CASSON FLUID FLOW PAST A SHRINKING SURFACE WITH HEAT AND MASS TRANSFERS

In this study, we have numerically investigated the heat and mass transfers behaviour of Casson fluid flow past a porous shrinking sheet in existence of a magnetic field, thermal radiation, and suction or blowing at the surface. Applying suitable similarity transformations, the leading partial nonlinear differential equations of mass, flow, and heat transfer are converted into solvable ordinary differential equations, which can then be solved numerically with the help of the MATLAB bvp4c scheme. We have analyzed and shown graphically the implications of several non-dimensional controlling factors on the profiles of temperature, concentration, and velocity. Additionally, the Sherwood, Nusselt, and Skin friction for Casson fluids are examined and tabulated. The current study's findings for Casson fluid exhibit great consistency with previous research under specific circumstances.


INTRODUCTION
There are numerous uses for the study of non-Newtonian fluids in industries and engineering, particularly in the process of separating fossil oil from petroleum goods.A Casson fluid is a non-Newtonian fluid with yield stress.Additionally, due to the chain structure of blood cells and the materials they carry, such as protein, fibrinogen, rouleaux, etc., human blood may also be considered a Casson fluid.Thus, the Casson fluid plays a significant role in both biological science and engineering.The problem of the flow by reason of stretching or shrinking sheets has attracted many researchers, and it is a topic of attention in the literature (Grubka and Bobby [1], Banks [2], Crane [3], Keller and Magyari [4], Lio and Pop [5], etc.).Boundary layer flows have many significant applications in industrial manufacturing processes.Though there aren't enough works on the flow past on shrinking sheets, Wang [6] was the foremost to study the unstable viscous flow caused by a shrinking sheet.Mikalavcic and Wang [7] have examined the viscous hydrodynamic flow caused by the shrinking surface for particular values of the suction parameter and came to the conclusion that, for both two-dimensional and axi-symmetric flows, the shrinking sheet solution might not be unique at particular suction rates.After that, Fang and Zhang [8] have clarified how an external magnetic field affects the flow of a shrinking sheet and discovered that a high magnetic field ensures a constant flow of the boundary layer.Following that, several scholars [9][10][11][12][13][14][15][16][17][18][19][20][21] examine the non-Newtonian fluid flow past a diminishing sheet from a variety of physical angles.
Dey et al. [37] explored the stability of MHD Casson fluid over a porous elongating sheet.Bhattacharyya et al. [38] studied the MHD Casson fluid over a porous stretching/shrinking surface in the existence of wall mass transmission.Das et al. [39] have studied numerically to examine the nanofluid flow in permeable media past a vertical stretching surface with heat and mass transfers.
Pramanik [40] studied the characteristics of Casson fluid heat transfer via thermal radiation and porous media.Sarkar et al. [41] enlightened the significance of this fluid model in various contexts and in relation to heat radiation is investigated through the use of an inclined cylindrical surface.Elucidation of the non-Newtonian Casson fluid dynamics across a rotating non-uniform surface under the influence of coriolis force was enlightened by Oke et al. [42].Dey et al. [43] have investigated the energy transfer and entropy creation of hydro-magnetic stagnation point flow in micropolar fluids under uniform suction and injection.
Kinetic processes like heat and mass transmission can happen and be studied individually or together.While studying them separately is easier, in the case of diffusion and convection, both processes are modelled by comparable mathematical equations.In certain situations, such as evaporative cooling and ablation, mass transfer must be taken into account in addition to heat.Problems with combined mass and heat transfer are significant in many processes, and they have gained attention recently in the chemical industry, drying, evaporation on a water surface, and the process of connecting with thermal retrieval.
In this research paper, we have examined the heat and species concentration transmission of Casson fluid through a permeable medium past a shrinking sheet.This motion is mathematically controlled by a system of non-linear PDEs that, EEJP. 1 (2024) Rajesh Kumar Das, et al.
with the suitable transformation, are converted into non-linear ODEs.The velocity, temperature, and concentration profiles are obtained by numerically solving this system under the proper boundary conditions.The effects of the problem's physical characteristics on these results are explored graphically and numerically using a series of figures and tables.A quick analysis can produce a model that helps explain the mechanics of physiological fluxes.

