EFFECTS OF VARIABLE VISCOSITY ON MAGNETIC FIELD INTENSITY WITH HEAT, MASS TRANSFER OVER A VERTICAL PLATE

Heat and mass transfer flow along a vertical plate under the combined buoyancy force of thermal and species diffusion in the presence of a magnetic field is investigated. This paper examines the effect of variable viscosity with heat and mass transfer over a vertical layer on the magnetic field intensity. The governing equation of continuity, energy, momentum, and concentration is transformed into ordinary differential equations and then solved the governing transformed ordinary differential equations through the technique of perturbation. Expressions were generated for the distribution of velocity, temperature, and concentration. Tests for parameter values of importance are described graphically and addressed quantitatively from a practical point of view. The findings demonstrate that as the amount of Grashof (Gc) increases, the effects of the concentration bounce increase, and hence the velocity of the fluid decreases. Velocity and temperature decrease as the Prandtl number (Pr) increases. As parameter (K) of the magnetic field increases, the intensity decreases. The velocity and concentration decrease even as the Schmidt number increases.


INTRODUCTION
The viscosity of a fluid can be termed as friction involving a fluid.It is a measure of internal resistance to the flow of a fluid when subjected to shear stress or tensile stress.Fluids that show stress resistance are termed viscous fluids while those that do not exhibit resistance to shear stress are referred to as ideal fluids or inviscid fluids (Hassan, Lawal, & Amurawaye, 2020) [1].The viscosity of a fluid depends on various factors such as temperature, pressure, and shear rate.The temperature affects the viscosity of both liquids and gases.In the case of liquids, increasing the temperature of the liquid leads to a reduction in its viscosity as this can be explained using the particle theory.Heat transfer through thermal radiation is of significance to most engineering and industrial processes occurring at higher temperatures and its knowledge is important in the design of most engineering equipment [2 -5].Such equipment is used as devices for propelling air crafts, missiles, satellites, and hypersonic flights, in rocket combustion chambers, gas turbines, and gas-cooled nuclear reactors.Variable viscosity property of a fluid and energy transfer have been of vast interest and issues of discussion among researchers in petroleum and chemical engineering due to its technological and industrial application.Some of the applications of flow can be established in well-arranged chemical reactors, geothermal containers, physical transformation companies, convertible exhaust schemes to mention a few.Investigations on fluid behavior cannot be adequately described based on variable viscosity properties, convective cooling, thermal radiation, heat source/sink, porosity, among the few [1 -2, 6].
Many researchers have conducted thorough studies in this field, which include studies in [7 -11].However, in the preceding studies the radiation, chemical reaction, variable suction, and variable viscosity effects were neglected.However, exceptions are observed in [12] by analyzing the effect of radiation flows over an elastic stretching surface on heat and fluid.Mukhopadyay [13] examined the influence of thermal radiation on elastic boundary layers of combined convection heat transfer issue from a vertical stretching surface incorporated in porous media.Based on the above research the fluid's viscosity was believed to be stable.Several researchers studied and investigated the influence of vector properties for fluid viscosity and thermal conductivity over a continuous flowing surface at fluid heat transfer.In the case of turbulent flow, [14] measured the effect of radiation and variable viscosity on a free MHD convection flow through a semi-infinite flat plate with an associated magnetic field.The influence of chemical reaction and variable viscosity on hydromagnetic combined convection heat and mass transfer for Hiemenz flow across porous media in the presence of radiation and magnetic field has been studied in [15].Dandapat et al. [16] conducted a study on the effects of variable thermal conductivity, variable viscosity, and thermocapillary on flow and heat transfer on a horizontal stretch sheet in a laminar liquid film.Hayat et al. [17] experimented to examine the effect of variable viscosity and thermal conductivity in the three-dimensional mixed convection flow of viscous fluid across an exponentially stretching surface with heat transfer.Siddiqa et al. [18] studied the numerical solutions for the normal convection flow along a vertical wavy cone centered in the thermally radiating stream.Recently, Nadeem et al. ) [19] proposed exponential surface temperature and recommended exponential heat flux because of the movement of micropolar fluids on a Riga board.Krishna et al. [20] proposed a theoretical analysis of the influence of magnetohydrodynamic systems on classical heat transfer.However, there has been no study conducted on the effect of variable viscosity with heat and mass transfer over a vertical layer with particular reference to the magnetic field intensity.
The objective of this paper is to study the effects of variable viscosity on magnetic field intensity with heat, mass transfer over a vertical fixed plate, and other parameters on the flow such as modified Grashof number (Gc), Schmidt number (Sc), thermal Grashof number (Gr), Prandtl (Pr), permeability parameter (K 1 ,), and chemical reaction parameter (K).

