Shifted Legendre Collocation Analysis of Time-Dependent Casson Fluids and Carreau Fluids Conveying Tiny Particles and Gyrotactic Microorganisms: Dynamics on Static and Moving Surfaces

Abstract

With an emphasis on the dynamics of (a) time-dependent fluids exhibiting plastic dynamic viscosity, (b) time-dependent fluids exhibiting limiting viscosities at zero, and at infinite shear rate such that each transport phenomenon conveys tiny particles and gyrotactic microorganisms, there is little or no method of solution for further analyses of the transport phenomenon when the migration of the tiny particles due to temperature gradient and its haphazard movement is strongly influenced by the fluid's concentration. Shifted Legendre Collocation method was developed and used to obtain the solution of the coupled, nonlinear, and dimensionless form of the dimensional Partial Differential Equation that models the transport phenomenon mentioned above, starting with the closed-form of Legendre polynomials (LPs) and considering shifted LPs that are orthogonal over the interval [− 1,1] with weighting function equivalent to unity. Based on the analysis of the given data, it is reasonable to conclude that Casson fluid has high values for local skin friction coefficient in the static wedge situation. In the Carreau fluid for moving wedge scenario, maximum values for local Nusselt number, local Sherwood number, and local density of motile microbe number are also mentioned.

Appendix 1: Non-dimensional Parameters and Their Expression

Mathematical Expression	Name of the parameter	Mathematical Expression	Name of the parameter
$We=\sqrt{\frac{{\Gamma }^{2}\left(m+1\right){U}_{b}^{3}}{2\nu x}}$	Weissenberg number	$\xi =\frac{{U}_{w}}{{U}_{b}}$	Velocity ratio parameter
$A=\frac{2c}{\left(m+1\right)a{x}^{m-1}}$	Unsteadiness parameter	$\mathrm{Pr}=\frac{k}{\nu \rho {C}_{p}}$	Prandtl number
$\Lambda =\frac{2x{\beta }^{*}g\left(1-{C}_{\infty }\right)({T}_{w}-{T}_{\infty })}{{U}_{b}^{2}}$	Mixed convection parameter	$\omega =\frac{16{\sigma }^{*} {T}_{\infty }^{3}}{3{k}^{*}{k}_{f}}$	Radiation parameter
${N}_{r}=\frac{({\rho }_{p}-{\rho }_{f})({C}_{w}-{C}_{\infty })}{{\rho }_{f}{\beta }^{*}\left(1-{C}_{\infty }\right)({T}_{w}-{T}_{\infty })}$	Buoyancy ratio parameter	$Sc=\frac{\nu }{{D}_{B}}$	Schmidt number
${R}_{b}=\frac{\gamma ({\rho }_{m}-{\rho }_{f})({N}_{w}-{N}_{\infty })}{{\rho }_{f}{\beta }^{*}\left(1-{C}_{\infty }\right)({T}_{w}-{T}_{\infty })}$	Bio-convection Rayleigh parameter	$S{c}_{b}=\frac{\nu }{{D}_{m}}$	Bio-convective Schmidt number
${N}_{b}=\frac{\tau {D}_{B}}{\nu }$	Brownian motion parameter	$Pe=\frac{b{W}_{c}}{{D}_{m}}$	Bio-convective Peclet number
${N}_{t}=\frac{\tau {D}_{T}({T}_{w}-{T}_{\infty })}{{\alpha }_{m}{T}_{\infty }}$	Thermophoresis parameter	$\delta =\frac{{N}_{\infty }}{{N}_{w}-{N}_{\infty }}$	Microorganism concentration difference
$H{a}^{2}=\frac{\sigma {B}_{0}^{2}}{\rho a{x}^{m-1}}$	Hartmann number