NUMERICAL INVESTIGATION OF MHD CASSON NANOFLUID FLOW TOWARDS A NON LINEAR STRETCHING SHEET IN PRESENCE OF DOUBLE-DIFFUSIVE EFFECTS ALONG WITH VISCOUS AND OHMIC DISSIPATION

The intention behind carrying out this research work is to analyze the steady Magnetohydrodynamic (MHD) boundary layer flow with Casson nanofluid in presence of Viscous and Ohmic dissipation effects towards a non-linear stretching sheet. Two phase representation of nanofluid studied the consequence of Brownian motion along with thermophoresis. The major purpose of study is to investigates the significant role of prominent fluid parameters especially yield stress, slip velocity, thermophoresis, Brownian motion, Eckert number, Schmidt number, magnetic parameter and non-linear stretching parameter on profile of velocity, temperature distribution and concentration while keeping the other parameters under study constant. Runge-Kutta Fehlberg (RKF) approach was adopted to numerically solve the non-linear governing equations and the linked boundary conditions by use of shooting technique. In present study, we use MATLAB for finding the final outcomes and relating the concluding results for local Nusselt number −θδ ′ (0) with extant outcomes in literature as a limiting case in the absence of thermophoresis and Brownian motion and an excellent agreement is noted. Over all the consequence of prominent fluid parameters are explained via graphs, whereas distinction of several valuable engineering quantities like skin friction coefficient, local Nusselt number and local Sherwood number are also tabulated. The finding of present study helps to control the rate of heat transportation as well as fluid velocity in any manufacturing processes and industrial applications to make desired quality of final product.


INTRODUCTION
In various fields of science and technology rapid progress has urged the researchers to extend their study towards the regime of boundary layer flow over a stretching sheet. The boundary layer flow behavior towards a linearly or nonlinearly stretching sheet plays a significant role for solving engineering problems and possess vast applications in manufacturing and production processes including metal spinning, rubber sheet manufacturing, production of glass fibers, wire drawing, extrusion of polymer sheets, petroleum industries, polymer processing etc. In these cases, the final product of desired characteristics depends on the rate of cooling in the process and the process of stretching. The dynamics of the boundary layer fluid flow over a stretching surface originated from the pioneering work of Crane [1] and he examined the incompressible steady boundary layer fluid flow caused by stretching sheet which moves in its own plane with linear velocity due to the uniform stress applications. This problem is particularly interesting as Crane [1] obtained the exact solution of 2D Navier-Stokes equations. Afterthat, Gupta and Gupta [2] extend Crane [1] work over different mathematical geometries. Yoon et al. [3] studied the theoretical and experimental results using Coulomb friction model by considering punch-sheet interface. Also, Sarma and Rao [4] examined the viscoelastic fluid flow by considering stretched sheet. In view of this, Vajravelu [5] studied flow and heat transfer in a viscous fluid over a nonlinear stretching sheet without using the impact of viscous dissipation. Cortell [6] examined heat and fluid flow transportation over a nonlinear stretching sheet for two different types of thermal boundary conditions, prescribed surface temperature (PST) and constant surface temperature (CST). The influence of heat transfer on the stagnation point flow of a third-order fluid over a shrinking surface has been studied by Nadeem et al. [7]. Recently, Prasad et al.