Mathematical Formulation
Consider a steady, free convective flow of an incompressible viscous electrically conductive fluid through a semi-infinite vertical layer at a constant temperature T   and a uniform transverse magnetic field.The fluid temperature far from the plate is T   .The axis is taken upwards around the wall, and y-axis is taken to it as normal.The gravitational force descends almost vertically.As the fluid's density is small the viscous dissipative heat is believed to be zero.It is often assumed that the magnetic field of constant strength is naturally applied to the vertical layer, so that the electrical conductivity of the fluid is supposed to be minimal so that the magnetic field generated can be ignored relative to the magnetic field applied.The temperature and concentration of the fluid at infinity are considered in this paper to be T   and C   everywhere and the temperature and concentration at the plate are T  and C  respectively.
The plate is taken along ′-axis in vertically upward direction and ′-axis is taken normal to the plate as shown in Fig. 1 where ' u and ' v refer to the velocity components in the directions of x and y , respectively, g is defined as the acceleration due to gravity,  and 2  denote the volume expansion coefficient, 0 k and  represent the kinematic viscoelasticity and the kinematic viscosity, respectively,  is defined as the density,  represents the viscosity, K T is described as the thermal conductivity, Cp is defined as a fluid heat at constant pressure, refers to the electrical conductivity of the fluid, e  is defined as the magnetic permeability, D is defined as the molecular diffusivity, ' T  and ' T  represent the temperature of the plate and the temperature of the fluid far away from plate, respectively.C   and ' C  refer to the concentration of the plate and the concentration of the fluid far away from the plate, respectively.For 0 0 v  , it is indicated that the negative sign suction is towards the plate.
The boundary conditions of the problem are Introducing the following non-dimensionless quantities The dimensionless variables are substituted in ( 6) into (1) to (5), we get (dropping the bars) Pr with the following boundary conditions where Gr is defined as the thermal Grashof number, Gc the mass Grashof number, Sc the Schmidt number, Pr the Prandtl number, Rm the Reynold number, K the viscoelastic Parameter, and S the permeability.

The Boussinesq approximation Model
The reason of this assumption is that there are flows in which all the temperature varies slightly and thus the density varies slightly, but in which the buoyancy controls the motion.For certain natural convection flows, you can achieve easier convergence with the Boussinesq approximation model by creating a fluid density problem as a function of temperature (Cherkasov et al. 2018).This model represents density as a constant value in all the equations that have been resolved, except for the buoyancy term in the momentum as follows: where 0  describes the (constant) density of the flow, 0 T represents operating temperature and  describes the thermal expansion coefficient.Equation ( 12) is generated by obeying the Boussinesq approximation 0 (1 to eliminate  the buoyancy term.This approximation is valid as long as the original density changes are low; specifically, the Boussinesq approximation is correct when equation ( 13) is satisfied.

