The basic objective of this work is to study the heat transfer of Casson fluid of non-Newtonian nature. The fluid is considered over a vertical plate such that the plate exhibits arbitrary wall shear stress at the boundary. Heat transfers due to exponential plate heating and natural convection are due to buoyancy force. Magnetohydrodynamic (MHD) analysis in the occurrence of a uniform magnetic field is also considered. The medium over the plate is porous and hence Darcy’s law is applied. The governing equations are established for the velocity and temperature fields by the usual Boussinesq approximation. The problem is first written in dimensionless form using some useful non-dimensional quantities and then solved. The exact analysis is performed and hence solutions via integral transform are established. The analysis of various pertinent parameters on temperature distribution and velocity field are reported graphically. It is found that pours medium permeability parameter retards the fluid motion whereas, velocity decreases with increasing magnetic parameter. Velocity and temperature decrease with increasing Prandtl number whereas the Grashof number enhances the fluid motion. Further, it is concluded from this study that the results obtained here are more general and in a limiting sense several other solutions can be recovered. The Newtonian fluid results can be easily established by taking the Casson parameter infinitely large i.e., when

Non-Newtonian fluids have various important industrial applications in many fields [

The goal of this paper is to inspect the exact study of heat transfer of Casson fluid through a perpendicular plate with random shear stress at the bounding wall which at the same time executing exponential heating. The problem is solved for exact solution via integral transform of Laplace. The special cases are also derived from the general solution. The outcomes for temperature and velocity are schemed and discussed graphically.

Unsteady Casson fluid with mixed convection is considered. The flow is due to the arbitrary shear stress and natural convection through an unbounded vertical plate. The plate is heated exponentially.

At beginning the fluid as well as plate are at rest with constant temperature

The physical boundary conditions are:

where _{p}

The appropriate dimensionless variables are,

Into

where _{1}_{p}

_{p}

By Laplace transform the exact solution of the

And the corresponding Nusselt number is obtained as

and the velocity profile is

where

where

The different physical parameters analysis for velocity and temperature is schemed graphically and discussed by using Mathcad-15 software. The influence of Casson parameter

The _{p}_{p}

_{p}

The exact analysis for mixed convection Casson fluid is obtained. Where the plate exhibits arbitrary wall shear stress at the boundary over a vertical plate. Heat transfers because of exponential plate heating is also considered. The main equations with suitable initial and boundary condition is recognized for temperature and velocity. The Laplace technique is used to find the exact solution for velocity and temperature profile. The effects of different parameters are discussed graphically for temperature and velocity. The main conclusions are derived from this article are:

The embedded parameters _{p}

The