Journal of Thermal Engineering, Research Article, Vol. 7, No. 2, Special Issue 13, pp. 1-17, February, 2021 2 [8] examined the mixed convection heat transfer aspects with variable fluid flow properties over a non-linear stretching surface. Fluid heating and cooling are important in many industries such as power, manufacturing and transportation. Effective cooling techniques are greatly needed for cooling any sort of high energy device. Common heat transfer fluids such as water, ethylene glycol, and engine oil have limited heat transfer capabilities due to their low heat transfer properties. In contrast, metals have thermal conductivities up to three times higher than these fluids, so it is naturally desirable to combine the two substances to produce a heat transfer medium that behaves like a fluid, but has the thermal of a metal. Since last two decades, study of nanofluid has urged the researcher's attention due to their heat transportation rate. Nanofluid comes in existence when we add a small quantity of nano-sized 10 −9 − 10 −7 particles to the base fluids. Low heat transportation fluids like fluorocarbons, glycol, deionized water, etc. have badly thermal conductivity and therefore deliberated necessary for heat transfer coefficient surrounded by heat transfer medium and surface. The nanoparticles are typically made up of metals ( , ), nitrides ( , ), carbides ( ), oxides ( 2 3 ), or nonmetals (carbon nanotubes, Graphite, etc.) and the base fluid (conductive fluid) is usually water or ethylene glycol. Also, it has been experimentally proved that rate of heat conduction of nanofluids is more than rate of heat conduction of the base fluids. The concept of nanofluid was initially proposed by Choi and Eastman [9] to indicate engineered colloids composed of nanoparticles dispersed in a base fluid. An MIT based comprehensive survey has been done by Buongiorno [10] for convective transportation in nanofluids by considering seven slip conditions that may produce a relative velocity within the base fluid and nanoparticles. Only two (Brownian motion and thermophoresis) out of these seven slip mechanisms were found to be important mechanisms. By adopting Buogiorno's model, Kuznetsov and Nield [11] explored the nanofluid boundary layer uniform convecting fluid flow.
In recent years, MHD fluid flow has gained researchers attention due to its controllable heat transfer rate. Magnetohydrodynamics (MHD) effect also play and influential role in controlling the rate of cooling as well as segregation of molten metal's from various non-metallic impurities. Magnetohydrodynamic (MHD) fluid flow has enormous utilization in manufacturing processes, even in the industrial areas as well. The terminology "Magnetohydrodynamic" is combination of three elementary terms magneto that stands for magnetic field, hydro that stands for fluid / liquid and dynamics that stands for evolution of particles. The existence of external magnetic field gives rise to Lorentz drag force which acts on the fluid, so potentially altering the characteristics of fluid flow especially velocity, temperature and concentration. Grouping of electromagnetism Maxwell's equation and fluid mechanics Navier's stokes equations therefore provides Magnetohydrodynamic (MHD) relation [12,13]. Hayat et al. [14] studied the MHD fluid flow transportation over stretching surfaces. Later, the influence of viscous and Ohmic dissipation (i.e. joule heating) in nanofluid has been presented by Hussain et al. [15]. Vajravelu and Canon [16] studied the flow behavior of fluid towards a non-linear stretching sheet. Further, Matin et al. [17] analyzed the entropy effect in MHD nanofluid flow over stretching surface. Shawky et al. [18] studied the Williamson nanofluid flow in porous medium and he acknowledged that enhancement in non-Newtonian parameter escalates skin friction coefficient along with the rate of heat transfer. Basir et al. [19] examined the consequences of Peclet and Schmidt number in existence of partial slip towards a stretching surface. After that, rate of heat transfer along with partial slip condition was generalized by Pandey and Kumar [20]. Recently, Vinita and Poply [21] discussed MHD slip fluid flow of nanofluid in the existence of free stream velocity or outer velocity towards a stretching surface. Vinita et al. [22] studied MHD fluid flow with variable slip conditions over non-linear stretching surface. Furthermore, non-linearity effect towards the stretching surface under different physical circumstances has been examined by researchers in [6,[23][24][25].