METHODS OF SOLUTION
Magnetic field strength, also called magnetic intensity or magnetic field intensity, is the part of the magnetic field in a material that arises from an external current and is not intrinsic to the material itself.It is expressed as a vector and is measured in units of amperes per meter.In order to solve (7) to (10) according to the boundary conditions (11), we assume the following equations: Cy can be determined.
Substituting (14) to ( 17) into ( 7) to (11), respectively, and comparing harmonic terms with non-harmonic terms, yield the following equations: 00 Pr 0 with the following boundary conditions where the primes denote differentiation concerning y.
Solving equations ( 18) to (25) according to boundary conditions (26), and substituting the solutions obtained with ( 14) and ( 16), respectively, the field of velocity can be represented as follows: Rm

RESULTS AND DISCUSSION
The results on velocity, temperature, and concentration distributions have been discussed for the following parameters: modified Grashof number   Gc , Thermal Grash of number         The influence of the modified Gc on the velocity is displayed in Figure 2. It is revealed that when all other parameters are kept constant, the velocity increases with increasing the modified Gc .The influence of the thermal Gr on the velocity is displayed in Figure 3, showing that increasing Gr leads to an increase in efficiency as all other parameters occurring in the area of velocity are kept constant.Figure 4 presents the influence of Sc on the velocity.It is found that when all the competing parameters are kept constant, and an increase in Sc reduces the velocity.The effect of Pr on the velocity field is illustrated in Figure 5, where the velocity field is observed to increase with Pr when all other parameters occurring in the velocity field are kept constant.Figure 6 illustrates the speed profiles for various parameters of K 1 .It is clear that the peak value of the velocity continues to decrease as K 1 increases when all other contributing parameters are kept constant.It can be seen from Figure 7, describing the effect of K on the velocity, that when all other participating parameters are kept constant, the chemical reaction parameter K is decreased, the velocity, in general, decreases with reducing the chemical reaction parameter K .It's also found that as we're heading away from the ground, the velocity of the fluid is going down.

Temperature and Concentration profiles
The temperature and concentration profiles are presented in Figure 8 and Figure 9, respectively.The effect of Pr on the temperature field is illustrated in Figure 8, from which it is observed that the temperature field decreases with increasing Pr when all other parameters that appear in the temperature field are held constant.Figure 9 displays the influence of Sc on the concentration, where one can see that as all other participating parameters are held constant, increasing Sc leads to a decrease in the concentration.

CONCLUSIONN
In this paper, the governing equations for variable viscosity effects on magnetic heat and mass transfer equations over a vertical plate have been studied.The resulting equations have been analytically solved using the perturbation method.The results displayed illustrate the flow characteristics with respect to velocity, temperature, and concentration.It is found that when the Grashof number increases, the concentration buoyancy effect is enhanced and thus, the fluid velocity increases.The velocity as well as the temperature decrease with an increase in the Prandtl number Pr.It is revealed that, with the increase of chemical reaction parameters, the velocity decreases as well.The velocity decrease with the increase in the Schmidt numbers.An increase in buoyancy force decreases the skin friction, whereas there is no effect on the Nusselt number and Sherwood number.An increase in the magnetic field increases the skin friction and a reverse trend is seen for higher time on the Nusselt number.The Prandtl number has a sign on the Nusselt number.The Sherwood number decreases due to higher values of the Schmidt number.

Declaration of competing interest.
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Fig 1 .
Fig 1.The physical configuration of the model.The governing equations of the fluid flow can be written as follows

Figures 2
Figures 2 to 7 reflect the respective velocity profiles with changing parameters.

Figure 2 .
Figure 2. Profile velocity of various values of Gc.

Figure 3 .
Figure 3. Profile velocity of various values of Gr.

Figure 4 .
Figure 4. Profile velocity of various values of Sc.

Figure 5 .
Figure 5. Profile velocity of various values of Pr.

Figure 6 .
Figure 6.Profile velocity of various values of K1.

Figure 7 .
Figure 7. Profile velocity of various values of K.

Figure 8 .
Figure 8. Temperature profile for different values of Pr.

Figure 9 .
Figure 9. Concentration profile for different values of Sc.