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Casson fluid flow over a vertical porous surface with chemical reaction in the presence of magnetic field has been studied. A similarity analysis was used to transform the system of partial differential equations describing the problem into ordinary differential equations. The reduced system was solved using the Newton Raphson shooting method alongside the Forth-order Runge-Kutta algorithm. The results are presented graphically and in tabular form for various controlling parameters.

A fluid in which the viscous stresses arising from its flow at every point are linearly proportional to the rate of change in its deformation over time is called Newtonian fluid. This means that in a Newtonian fluid, the relationship between the shear stress and the shear rate is linear with the proportionality constant to refer to as the coefficient of viscosity. On the other hand, a fluid whose flow properties are different in any way from that of the Newtonian fluid is called a non-Newtonian fluid. Unlike the Newtonian fluids, the viscosity of non-Newto- nian fluid is dependent on shear rate history. That is to say, in a non-Newtonian fluid, the relationship between the shear stress and the shear rate is different and can even be time dependent. Thus a constant coefficient of viscosity cannot be defined. Some examples of non-Newtonian fluids are salt solutions, molten polymers, ketchup, custard, toothpaste, starch suspensions, paints, blood and shampoo.

It is important to note here that, many fluids of industrial importance are non-Newtonian. It is now generally recognized that, in real industrial applications, non-Newtonian fluids are more appropriate than Newtonian fluids, due to their applications in petroleum drilling, polymer engineering, certain separation processes, manufacturing of foods and paper and some other industrial processes [

Casson fluid can be defined as a shear thinning liquid which is assumed to have an infinite viscosity at zero rate of shear, a yield stress below which no flow occurs and a zero viscosity at an infinite rate of shear [

Fredrickson [

The viscous fluid flow due to a stretching flat sheet was first investigated by Crane [

From literature, it can be found that not much attention is given to the Casson fluid flow over a porous vertical surface with chemical reaction in the presence of magnetic field. The increasing use of several non-Newtonian fluids in processing industries has motivated a study to understand their behaviour in several transport processes. Therefore, in this investigation, the steady incompressible Casson fluid flow and mass transfer towards a porous vertical stretching sheet are studied. The governing partial differential equations are converted into systems of nonlinear ordinary differential equations (ODE) using the suitable similarity transformations. The transformed self-similar ODEs are solved by shooting method: an efficient numerical method for solving boundary value problem [

Consider a two-dimensional steady incompressible Casson fluid flow over a vertical porous stretching surface at y = 0 in the presence of a transverse magnetic field, as shown in _{0} in the y direction. The magnetic Reynolds number is considered to be very small so that the induced magnetic field is

negligible in comparison to the applied magnetic field. The tangential velocity u_{w}, due to the stretching surface is assumed to vary proportionally to the distance x so that u_{w} = ax, where a is a constant.

The rheological equation of state for an isotropic flow of a Casson fluid [

In Equation (1),_{;} and e_{ij} is the (i, j)^{th} component of the deformation rate, π is the product of the component of deformation rate with itself, π_{c} is a critical value of this product based on the non-Newtonian model, μ_{B} the plastic dynamic viscosity of the non-Newtonian fluid and P_{y} is the yield stress of the fluid. If u and v are the fluid x- and y-components of velocity respectively; and C being the concentration field; then the equations governing the steady boundary layer flow of the Casson fluid are:

Subject to the following boundary conditions:

where _{m} is

the mass diffusion, γ is the reaction rate, v_{0}(x) is the suction velocity from the surface, C_{w} is the concentration at the surface, C_{∞} is the free stream concentration, β_{c} is the solutal expansion coefficient, ρ is the fluid density, g is gravitational acceleration, and σ is the electrical conductivity.

The following dimensionless quantities are introduced:

Substituting Equation (5) in (2)-(4) yields:

The transformed boundary conditions are

The prime symbol denotes differentiation with respect to the similarity variable η, where

parameter,

The numerical technique chosen for the solution of the coupled ordinary differential Equations (7)-(8) together with the associated transformed boundary conditions (9) is the standard Newton-Raphson shooting method alongside the fourth-order Runge-Kutta integration algorithm. From the process of numerical computation, the plate surface temperature, the local skin-friction coefficient, the local Nusselt number and the local Sherwood number, which are respectively proportional to −f″(0) and −ϕ′(0) are computed and their numerical values presented in a tabular form.

_{C}. This means that the combined effect of magnetic field, Casson parameter, Schmidt number, reaction rate parameter and suction parameter is to increase the local skin friction; whereas that of the buoyancy force is to decrease the local skin friction at the surface of the plate. Moreover, it is observed that the rate of mass transfer increases with increasing values of fw, Gc, Sc and B; and decreases with increasing values of M and β.

