at increasing Prandtl numbers among others findings. Parkash et al. [23] buttressed that suspension stability enhancement is significant for the lower Prandtl number range and diminishes for the higher Prandtl number range while investigating various values of Prandt numbers CFD modeling of the slurry pipeline. Motivated by the existing work in the literature and the application of the micropolar fluid in science related disciplines, this work aims to provide more insight into the field by considering the behaviors of heat generation/ absorption in an extension to the work in Baharifard et al. [8]. The objective is to examine the combined impact of thermal radiation and heat generation/absorption on a viscous Micropolar fluid over a stretching sheet with suction/ injection.

Problem Formulation

Consider a steady case of two-dimensional boundary layers’ transport of a magnetohydrodynamics Micropolar fluid through a porous medium of a horizontal stretching sheet with suction/injection. The fluid exhibits the constant properties of a viscous fluid and the magnetic field Bo of uniform strength is executed in a perpendicular direction to the flow as shown in Figure 1. On the account of the small magnetic Reynolds number, the induced magnetic parameter is negligible. No electric field is assumed to exist and the hall effect is not taken into account. The heat transport is considered with the combined impact of thermal radiation, viscous dissipation and heat generation/absorption.

conductivity. The coupling constant where s is a constant characteristic of the fluid and ρ is the density of the fluid. G1 is the microrotation constant, Qo is the heat generation/absorption coefficient, k is the thermal conductivity, Cp is the specific heat of the fluid at constant pressure and ∞ describes far field condition. Since the boundary layer is optically thick, the Rosseland approximation for heat transport is taken into consideration. The radiative heat flux (σ* and k* are the Stefan-Boltzmann constant and the mean absorption coefficient, respectively). Assuming temperature varies as the flow progresses, we linearize by using the Taylor’s series to expand the term T4 about T∞. Therefore (overlooking the higher order parts). Applying the transformation techniques components expressed below

Figure 1. Model geometry. The fluid temperature is taken as T while the surface and ambient temperatures are denoted by Tw and T∞. Velocity of the stretching sheet is taken as uw(x) = bx. Taking into account the aforementioned expression, the model equations for transport phenomena under the usual boundary layer approximation are stated for continuity, momentum, microrotation, and energy equations (Baharifard et al. [8], Sedeek et al. [24]) respectively as follows

And the boundary conditions on a semi-infinite computational domain is as follows (10)

(4) Some important physical parameters applied are: The boundary conditions are: (5) (6) Velocities in x and y directions are u and v respectively, vw is velocity of suction/injection at the wall, b is a constant, v is the kinematic viscosity, K is the permeability of porous medium, N represents microrotation, σ is the electrical

Method of Solution Differential equations are generally inevitable in the field of applied Mathematics. Their solutions can be obtained by several methods. Here, Galerkin Weighted Residual Method (GWRM) is adopted for this work as it is very efficient in providing accurate result for non-linear differential equations. Following Akinbo and Olajuwon [25], Eqs. (7) - (9) and (10), followed from trial functions.

The unbounded interval f(∞), g(∞) and θ(∞) automatically agreed. Following the rule of the solution, the functions expressed in Eqn. (11) and non-dimensional eqns. (7 - 9) produce the residual functions Rf, Rg and Rθ multiplied by and integrated under the associated boundary conditions which were carried out through MATHEMATICA package

Results And Discussion

In order to gain physical understanding to the study, the embedded parameters: R = 0.1, P = 1, G = 2, K = 1, γ = 1, Ec = 0.02, Pr = 5.2, Fw = 0.1, Q = 0.02 are discussed by varying each parameter while keeping the others constant. Tables 1–5 demonstrated agreement with the existing literature when compared with Sedeek et al. [24] and Baharifard et al. [8] for skin fiction f ''(0, wall couple stress g'(0) and Nusselt number θ'(0). Table 6 presents the behaviours of various parameters on skin friction, wall couple stress and Nusselt number.

Table 3. Coupling constant parameter for f ''(0), g'(0), θ'(0) K

Table 4. Thermal radiation parameter for f ''(0), g'(0), θ'(0) R f ''(0) 0.1 0.5 1 3 g'(0) 0.1 0.5 1 3 θ'(0) 0.1 0.5 1 3

Table 5. Suction/Injection parameter for f ''(0), g'(0), θ'(0) Fw

Obviously from the table, the skin friction produced negative values, indicating the presence of a drag force exerted on the fluid by the plate. However, the Prandtl number and radiation parameter are increasing functions of the Nusselt number which enhances the rate of heat transfer. Figures 2 and 3 illustrate the behaviour of magnetic parameter (M) via velocity field f ‘(η) and Rotational velocity profile g(η). As expected, a drag-like force known as Lorentz force that opposes the fluid motion is produced by an increase in M causing a reduction in the velocity and microrotation velocity of the fluid. This in turn results in

a decrease of the momentum boundary layer thickness. The reverse is the case as observed in Figure 4 where an advancement in the value of the magnetic parameter led to frictional heating which contributed to a higher temperature distribution within the layer that consequently increased the thickness of the thermal boundary layer. Figures 5 and 6 illustrate the behaviour of the permeability parameter (P) via f ‘(η) and g(η). As expected, an increase in P produces a high resistance to the velocity distribution of the fluid thereby causing a reduction in f ‘(η) and g(η). The resistance against the motion of the fluid particles pioneer

Table 6. Skin friction, wall couple stress and Nusselt number with γ = 1.0 f''(0)

Conclusion

In this work, analytical techniques have been carried out via the Galerkin Weighted Residual method to investigate the combined impact of thermal radiation and heat generation/absorption on a viscous hydromagnetic micropolar fluid and the following results are generated out of the results obtained.

1. Fluid molecules move more rapidly in their random

manner at lower variation in Q at 0.01 ≤ Q ≤ 0.06, but get escalated as Q > 0.06. This makes the thermal effect have a greater impact on the fluids in distant fields.

3. The temperature field is boosted over a conversion rate

of kinetic energy to internal energy that invokes heat addition by means of frictional heating as 0.1 ≤ Q ≤ 6.0

4. Greater resistance is produced against the motion of the

fluid on the variation of the porosity parameter, which declines its associated boundary layer thickness. 5. Different values of the Prandtl number and Eckert number improve the Nusselt number which consequently strengthens the rate of heat transfer.

Data Availability Statement

The authors confirm that the data that supports the findings of this study are available within the article. Raw data that support the finding of this study are available from the corresponding author, upon reasonable request.

Conflict Of Interest

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Ethics

There are no ethical issues with the publication of this manuscript.

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