Casson fluid is classified as a non-Newtonian fluid due to its rheological characteristics in relation to the shear stress-strain relationship. It behaves like an elastic solid at low shear strain and above a critical stress value; it behaves like a Newtonian fluid. A Casson fluid can best be described as a shear thinning liquid with infinite viscosity at zero shear rates, and zero viscosity at an infinite rate of shear. Some common examples of liquids that exhibit Casson fluid characteristics include tomato sauce, honey, soup, orange juice and human blood. In 1959, Casson fluid model was pioneered by Casson [26] for prognosis pigment-oil intermission flow. Furthermore, nanofluid boundary layer fluid flow over disparate geometries was premeditated by numerous researchers in [27][28][29][30][31][32][33][34][35][36]. Fluids behave as a solid, if the yield stress is more than applied shear stress. On the other hand, fluid behaves as a liquid for lesser yield stress. Kameshwaran et al. [37] investigated the dual solution in shrinking / stretching surface with Casson fluid flow to 3 determine Lorentz force effect on heat and flow transportation. Also, Khan et al. [38] studied consequence of viscous dissipation on MHD nanofluid flow past a stretching sheet. Afify [39] investigated influence of partial slip chemical reaction and viscous dissipation over Casson nanofluid by considering stretched sheet.
Motivated by the above mentioned literature survey and widespread biological, industrial as well as engineering applications, it is of prime importance to explain the importance of mass and heat transportation of MHD flow past a non-linear stretching sheet with viscous and Ohmic dissipation effects by considering Navier velocity slip conditions. A precise review of already published research discloses that no such attempt has been made earlier even though the fruitfulness of thoughts and phenomenon explained in the present work can be expected to lead to the extremely productive interaction across disciplines. The mathematical difficulties and non-linearity appearing in the Casson nanofluid equations led us to use the numerical approach. The methodology adopted will be to solve the non-linear differential system through Runge-Kutta Fehlberg method. Behaviors of arising variables on concentration, temperature and velocity are graphically interpreted. The outcomes of current study reveal that non-linearity behavior of stretching sheet enhances the values of Sherwood number and skin friction coefficient while declination in value of Nusselt number is noticed. We give qualitative and quantitative comparisons with previously published work to show that our results have highly accurate. This study holds important industrial application, particularly in the field of extrusion where the fluid dispersed with particles is used to augment the strength and durability of the material.

MATERIALS AND METHODS
In present analysis, 2-D incompressible fluid flow in MHD Casson nanofluid over non-linear stretching sheet has been considered. Non-linear behavior generates flow and sheet is stretched in both direction of x axis with stretching velocity = , where , and denotes a constant, stretching surface coordinate and non-linear stretching parameter respectively.
= ∞ + at = , where refers to the positive constant, ∞ refers to the ambient temperature attained and m refers to the physical parameter known as surface temperature parameter. Also, by introducing = , we have a special case of constant surface temperature (CST). Figure 1 represents the physical model of the current study. The continuity, momentum, energy and concentration equations of the incompressible Casson nanofluid boundary layer flow are as follows (see Buongiorno [10], Mustafa and Khan [40], Usman et al. [41], Gireesha et al. [42], Seth et al. [43]) Boundary conditions are given as: Here horizontal and vertical velocities are represented by and , respectively. Also denotes kinematic viscosity, is the density of particle, = ( ) is the thermal diffusivity, is a positive constant, = ( ) ( ) defines a proportion of heat capacities, denotes Casson fluid parameter, reflects thermophoretic diffusion coefficient, refers to slip velocity that is considered to be proportional to wall stress described as: here, refers to the slip length which is proportionality constant for slip velocity. The fundamental equations (1) to (4) with boundary conditions (5) to (6) are transformed using similarity variables [25]  Inserting equation (8) into equations (2) to (4), the governing equations (1)-(4) takes the form The relevant boundary conditions are reduced to where prime denotes derivative with respect to and the key crucial parameters are defined by: Here is the magnetic parameter, is the thermophoresis parameter, is the Schmidt number, is Prandtl number, is the velocity slip parameter, is the Eckert number and is the Brownian motion parameter. It is notes significantly that this boundary value problem condenses into the classical problem of mass, heat and flow transportation because of the stretching surface in a viscous fluid when = 1 and in equations (10) and (11); , are zero (The boundary value problem is of no physical significance and condenses into the classical problem for becomes ill-posed).