MATHEMATICAL FORMULATION
Consider a two-dimensional, incompressible, electrically conducting hydromagnetic Casson fluid flow over a permeable, shrinking sheet with heat and mass transfer.To scrutinize the suction and blowing processes, the wall permeability characteristics have been used.Figure 1 illustrates how the non-Newtonian fluid drenches the porous material y>0 while the flow dynamics take place in the area y<0.A magnetic field B(x) has been implemented in the flow.Following Dey et al. [37], the rheological equation of an incompressible and isotropic Casson fluid is as follows: where,    and  is the ,  ℎ component of deformation rate.
Considering the above assumptions, the leading equations for the projected fluidic model are given by: The relevant boundary restrictions are  −,   ,   ,   at  0, where ,  denotes velocity components along x and y directions,  stands for kinematic viscosity,  represents casson fluid parameter,  is the electrical conductivity,  stands for fluid density,  is the porous medium permeability,  represents fluid temperature,  denotes thermal conductivity,  describes species concentration,  is the coefficient of mass diffusion,  0 and  0 are assumed as blowing and suction conditions, respectively.
Using equation (6) in equations ( 2)-( 5), it transforms as follows: +  = 0 (8) The relevant boundary restrictions are given by The physical parameters that are important for our studies are Sherwood number, Nusselt number, skin friction coefficient.They are expressed as

RESULT AND DISCUSSION
The resultant ordinary differential equations and related surface limitations are numerically computed via the bvp4c MATLAB software.The flow behaviour patterns for diverse values of the leading parameters are displayed in both tabular and graphical form.Figure 2 exhibits the velocity profile for various values of magnetic field parameter (M).It is found that with increment in M, the magnitude of the velocity profile increase.This is because, with increment of M will dominate the Lorentz force and lessen the impact of the fluid's viscosity at the surface, increasing the fluid's speed.).Here we have seen that with enhancement of K 1, the velocity profile increases.As K 1 grows, the fluid has a larger area to move, which causes its velocity to rise. Figure 4 displays the thermal profile for various values of Prandtl number (Pr).From the figure it is clear that with rise in Pr, the thermal profile decreases.As Pr increases, the temperature drops.The thickness of the thermal boundary layer decreases with an increase in Prandtl number.The ratio of momentum diffusivity to heat diffusivity is known as the Prandtl number.Pr regulates the relative thickness of the thermal and momentum boundary layers in heat transfer issues.Figure 5 elucidates the variation of Schmidt number for species concentration profile.It is seen that the concentration falls, as Sc intensifies.The mass transfer rate rises with a higher Schmidt number, which causes the concentration profiles to fall.

VALIDATION OF RESULTS
To validate our numerical scheme, we compare our results with Bhattacharyya et al. [17], and they are found to be in good agreement with the result.(Table 1.) In our experiment, Table 2 reflects the influence of the Magnetic parameter on the skin friction coefficient (′′(0)), Sherwood number (−′(0)), and Nusselt number (−′(0)).We have seen that Nusselt number, and Sherwood number exhibit a declining trend while skin friction increases as the Magnetic parameter increases.Table 3 elucidates the influence of porosity parameter ( ) on the skin friction coefficient (′′(0)), Sherwood number (−′(0)), and Nusselt number − (0) .From the table it is evident that as  enhances, the skin friction coefficient trends enhancement on the other hand the Nusselt number, and Sherwood number exhibit a declining trend.Table 4 depicts the impact of the Casson fluid variable on Sherwood number (− (0)), Nusselt number − (0) , and skin friction coefficient ′′(0) accordingly.We have identified that with an increase in the Casson fluid parameter, the skin friction coefficient experiences growth, but Nusselt number and Sherwood number experience a decline trend.Table 5 exhibits the influence of suction parameter () on the skin friction coefficient (′′(0)), Sherwood number (−′(0)), and Nusselt number (−′(0)).From the table, it is clear that as  increases, ′′(0) increases but Sherwood number and Nusselt number decrease.