Figures 2-5 show the effects of the magnetic parameter (M), suction parameter (fw), Casson parameter (β), and local solutal Grashof number (Gc), respectively, on the velocity profile, f′(η). Generally, the fluid velocity is minimal at the plate surface and increases to the free stream value satisfying the far field boundary conditions.The effects of magnetic parameter (M) and the suction parameter (fw) on velocity profiles are seen in

B | Sc | [ | [ | [ | Present Study |
---|---|---|---|---|---|

0.01 | 1.0 | 0.59157 | 0.592 | 0.59136 | 0.591382 |

0.10 | 1.0 | 0.66902 | 0.669 | 0.66898 | 0.668983 |

1.00 | 1.0 | 1.17649 | 1.177 | 1.17650 | 1.176499 |

10.00 | 1.0 | 3.23122 | 3.232 | 3.23175 | 3.231228 |

M | Β | Gc | Sc | B | fw | −f″(0) | -ϕ′(0) |
---|---|---|---|---|---|---|---|

0.5 | 0.5 | 0.1 | 0.6 | 0.3 | 0.1 | 0.701894 | 0.675765 |

0.7 | 0.5 | 0.1 | 0.6 | 0.3 | 0.1 | 0.747866 | 0.670528 |

1.0 | 0.5 | 0.1 | 0.6 | 0.3 | 0.1 | 0.812075 | 0.663476 |

1.5 | 0.5 | 0.1 | 0.6 | 0.3 | 0.1 | 0.909246 | 0.653385 |

0.5 | 0.3 | 0.1 | 0.6 | 0.3 | 0.1 | 0.584131 | 0.690091 |

0.5 | 1.5 | 0.1 | 0.6 | 0.3 | 0.1 | 0.942483 | 0.650258 |

0.5 | 2.0 | 0.1 | 0.6 | 0.3 | 0.1 | 0.993805 | 0.645458 |

0.5 | 0.5 | 0.5 | 0.6 | 0.3 | 0.1 | 0.615589 | 0.684487 |

0.5 | 0.5 | 1.0 | 0.6 | 0.3 | 0.1 | 0.511641 | 0.694304 |

0.5 | 0.5 | 1.5 | 0.6 | 0.3 | 0.1 | 0.411199 | 0.703204 |

0.5 | 0.5 | 0.1 | 0.5 | 0.3 | 0.1 | 0.700321 | 0.607036 |

0.5 | 0.5 | 0.1 | 1.0 | 0.3 | 0.1 | 0.705957 | 0.911669 |

0.5 | 0.5 | 0.1 | 1.5 | 0.3 | 0.1 | 0.708792 | 1.155165 |

0.5 | 0.5 | 0.1 | 0.6 | 0.5 | 0.1 | 0.703184 | 0.764950 |

0.5 | 0.5 | 0.1 | 0.6 | 1.0 | 0.1 | 0.705420 | 0.949749 |

0.5 | 0.5 | 0.1 | 0.6 | 1.5 | 0.1 | 0.706927 | 1.102396 |

0.5 | 0.5 | 0.1 | 0.6 | 0.3 | 0.5 | 0.774640 | 0.831437 |

0.5 | 0.5 | 0.1 | 0.6 | 0.3 | 1.0 | 0.874223 | 1.047186 |

0.5 | 0.5 | 0.1 | 0.6 | 0.3 | 1.5 | 0.982768 | 1.281483 |

It is observed in

Figures 6-11 show the plots of the effects of the magnetic parameter (M), suction parameter (fw), Casson parameter (β), Schmidt number (Sc) and chemical reaction parameter (B) on the concentration profile, ϕ (η) respect-tively. It is observed in

to increase concentration thereof. The concentration profile decreases with increasing fw as shown in

At a point in the flow where B is zero implies no chemical reaction. On the other hand, an increase in B means an increase in the chemical reaction rate which causes a reduction in concentration.

An analysis of Casson fluid flow over a vertical porous surface with chemical reaction in the presence of a transverse magnetic field has been presented. Numerical results have been compared to earlier results published in the literature and a perfect agreement was achieved. Among others, our results reveal that:

1) The velocity decreases with the increase in values of M, fw and β; and increases with increasing values of Gc.

2) The concentration boundary layer decreases with increasing values of fw, Gc, Sc and B; and increases with increasing values of M and β.

3) The skin friction at the surface increases with increasing values of M, fw, β, Sc and B; and decreases for increasing values of Gc.

4) The rate of mass transfer at the surface increases with increasing values of fw, Gc, Sc and B; and decreases with increasing values of M and β.