It is worth mentioning that equation (9) with the boundary conditions (12), (13); where = 0, are the classical Blasius flat-plate flow problem and a comprehensive numerical analysis of that problem has been conceded by the author of this work. For the linear stretching problem (i.e., when = 1), the exact solution for is ( ) = 1 − − , which was first obtained by the great scientist Crane [1] and this exact solution is unique, while for the non-linear stretching problem (i.e., when ≠ 1), there is no exact solution. Also, the physical quantities of interest skin friction coefficient, local Nusselt number and local Sherwood number are respectively defined as: where , and are wall shear stress, local heat flux and local mass flux at the stretching surface serially given as: The system of partial differential equations (9)-(11) are highly non-linear, hence cannot be solved by using analytical methods. The variational Runge Kutta Fehlberg method has been implemented.

NUMERICAL PROCEDURE
The system of differential equations (9)- (11) are not solved analytically due to extremely non-linear behaviour. To solve this system of non-linear differential equations (9)-(11) together with boundary conditions (12)-(13), we adopt Runge Kutta Fehlberg approach (numerical procedure) and solution is investigated against the variation in Eckert 6 number , Casson fluid parameter , velocity slip parameter , thermophoresis , Brownian motion parameter , Schmidt number , non-linear stretching parameter and magnetic parameter . The value of horizontal axis -is chosen in such a way that a velocity, temperature and concentration profiles asymptotically tends to the boundary condition. All the simulations are conceded with = 10. However, to illustrate the characterization of curve efficiently, much lower values of are used. The major advantage of shooting method over many numerical techniques is that it has fifth order truncation error. Also the process of solution computation is easier as compared to other numerical methods or techniques. For this purpose, the coupled differential equations (9)-(11) are rehabilitated into a set of seven initial value problems by launching the new set of dependent variables (i=1 to 7) as: 3 ′ = 1 The subjected boundary conditions (12) and (13) To solve the above system of differential equations along with the boundary conditions, shooting technique is implemented which consists of the following steps [15]: 1. Choose the appropriate value for the limit ∞ . 2. Next important step is to select good initial approximations for 3 (0), 5 (0) and 7 (0). 3. Therefore the above system of differential equations (19)-(25) is converted into the initial value problem and then it is solved with the help of Runge-Kutta-Fehlberg method. 4. The computed solution will converge if absolute difference of given and computed values of 2 (∞), 4

Validation of Numerical Scheme
In current study, we use RKF method for finding the final outcomes and the adopting numerical process (i.e. shooting technique) is valid. We take constant vale of nanoparticle concentration and temperature at the boundary. For the purpose of verification of the code, we compare our results for local Nusselt number with those of Rana and Bhargava [25] and Cortell [6] for different values of Prandtl number and non-linear stretching parameter by taking the values of Brownian motion parameter and thermophoresis parameter as zero as shown via Table 1. A very good agreement in the results can be seen which strengthens the reliability of the present work (see Figures 3  and 4).

