MHD Casson non-Newtonian nanofluid over a nonlinear penetrable elongated sheet with thermal radiation and chemical reaction

Consider a steady flow in two-dimensional of a viscous, incompressible Casson nano liquid over a nonlinear penetrable elongated sheet with radiation and chemical reaction. The Casson liquid rheological model is used to explain the non-Newtonian liquid attributes. Similarity variables are utilized to evaluate the governing flow model into set of nonlinear total differential equations. The outcomes of the flow equations were gotten by using Runge-Kutta alongside the shooting techniques. In other to explain the physics of the problem, impact of flow parameters are presented in graphs while computations on engineering curiosity are presented in table. Ahike in the Casson liquid term is observed to degenerate the fluid velocity alongside the momentum layer thickness. The impact of the imposed magnetic is felt by decreasing the velocity owning to the Lorentz force.


Introduction
Fluids are generally classified as Newtonian and non-Newtonian. The Newtonian fluids signifies the linear relationship that exist between shear stress as well as shear rate. On the contrary, non-Newtonian fluids portrays the nonlinear relationship existing between the shear stress and strain. Examples of Newtonian fluids are water, oil etc while Casson, Walters-B liquid, micropolar, power law liquids etc are examples on non-Newtonian liquids. One of the most popular non-Newtonian liquid is Casson liquid. It finds applications in the manufacturing of food, biological engineering and in drilling operations. Mahanta and Shaw [1] elucidate the flow of Casson liquid in 3D past a penetrable linearly stretchable sheet. Samrat et al. [2] elucidate the attribute of thermal radiation with unsteadiness flow of Casson nano liquid past a stretchable surface. Raza et al. [3] studied the impact of multiple slip on MHD non-Newtonian nano liquid flow past an elongated sheet. Mukhopadhyay et al.[4] gave detailed behaviour of Casson liquid over a stretchable surface. The flow of Casson liquid with MHD past an exponentially and shrinking sheet has been elucidated by Nadeem et al. [5]. Vijaya and Reddy [6] discussed Casson liquid flow past an upward penetrable plate with radiation as well as Soret impact. Krishna et al. [7] elucidate chemically reactive MHD flow of Casson fluid with penetrable stretchable sheet. The recent study of Idowu and Falodun [8] discussed the simultaneous flow of two non-Newtonian liquids by varying viscosity as well as thermal conductivity. In another study of Idowu and Falodun [9] presented discussion on MHD non-Newtonian nano liquid past a slanting plate with thermophoresis. Idowu et al. [10] studied Casson dissipative liquid by varying viscosity alongside thermal conductivity.
The thermal radiation acts a major role in industrial engineering. It major importance on heat transport process is in design of energy system utilized at a very higher temperature. Examples are missiles, gas turbines, space vehicles, nuclear power plants and satellites. Fagbade et al. [11] employed spectral approach to the problem of MHD flow of Walters-B liquid with thermal radiation alongside heat source.
PSYCHOLOGY AND EDUCATION (2021)  The consideration of heat alongside mass transport with MHD have received much attention by many scholars owning to their numerous applications in chemical engineering and sciences. Liquids such as ethylene glycol, water and oil are poor conductivity of heat, they are found very useful as a cooling materials for boosting manufacturing. The presence of MHD drags the flow owning to the production of Lorentz force. This indicate the usefulness of MHD in reducing the turbulence of fluid flow.
Inspired by the literature discussed above to study a steady two-dimensional flow of a viscous, incompressible non-Newtonian nano liquid over a nonlinear penetrable elongated sheet with radiation and chemical reaction. Similarity transformation has been utilized to simplify the governing equations, and the reduced boundary value problems are solved using Runge-kutta method alongside shooting techniques.

Mathematical analysis
Consider steady, laminar, viscous and incompressible boundary layer flow of nano liquid past a nonlinear penetrable and stretchable surface. The y -coordinate is taken normal to the stretching surface while the flow occurs immediately 0 y  . The Casson liquid flow arises owning to the stretching sheet that take place out of the shit at =0 x and =0 y . Let us assume that a certain point at the plate, the speed is equivalent to the distance from the slit. Thus, the approximation of the boundary layer is valid on the flow. Starting from the wall where (14) With respect to equation (14) above, the governing flow equations are simplified to obtain and w  signifies wall shear stress, w q signifies heat flux while m q signifies mass flux. They are Employing equation (14) on the above to obtain

Numerical approach
The converted nonlinear differential Equations (15)-(17) with the boundary conditions (18) and (19) are elucidated by Runge-Kutta Fehlberg method along with shooting technique. This method has been proven to be adequate and gives accurate results for the boundary layer equations

