3D MHD Radiation Flow of Unsteady Casson Fluid with Viscous Dissipation Effect

A numerical evaluation of the crucial physical properties of a 3D unsteady MHD flow along a stretching sheet for a Casson fluid in the presence of radiation and viscous dissipation has been carried out. Meanwhile, by applying similarity transformations, the nonlinear partial differential equations (PDEs) are transformed into a system of ordinary differential equations (ODEs). Furthermore, in the numerical solution of nonlinear ODEs, the shooting method along with Adams Moulton method of order four has been used. The obtained numerical results are computed with the help of FORTRAN. The tables and graphs describe the numerical results for different physical parameters which affect the velocity and temperature profiles.


Introduction
During the past few decades, the boundary layer problems related to a stretching surface have attracted extensive attention of researchers, because the number of applications related to this area is found in engineering and industrial manufacturing processes. Actually, the boundary layer has a meaningful concept in physics and fluid mechanics, which is introduced as the layer of the fluid in the region of a bounded area where the effects of viscosity are powerful. Moreover, it is also a region in the flow field in which the fluid deforms with a relative velocity. Each primary fluid has some basic important properties which play an essential role in its dynamics. The stretching and cooling rates both are significant in the manufacturing process for the effective results of the final product. A speedy change in stretching damages the final product because of sudden solidification, so it is essential to maintain the stretching rate. The 2D flow of an incompressible liquid within the boundary layer along the stretching surface was first presented by Crane [1]. Various researchers have studied the interesting fluid flow along a stretching sheet [2][3][4][5].
Magnetohydrodynamics (MHD) is a combination of three words, magneto means magnetic field, hydro means water and dynamics means movements. Meanwhile hydromagnetic flow is the analysis of the magnetic properties of an electrically conducting fluid. Plasmas, liquid metals, saltwater, and electrolytes are considered magneto fluids. MHD flow has a wide range of applications in engineering devices such as the design of heat exchanges, blood pumping machines and the MHD electric power generators. The main role of a magnetic parameter in the flow field is to produce a resistive force which maintains the flow and detains the boundary layer separation. A number of researchers investigated the flow models which contain the hydromagnetic effects. On top of that, Pavlov [6] examined the MHD flow of viscous fluids along a stretching sheet. Alfven [7] established the existence of electromagnetic-hydro-dynamic waves. Sarpakaya [8] studied the flow of specific types of fluids in the magnetic field.
The time-dependent flows are considered as unsteady flow. Wang [9] investigated the time dependent flow problems. Furthermore, various researchers considered the impacts of induced magnetic field on the time dependent MHD flow within the boundary layer region [10][11][12]. Ishak et al. [13] studied the heat transfer of a time dependent flow. The temperature variation between the surrounding and the ambient fluid, produce the radiation.
The complicated behavior of stress-strain can be found in a type of fluids which is called non-Newtonian fluids. Moreover, non-Newtonian fluids have earned considerable attention because a number of applications of these fluids are found in engineering and industry. The Casson fluid is one of the most important non-Newtonian fluids, which is used in metallurgy, food processing etc. Casson [14] introduced the Casson fluid model for the pigment-oil suspensions. Casson fluid exhibits the properties of yield stresses. Whenever the shear stress is greater than the yield stress, the fluid acts like a liquid. Likewise, if shear stress is less than the yield stresses the fluid acts like a solid. In the category of Casson fluids, Jelly, shampoo, toothpaste, ketchup, tomato sauce, honey, soup and juices are founded. Actually, yield stress analysis is important for all complex structured fluids. Dash et al. [15] examined the Casson fluid inside a pipe containing a porous medium Later on Eldabe et al. [16] investigated the hydromagnetic flow of a Casson fluid bounded between two cylinders in a rotating position. Later on, several researchers worked on the free convective electromagnetic flow of Casson fluid in various conditions [17][18][19][20][21]. Maleque [22] investigated the MHD flow of Casson liquid along with a rotating disk. Kataria and Patel [23] considered the ramped wall temperature with heat and mass transfer in the hydromagnetic flow of Casson liquid through a porous medium. The MHD Casson fluid with the effects of Hall, Dufour and thermal radiation was analyzed by Vijayaragavan and Karthikeyan [24]. G. Narender et al. [25] studied the impact of the radiation effects in the presence of heat generation/absorption and magnetic field on the magnetohydrodynamics (MHD) stagnation point flow over a radially stretching sheet using a Casson nanofluid. G. Narender et al. [26] examined the viscous dissipation and thermal radiation effects on the MHD mixed convection stagnation point flow of Maxwell nanofluid over a stretching surface. G. Narender et al. [27] explored the impacts of external magnetic field inclinations and viscous dissipation due to heat generation or absorption parameter on MHD mixed convective flow of Casson nanofluid.
The detailed review work of Parshu and Nankeolyar [28] is explained in this article and extended by considering viscous dissipation. The numerical solution of various parameters has been discussed which impact the skin friction coefficients, Nusselt number, velocity and temperature. Investigation of obtained numerical results is given through tables and graphs.

