Unsteady squeezing flow of a magnetized dissipative non-Newtonian nanofluid with radiative heat transfer and Fourier-type boundary conditions: numerical study

Abstract

Improved thermal management in high-temperature tribological systems requires novel developments in lubricants. Motivated by combining nanoparticle and magnetorheological plastomer features, this research paper deals with the analysis of the high-temperature magnetohydrodynamic squeeze flow of a Casson nanofluid between parallel disks with the Fourier-type boundary conditions including radiation. Rosseland’s diffusion flux and the Buongiorno nanoscale model are used. Suction and injection effects at the disks are also considered as is viscous heating. Robin (Fourier) boundary conditions are included, and the Buongiorno nanoscale model is used which enables the simulation of nanoparticle mass diffusion, Brownian motion and thermophoresis. The emerging nonlinear boundary value problem is solved with the bvp4c routine in MATLAB routine with appropriate boundary conditions at the disks. The effects of squeeze number, Hartmann number, Brownian motion parameter, Prandtl number, Eckert number, thermophoresis parameter, Casson viscoplastic rheological parameter and thermal radiation parameter for both disk suction and injection cases and also with equivalent and different Biot numbers at the disks are presented graphically. MATLAB solutions are validated with earlier published results. Drag force increases with greater magnetic field strength. Increasing squeezing parameter substantially modifies the velocity distribution, causing a deceleration near the disk surfaces but an acceleration further from the disks. Elevation in Prandtl number and Eckert number results in a significant enhancement in temperature but a strong depletion in nanoparticle concentration for both equal and unequal Biot numbers at the disk surfaces. Nanoparticle concentration is depleted at the disk surfaces with increasing Brownian motion parameter values. With an increase in the Casson viscoplastic parameter, temperature decreases, i.e., cooling is induced, whereas nanoparticle concentration increases. The simulations show that significant temperature elevation is produced with increasing Brownian diffusion, viscous dissipation and radiative flux effects and that combining nanoparticles and viscoplastic effects offers a good thermal management mechanism in squeezing lubrication.

Appendix

The rheological equation of the Casson fluid is defined (following Mohyud-Din and Khan [25] and Khan et al. [26]) as follows:

$$ \tau_{ij} \,\, = \,\,\left[ {\mu_{B} \,\, + \,\,\left( {\frac{{p_{y} }}{2\pi }} \right)^{1/n} } \right]^{n} \,\,2\,\,e_{ij} $$

(A1)

where $\mu_{B}$ is the dynamic viscosity of the non-Newtonian fluid, $p_{y}$ is the yield stress of the fluid and $\pi$ is the product of the component of deformation rate with itself, i.e., $\pi \, = \,e_{ij} \,e_{ij}$ (self-product of component of deformation rate with itself) where $e_{ij}$ is the $\left( {i,j} \right)^{th}$ component of the deformation rate. For n < 1, the fluid is pseudoplastic (shear-thinning); for n > 1 it is dilatant (shear-thickening). Further:

$$ \tau_{ij} \,\, = \,\,\left\{ \begin{gathered} 2\,e_{ij} \,\,\left[ {\frac{{p_{y} }}{{\sqrt {2\pi } }}\,\, + \,\,\mu_{B} } \right]\,\,\,\pi \,\, > \,\,\pi_{c} \, \hfill \\ 2\,e_{ij} \,\,\left[ {\frac{{p_{y} }}{{\sqrt {2\pi_{c} } }}\,\, + \,\,\mu_{B} } \right]\,\,\pi \,\, < \,\,\pi_{c} \hfill \\ \end{gathered} \right\}\,\, $$

(A2)

Here, $\pi_{c}$ is the critical value of the said self-product. If shear stress is less than the yield stress applied to the fluid, the fluid acts like a solid, whereas if shear stress exceeds the yield stress, motion is initiated. Examples of Casson fluid are jelly, foodstuffs, tomato sauce, gels, honey, certain polymers, soup, blood under certain shear rates, etc.