Aligned magnetic field of two-phase mixed convection flow in dusty Casson fluid over a stretching sheet with Newtonian heating

The effect of aligned magnetic field is numerically investigated for mixed convection flow of dusty Casson fluid over a stretching sheet. The governing equations of flow and heat transfer for the two-phase model (fluid and dust) with an appropriate thermal boundary condition which is Newtonian heating (NH) is presented. The similarity transformation is employed to transform the nonlinear governing equations for each phase into the ordinary differential equations which then solved numerically using Runge-Kutta Fehlberg (RKF45) method. Numerical solutions obtained for velocity and temperature distributions are illustrated through graph by varying several physical parameters. It is observed that the fluid velocity decreases with an increase in aligned magnetic field and particle-fluid interaction parameter.


Introduction
Engineering devices has many applications which embedded of mixed convection flow involving the heat transfer processes. This process also can be seen on atmospheric boundary layer flow, heat exchangers, solar collector, nuclear reactors and electronic equipment [1]. If the free convection and forced convection behave together then the mixed convection is arises in that flow. Such flow happened due to the difference temperature at the surface and ambient fluid which develop the buoyancy force. There have been several studies in the literature reporting the mixed convection flow for various physical aspects and geometry [2][3][4][5][6][7].
The above existing literatures are only restricted on the single flow (pure fluid). Actually, the foreign bodies or impurities in the form of shoot or ash may contain in a fluid. Thus, the development of mathematical model for predicting the accumulation of particles in fluid will help to determine its rheology for investigation the flow and heat transfer process of fluid. [8] studied the non-interacting particles in fluids over a semi-infinite flat plate. [9] and [10] discussed the effect of magnetic on the boundary layer flow of viscous fluid with dust particle, meanwhile [11] focuses on dusty nanofluid flow with the influence of buoyancy parameter. The non-Newtonian fluid of Casson model has been extensively used in food industries and is generally applied in manufacturing of chocolate [12]. The objective of the present paper is to determine whether the present of dust particles influences the behaviour of Casson fluid on the mixed convection flow over a vertical stretching sheet. Here, the Casson fluid is interpreted as chocolate while the dust particle as copper. The mixture of chocolate and copper is numerically studied and hence it is expected to provide the understanding of two-phase boundary layer flow of non-Newtonian fluid contains dust particles.  Under the boundary layer and Boussinesq approximations, the governing equations for two-phase flow as adopted from [13] and [14] can be written as: Fluid phase:

Mathematical Model
Dust phase: where ( , ) u v and ( , ) where a is positive constant and s h is heat transfer parameter. The following similarity transformations were applied on equations (1)-(6) ( ) where ψ is the stream function defined as ψ Equations (1)-(6) are now transformed to ordinary differential equations as follow and the boundary conditions (7) where a prime ( ) ′ denotes differentiation with respect to η .
is the mass concentration of particle phase, The exact solution of equation (9) which considering only Casson parameter is obtained by [15] and can be expressed as ( ) The exact expression for temperature profile by using equation (14) is

Numerical procedure
The numerical computation of equations (9)

Results and discussion
The mixture of chocolate with copper is discussed and their thermophysical properties are shown in table 1. The physical parameters of aligned angle 1 α , magnetic field parameter M , particle-fluid interaction parameter β and buoyancy parameter λ are keeping to be fixed throughout the study in   where both parameters are influencing each other as denoted in fifth term of equation (9), so it can be seen from the figures that the curves at that respective value are overlapping.
The variation of particle-fluid interaction parameter β on the velocity and temperature for fluid and dust phases are presented in figures 4-5. The decreasing pattern of velocity and temperatures profiles is revealed in these figures as the β increases (decreases) for dust (fluid) phase. Theoretically, a large β reduces the relaxation time of particle phase and turns to develop the drag force on the fluid in contact with it. It is revealed from the figures that the dust phase will become equilibrium with the fluid phase for the bigger β . At this stage, the velocity and temperatures profiles for both phases become parallel.

Conclusions
The numerical solution of dusty Cassson fluid model over a heated stretching sheet subjected to NH is presented with the applied aligned magnetic field and buoyance force on the flow region. The governing equations for the fluid and dust phases are studied separately and the obtained numerical results of flow characteristic and heat transfer process for each phases are presented graphically. Based on the present findings, the flow of Casson fluid is affected by the present of dust particle. The understanding of the behaviour of two-phase flow can help to predict the flow processes of the contaminated fluid.