Mathematical Modelling
A 3D time-dependent, magnetohydrodynamic flow of an incompressible Casson fluid along a linearly stretchable surface has been examined. Meanwhile, the surface is along the plane, Reddy et al.
Iraqi Journal of Science, 2023, Vol. 64, No. 7, pp: 4018-4033 4020 which means that 0 y = and the fluid confined along the positive direction of y − axis has been considered. Furthermore, the sheet is considered to be stretched along x − axis. The time dependent magnetic field has been assumed to act along y-axis which is normal to the surface of the sheet. The physical model of flow is given below in Figure 1. Here w u is the stretching sheet velocity along the x − direction, the surface temperature is w T and the disposition temperature is T  . The system of equations describing the flow has been given below, which contains the PDEs of continuity equation, momentum, and energy transfer [28].
The associated boundary conditions can be written as: The Casson liquid parameter is denoted by  , the electrical conductivity by  , density by  , the kinematic viscosity by  , Hall current by m , the temperature by T , the thermal diffusivity by .
where a and  are constants and 0 B the magnetic strength.
The radiative heat flux r q [28] can be written as where the Stefan-Boltzmann constant is *  and the coefficient of Rosseland mean absorption is *  . From (6), putting r q into (4) and after a little simplification, we get 2 *3 For the conversion of the mathematical model (2)-(4) into the dimensionless form, the following similarity transformation [28] has been introduced, , The final dimensionless form of the governing model, is The associated BCs (5) shown as Different parameters used in (9)-(11) are defined as follow [28]: The skin friction coefficient along x − direction, z − direction and the local Nusselt number are defined as [28]: ( ) 22 ,, Given below are the formulae for w  , w q and m q [28].
The transformation of the above formulae into the dimensionless form has been carried out as: where, wx elucidates the local Reynolds number.

Method of Solution
For the solution of ODEs (9)-(11), the shooting method has been used. The dimensionless equations (9) and (10) are coupled in f and g . These two equations will be solved separately by the shooting method. Later on, the solution of (9) and (10) will be used in (11) 11  12  13  14  15 , , , .
The above mathematical models (9)-(10), now be listed in the form of the following first order coupled ODEs. The above IVP will be solved numerically by the Adams-Moulton method. To get the approximate solution, the domain of the problem has been taken as   where   is an appropriate finite positive real number. In the above system of equations, the missing conditions r and s , must be chosen in such a way that ( ) ( ) 24 , , 0 For the improvement of the missing condition, Newton's method has been implemented which is conducted by the following iterative scheme: The following steps are involved for the accomplishment of the shooting method.  ( ) iv Repeat ( ) i and ( ) ii . In a similar manner, the ODE (11) along with the associated BCs can be solved by considering f as a known function.

Representation of Graphs and Tables
The physical impacts of significant parameters on the skin friction coefficients and Nusselt number have been explained through graphs and tables. Prashu and Nankeolyar [28] used the spectral Quasilinearization method (SQLM) for the numerical solution of the discussed model. In the present survey, the shooting method along with Adams-Moulton Method has been opted for reproducing the solution of [28]. The results discussed in Table 4.1, reflect the impacts of significant parameters on the skin friction coefficients       The influence of the significant parameters on the temperature behavior is shown in  respectively. In Figure 4.8, the rising values of radiative parameter Nr , the temperature is reduced because the temperature distribution is inversely proportion to the Nr . In Figure 4.9 shows that the greater values of Prandtl number Pr has shrink the temperature profile. However, Pr is the ratio of the viscous diffusion to the thermal diffusion. Figure 4.10 portrays that the increasing values of temperature ratio tr, the temperature profile shows a growing behavior. Actually, tr is a ratio between temperature behavior at the surface to the temperature behavior beyond the surface. Further, Figure 4.11 is delineated to show the impact of Ec on the temperature field ( ) . This graph exhibits that by enhancing the estimations of Ec , the temperature field ( )  is also increased.

Conclusion
Some interesting findings have been listed below. ▪ The fluid flow model presented in the paper has applications in silicon suspensions, blood flow, polymer engineering, and the printing industry.