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Unsteady flow of Casson fluid past through a vertical channel has been studied by some researchers due to its importance of applications in science and technology. Therefore, the main purpose of this paper is to obtain exact solutions for unsteady free convection flows of Casson fluid with effects of magnetohydrodynamics (MHD) past through vertical channel. This paper is continued study from published article [18] with additional effects of magnetohydrodynamics (MHD). Dimensional governing equations are converted into dimensionless forms by using appropriate dimensionless variables. Dimensionless parameters are obtained through dimensionless process such as Casson fluid, time, Prandtl number, Grashof number and magnetic field. Laplace transform method is used to solve the dimensionless equations with associated initial and boundary conditions. Solutions for velocity and temperature profiles are obtained. Skin friction and Nusselt number are also calculated. The obtained analytical results for velocity and temperature are plotted graphically to discuss the influence of dimensionless parameters on profiles. It is observed that fluid velocity increases with increases of Grashof number, Gr and time, t whereas it decreases with increases of Casson parameter, γ, magnetic field parameter, M and Prandtl number, Pr. Besides that, it is found that temperature profiles decrease with high value of Prandtl number, Pr while increases with high value of time, t. In order to validate the results, the obtained results in limiting cases are compared with the published results and it is found to be in a mutual agreement.

This study investigates radiation effect on unsteady oscillatory free convection between two parallel plates with mass diffusion. Appropriate non-dimensional variables are used to reduce the dimensional governing equations to dimensionless equation along with imposed initial and boundary conditions. The solutions for velocity, temperature, and concentration profiles are obtained by using Laplace transform method. To illustrate the behavior of the fluid flow, graphical results are presented with influenced of Schmidt number, Prandtl number, radiation parameter, oscillating parameter, Grashof, and mass Grashof number. The corresponding expressions for skin friction, Nusselt number, and Sherwood number are also calculated. It is observed that increasing radiation parameter, Prandtl, and Schmidt number will increase the Nusselt number but the skin friction will be reduced.

Free convection flow in a boundary layer region is a motion that results from the interaction of gravity with density differences within a fluid. These differences occur due to temperature or concentration gradients or due to their composition. Studies pertaining free convection flows of incompressible viscous fluids have received much attention in recent years both theoretically (exact or approximate solutions) and experimentally. The situation where the heat be transported to the convective fluid via a bounding surface having finite heat capacity is known as Newtonian heating (or conjugate convective flows). In this paper, the unsteady free convection flow of an incompressible viscous fluid between two parallel plates with Newtonian heating is studied. Appropriate non-dimensional variables are used to reduce the dimensional governing equations along with imposed initial and boundary conditions into dimensionless forms. The exact solutionsfor velocity and temperature are obtained using the Laplace transform technique. The corresponding expressions for skin friction and Nusselt number are also calculated. The graphical results are displayed to illustrate the influence of various embedded parameters such as Newtonian heating parameter and Grashof number. The results show that the effect of Newtonian heating parameter increases the Nusselt number but reduces the skin friction.

The effects of ramped wall temperature, rotation and porosity on mixed convection flow of incompressible second grade fluid are studied. The momentum equation is modelled in a problem of rotating fluid with constant angular velocity subjected to initial and oscillating boundary conditions. The energy equation is also introduced. Some suitable non-dimensional variables are used to write equations into non-dimensional form. Laplace transform method is used to solve these equations in order to obtain the analytical solutions of velocity and temperature profiles. Computations are carried out and presented graphically to analyse the effect of second grade fluid parameter, rotation parameter, porosity parameter, Prandtl number and Grashof number on the profiles. It is found that, for larger values of porosity parameter, the fluid velocity will increase for both primary and secondary velocities. The results also show that, velocity for ramped wall temperature is lower compared to isothermal temperature. It is worth to mention that, the exact solutions obtained in this study can be used to check correctness of the results obtained through numerical schemes.

Heat transfer on an unsteady free convection rotating second grade fluid flow, which is flowing through an accelerated plate, is analyzed. The physical problem investigated is described by a coupled, linear system of partial differential equations, with appropriate boundary conditions. Laplace transform technique is applied to determine the analytical solutions of the dimensionless governing equations. The effects of various embedded parameters to the velocity and temperature distribution in the fluid are graphically illustrated and analyzed. The obtained analytical results constitutes a good verification to verify a more advance situation of the physical problem, which is described by a non-linear system and only can be solved by using approximation method.

The unsteady magnetohydrodynamic (MHD) free convection flow of Jeffrey fluid embedded in porous medium past an oscillating vertical plate generated by thermal radiation with ramped wall temperature is investigated. The incompressible fluid is taken electrically conducting under the action of transverse magnetic field towards the flow. Constitutive relation of Jeffrey fluid is employed to model the governing equations in terms of partial differential equations with some physical conditions. The transformed dimensionless governing equations are solved analytically using Laplace transform technique. The impact of various pertinent parameters namely material parameter of Jeffrey fluid , dimensionless parameter of Jeffrey fluid , phase angle , Hartmann number , permeability parameter , Grashof number , Prandtl number , radiation parameter and dimensionless time on velocity and temperature distributions are presented graphically and discussed in details. It is observed that, the permeability parameter tend to retard the fluid velocity for ramped wall temperature but enhance the velocity for an isothermal plate. Besides that, this study shows, the amplitude of velocity and temperature fields for ramped wall temperature are always lower than isothermal plate. A comparison with the existing published work is also provided to confirm the validity of the present results and an excellent agreement are found.

In this paper, the exact solutions for unsteady free convection flow of rotating second grade fluid over an isothermal oscillating vertical plate are investigated. This phenomenon is modeled in the form of partial differential equations with initial and boundary conditions. Some suitable non dimensional variables are introduced. The corresponding non-dimensional equations with conditions are solved using Laplace transform technique. Exact solutions for velocity and energy profiles are obtained. They are expressed in simple forms in terms of exponential and complementary error functions of Gauss. It is found that they satisfy governing equations and conditions imposed. Computations are carried out and the results are analyzed for various emerging parameters.