Pulsating flow of Casson nanofluid through a channel with thermal radiation and applied magnetic field

The pulsating flow of MHD Casson nanofluid flow through a channel with the influence of thermal radiation is studied, considering the nanofluid flow in the vertically upward direction, maintaining the same velocity at the lower wall and upper wall. The perturbation technique was employed to solve the dimensionless expressions of velocity and temperature. The effect of various parameters such as Prandtl number and Casson fluid parameter on flow variables has been discussed.


Introduction
The Casson fluid is a non-Newtonian fluid with yield pseudo plastic behaviour. It displays an infinite shear rate with low apparent viscosity and, zero shear rates at infinite apparent viscosity. The flow of MHD Casson fluid in a channel with free convection investigated by Sheikh et al. [1]. the characteristics of Casson fluid on a permeable Riga-plate was analysed by Loganathan et al. [2]. The term 'nano fluid' is a fluid containing a suspension of solid nanoparticles which enhances or reduces the thermal conductivity of base fluid. Pulsating flow of Casson nano-fluid in a channel has enormous application in the process of flow of blood in heart, transpiration cooling. Combined forces and a flow free convection past a porous channel with a pulsating pressure is discussed by Bestman [3]. The emerging applications and advanced studies are reported in [4][5][6].
Magnetohydrodynamic fluid flow has enormous application in various engineering and industrial processes. Shehzad et al. [7] studied the effects of mass transfer on MHD flow of Casson fluid over a stretching surface. The study related to mass and heat transfer with thermal radiation has abundant appliance in much industrial and engineering process. This work deals with the effect of various parameters on the pulsating flow of MHD Casson nanofluid. The dimensionless expressions of velocity and temperature are solved by applying perturbation techniques and discussed the influence of some parameters on velocity and temperature.

Mathematical formulation:
Consider the uniform traverse applied magnetic field on the pulsatile flow of an incompressible Casson nano fluid with a time dependent pressure gradient in a channel between two fixed plates. [9,10,11]  where, ߝ ≪ 1 is suitably chosen positive quantity, A is a constant and ω is the frequency. We take a Cartesian coordinate system with ܺ −axis taken along the lower wall and ܻ −axis perpendicular to it. The lower wall maintaining the temperature ܶ and the upper wall maintains the temperature ܶ ଵ (ܶ < ܶ ଵ ). The nano fluid is injected and sucked out from the lower wall and upper wall respectively with the same velocity v0. The magnetic field is taken along the normal direction. The rheological equation for the Casson nano fluid is defined as follows [10].
The governing equation are defined below The boundary conditions for the present analysis are Here is restricted to spherical nano particles only, it does not account for other shapes of nano particles.
ܳ is the heat source / sink, ߚ = ߤ ඥ2π ୡ /p ୷ is Casson parameter, ܶ * is the temperature of the nanofluid. Applying Rosseland approximation for radioactive heat flux, ‫ݍ‬ is defined as: Where ‫ܭ‬ ഥ is the Rosseland mean absorption co-efficient.

Method of solution
The velocity u and temperature ϴ can be assumed to have the form Now substitute equations (11) and (12) into the equations (9) and (10) here m's and B's are given in Appendix.

Results and discussion
The  Figure. 3(a & b). From Figure.4(a) it is noticed that the unsteady temperature oscillating with declining Radiation parameter (ܴ݀) and attaining maximum near the walls. From Figure. 4(b) one can notice that the steady temperature declines for enhancement in ܴ݀.  Table. 1. The nano fluid is taken as ߮ = 0.1. One can notice from the table that for the base fluid and nano fluid, the Nusselt number distribution enhances at the lower wall as increasing ‫,ܯ‬ ܴ݁ and ܴ݀, but the opposite effect can be seen in ܳ. However, this behaviour is reversed at the upper wall. It is also observed that with the larger estimation of nano particle volume fraction (߮) the Nusselt ‫)ݑܰ(‬ increases at the upper wall, while it is diminishing at the lower wall.

Conclusion
The pulsating flow of Casson nanofluid with thermal radiation and applied magnetic field in a channel between two fixed plates in the presence of heat sink/source has been studied. The analytic perturbation technique was applied to solve the flow equations. The major results of the present investigation are as follows: The velocity of the nano fluid increases with larger approximation of frequency parameter (H), Casson parameter ( ), nano particle volume fraction (߮), but the opposite effect can be seen in M. The temperature of the nano fluid increases with a rise in nano particle volume fraction (߮) and Prandtl number ‫.)ݎܲ(‬ While the results are reversed for the rise in cross Reynolds number (Re) and the radiation parameter (Rd). The steady and unsteady temperature distributions of the nanofluid is an increasing function of ߮, while steady temperature is decreasing, and unsteady temperature oscillates for given rise in Rd. The Nusselt number (Nu) decrease at the lower wall and enhances at the upper wall with an increase in the nanoparticle volume fraction (߮), and the heat source.