Results and Discussion
This section explains the outcomes of the results from solving equation (15)-(17) subject to (18) and (19) numerically. To describe the physics of the problem, results are presented in graphs. Figure 2 portrays the effect of porosity term (K) on the velocity plot. It is noticed that an incremental value of K leads to degeneration in the velocity plot as well as the momentum layer thickness. Physically, a hike in K makes very hard for the liquids particles to move freely and hereby resist the velocity gradually. Figure 3 portrays the behaviour of the imposed magnetic term (M) on the velocity plot. An increment in the value of M as shown in Figure 3 declines the velocity plot. Practically, imposing the magnetism in the flow direction of an electrically conducting liquid such as Casson produces Lorentz force. This force has www.psychologyandeducation.net the tendency to drag the Casson flow by decreasing its velocity. hence, the fluid velocity alongside its momentum layer thickness is declined. This implies that, increase in M added more strength to the Lorentz force. Figure 4 presents the impact of varying the nonlinear stretching term (n) on the velocity plot. Obviously from figure 4, an increment in the stretching term degenerates the velocity plot alongside the momentum layer thickness. Figure 5 portrays the impact of the Casson ()  electrically conducting liquid on the velocity plot. A large value of  is detected to degenerate the velocity plot from 1 to 4. This indicates that the density alongside momentum layer thickness declines for a large value of  . Experimentally, when   it is detected that the behaviour of  is equivalent to a Newtonian behaviour. Physically, the Casson liquid possesses a plastic dynamic viscosity originates a resistance to the liquid flow by decreasing the velocity plot. Figure 6 portrays the role of thermal radiation () R on the temperature plot. A large value of R is observed to hike the temperature plot. This implies an enhancement in the thermal layer thickness. Physically, the thermal radiation helps to boost the thermal situation of the liquid environment. Hence, increasing R helps to boost the heat on the liquid and hereby lead to enhancement in the temperature of the liquid. Figure 7 represents the impact of the Prandtl number (Pr) on the temperature plot. A less fluid temperature alongside thermal layer thickness is noticeable for a large Pr. Hence, when <<1 Pr , it implies thermal diffusivity controls the flow behavior while implies momentum diffusivity controls the flow behavior. Therefore, is useful in controlling cooling rate of a conducting liquid electrically. In this study, there exist a wall temperature and free stream temperature which gives rise to w TT   along with thermal boundary layer thickness. Figure 8 shows the effect of Eckert number () Ec on the temperature plot. An increment in the value of Ec from 0.1 to 0.4 is observed to degenerate the temperature plot alongside the thermal layer thickness. Ec portray the flow kinetic energy and its associated enthalpy. The viscous dissipation term (Eckert number) hike heat energy to the fluid environment. Physically, Eckert number and thermal radiation in a fluid flow increase the rate of heat transfer. Now, considering Eckert number, thermal radiation with thermally-stratified medium assist to boast the heat transport rate by elevating the thickness of thermal boundary layer. Figure 9 represents the effect of the thermophoresis () Nt on the concentration plot. It is detected in figure 9 that a large value of Nt hike the heat transport rate by enhancing the fluid temperature plot. Figure 10 depicts the effect of the Brownian motion () Nb on the concentration plot.
A large value of Nb is observed to declines the concentration plot as well as the solutal layer thickness. The result in figure 10 shows the impact of Nb on concentration plot is opposite to that of Nt. Figure 11 shows the impact of the chemical reaction term () Kr on the concentration plot. An incremental value of R is observed to degenerate the concentration plot. Physically, it indicate a destructive chemical reaction. Figure 12 represents the effect of the Lewis number () Le on the concentration plot. An increment in Le is observed in figure 12 to degenerate the concentration plot. Figure 13 shows the slip effect on the velocity profile. The result in figure 13 shows a decrease in the velocity plot owning to a hike in the slip term. Figure 14 shows the slip effect on the temperature profile. The result in figure 14 shows a decrease in the temperature plot due to increase in the slip term. Figure 15 shows the slip effect on the species plot. The result in figure 15 shows a decrease in the concentration plot due to increase in the slip term. Table 1 shows the impact of various flow parameters on www.psychologyandeducation.net the engineering interest.
All pertinent flow parameters were found to have great effect on the skin friction, Nusselt and Sherwood number.

Conclusion
The study at hand provides the numerical outcomes for a steady, viscous flow of Casson liquid alongside heat transport past an elongated sheet with thermal radiation and chemical reaction. The velocity decreases significantly owning to increase in Casson term. The temperature also degenerate significantly owning to increase in Prandtl number. The effect of increasing values of the magnetic term is to suppress the velocity, whereas the temperature is enhanced with increasing thermal radiation parameter. Prandtl number can be used to increase the rate of cooling in conducting flows.