RESULTS AND DISCUSSION
Present study finds numerical solution of differential equations (9) to (11) subjected to the boundary conditions (12) to (13) Table  2  demonstrate  the  impact  of  fluid   A decrease in velocity profile is noticed for higher Casson fluid parameter . This graph elaborates that enhancement in the value of Casson fluid parameter will declines yield stress that hurdles the free movement of fluid particles and hence boundary layer thickness reduces. Thus, fluid velocity decreases. Consequently, declination in the value of skin friction coefficient is observes as seen in Table 2. Figure 6 deliberates the behavior of velocity variation for prominent values of velocity slip parameter (0.00, 0.05, 0.15). This graph exhibits that fluid velocity as well as slip velocity increases for higher values velocity slip parameter . This is due to the presence of velocity slip parameter. Hence, velocity of fluid flow close to the surface of sheet disparate the velocity of stretching sheet. Also, Table 2 gives valuable information that larger value of velocity slip parameter d falls down the numeric value of skin friction coefficient in absolute sense. 2). This figure shows that existence of magnetic parameter resists the fluid particle to move freely and main reason behind the resistance is that magnetic parameter produces Lorentz force and this magnetism behavior can be adopted for controlling the fluid movement. Thus, enhancement in the value of magnetic parameter causes the declination of velocity distribution. Figure 8 portraits the nanofluid velocity variation under the influence of non-linear stretching parameter (1.0, 1.1, 1.2). This figure prevailed that nanofluid velocity decreases due to more interruption surrounded by different layers of fluid and this is because of presence of non-linear stretching parameter . Consequently, velocity gradient increases i.e. increment in ′ ( ) is observed that can be seen in Table 2.  This figure indicates that as Casson fluid parameter increases, fluid temperature decreases and this graph elaborates that enhancement in the value of Casson fluid parameter declines velocity that will reduce the heat transfer rate and hence fluid temperature declines. Additionally, a slight decrease in temperature is noticed as seen in Figure 9. Consequently, enhancement in local Nusselt number Nux is noticed that can be seen in Table 2. Figure 10 shows the variation of velocity slip parameter (0.00, 0.05, 0.15) on temperature distribution ( ). This graph shows that as velocity slip parameter rises, boundary layer thickness declines and this is due to the presence of magnetic field as well as thermal jump of the fluid particles. Thus, nanoparticle temperature distribution declines for higher values of velocity slip parameter and hence rate of heat transportation increases that will increases the value of local Nusselt number. . The striking of atoms or molecules of the fluid particles with each other will create an arbitrary motion called Brownian motion of suspended (pendulous) particles and that will enhances width of boundary layer. Hence, fluid temperature increases for higher Brownian motion parameter and in consequence local Nusselt number decreases. Figure 12 deliberates the impact of fluid temperature under the consequence of thermophoresis parameter (0.2, 0.3, 0.4). Temperature gradient falls down for higher values of thermophoresis parameter that result in reduction of conduction of nanoparticles. Thus, width of boundary layer enhances due to reallocation of ultrafine particles from hotter to colder part and hence, temperature enhances for higher thermophoresis parameter that can be seen in Figure 12. Figure 13 demonstrate fluid temperature variation against Eckert number (0.0, 0.1, 0.2). A dimensionless quantity is the fraction of advective transportation and heat dissipation potential. As Eckert number enhances, thermal buoyancy effect raises that results in increasing temperature and that is the main reason behind the conversion of kinetic energy into thermal energy. Hence, fluid temperature enhances because of this conversion effect. Consequently, declination in Nusselt number is noticed that can be seen via Table 2. Figure 14 reflects variation of temperature distribution against magnetic parameter . With an increase in magnetic parameter , velocity profile decreases because of generation of Lorentz force that consequently intensify the boundary thickness and rate of heat transportation and hence fluid temperature enhances as shown via Figure 14.   Figure 16. Figure 16 prevailed that this effect is more pronounced for steady state flow. Figure 17 manifests the impact of velocity slip parameter (0.00, 0.05, 0.15) on nanoparticle concentration ( ). With an increase in the value of velocity slip parameter , nanoparticle concentration enhances that consequently declines the numeric value of local Sherwood number as seen in Table 2. 13  Table 2. Figure 19 portraits variation for nanoparticle volume fraction ( ) against thermophoresis parameter (0.2, 0.3, 0.4). This graph shows that with an increase in thermophoresis parameter, nanoparticle concentration increases. Basically, in case of thermophoresis force applied by a particle on the other particle will generates the movement of particles from hotter to colder part and hence fluid moves from hotter to colder region and hence intensification in the nanoparticle volume fraction is observed via Figure 19.  Figure 21 reflects the variation for nanoparticle concentration ( ) against the magnetic parameter (0.8, 1.0, 1.2). With increase in magnetic parameter , rate of mass transportation decreases that consequently increase nanoparticle concentration and hence reduction in the value of local Sherwood number is notice as seen in Table 2.