CONCLUSION
We have numerically investigated the thermal and species concentration transmission of Casson fluid model over a shrinking surface.The governing equations are transformed into solvable ODEs, and bvp4c solver scheme is used to solve them.The key points of our study are listed below.
• As the magnetic parameter enhances, the motion of the Casson fluid increases.
• The porosity parameter enhances the velocity profile.
• With rises in Prandtl number, the temperature profile falls.

•
The Casson fluid parameter is very important for controlling the fluid's temperature and concentration, which helps prevent damage to the system, as well as for helping the fluid develop its velocity.• Schmit number retards the concentration profile.
• Suction phenomenon increases the velocity of the fluids.

Figure 2 .Figure 3 .Figure 3
Figure 2. Velocity profile for M Figure 3. Velocity profile for K1 Figure 3 portrays the velocity profile for diverse values of porosity parameter (K 1).Here we have seen that with enhancement of K 1, the velocity profile increases.As K 1 grows, the fluid has a larger area to move, which causes its velocity to rise.Figure4displays the thermal profile for various values of Prandtl number (Pr).From the figure it is clear that with rise in Pr, the thermal profile decreases.As Pr increases, the temperature drops.The thickness of the thermal boundary layer decreases with an increase in Prandtl number.The ratio of momentum diffusivity to heat diffusivity is known as the Prandtl number.Pr regulates the relative thickness of the thermal and momentum boundary layers in heat transfer issues.Figure5elucidates the variation of Schmidt number for species concentration profile.It is seen that the concentration falls, as Sc intensifies.The mass transfer rate rises with a higher Schmidt number, which causes the concentration profiles to fall.Figures 6 gives the impression that the fluid's velocity is accelerated by increasing Casson fluid parameter () values.Enhancement of  generally causes the fluid's motion to slow down since it increases the plastic dynamic viscosity.However, throughout the flow, an opposite behaviour is shown, and this is only possible Figure3portrays the velocity profile for diverse values of porosity parameter (K 1 ).Here we have seen that with enhancement of K 1, the velocity profile increases.As K 1 grows, the fluid has a larger area to move, which causes its velocity to rise.Figure4displays the thermal profile for various values of Prandtl number (Pr).From the figure it is clear that with rise in Pr, the thermal profile decreases.As Pr increases, the temperature drops.The thickness of the thermal boundary layer decreases with an increase in Prandtl number.The ratio of momentum diffusivity to heat diffusivity is known as the Prandtl number.Pr regulates the relative thickness of the thermal and momentum boundary layers in heat transfer issues.Figure5elucidates the variation of Schmidt number for species concentration profile.It is seen that the concentration falls, as Sc intensifies.The mass transfer rate rises with a higher Schmidt number, which causes the concentration profiles to fall.Figures 6 gives the impression that the fluid's velocity is accelerated by increasing Casson fluid parameter () values.Enhancement of  generally causes the fluid's motion to slow down since it increases the plastic dynamic viscosity.However, throughout the flow, an opposite behaviour is shown, and this is only possible

Table 2 .
Numeric data of Sherwood number, Nusselt number and Skin fraction for magnetic parameter (M)

Table 3 .
Numeric data of Sherwood number, Nusselt number and Skin fraction for porosity parameter (K1)

Table 4 .
Numeric data of Sherwood number, Nusselt number and Skin fraction for Casson fluid parameter (β)

Table 5 .
Numeric data of Sherwood number, Nusselt number and Skin fraction for suction parameter (S)