# MHD mixed convection stagnation-point flow of a nanofluid over a vertical permeable surface: a comprehensive report of dual solutions

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## Abstract

The steady laminar magnetohydrodynamic mixed convection boundary layer flow of a nanofluid near the stagnation-point on a vertical permeable plate with prescribed external flow and surface temperature is investigated in this study. Here, both assisting and opposing flows are considered and studied. Using appropriate similarity variables, the governing equations are transformed into nonlinear ordinary differential equations in the dimensionless stream function, which is solved numerically using the Runge–Kutta scheme coupled with a conventional shooting procedure. Three different types of nanoparticles, namely copper Cu, alumina Al_{2}O_{3} and titania TiO_{2} with water as the base fluid are considered. Numerical results are obtained for the skin-friction coefficient and Nusselt number as well as for the velocity and temperature profiles for some values of the governing parameters, namely, the volume fraction of nanoparticles *ϕ*, permeability parameter *f* _{ o }, magnetic parameter *M* and mixed convection parameter *λ*. It is found that dual solutions exist for both assisting and opposing flows, and the range of the mixed convection parameter for which the solution exists, increases with suction, magnetic field and volume fraction of nanoparticles.

## Keywords

Mixed Convection Base Fluid Thermal Boundary Layer Heat Transfer Characteristic Dual Solution## List of symbols

*a, b*Constant

*C*_{f}Skin friction coefficient

*g*Acceleration due to gravity

*k*Thermal conductivity

- \( Nu_{x} \)
Local Nusselt number

- \( Gr_{x} \)
Local Grashof number

- \( Re_{x} \)
Local Reynolds number

*Pr*Prandtl number

*q*_{w}Surface heat flux

*T*Fluid temperature

*T*_{w}Surface temperature

- \( T_{\infty } \)
Ambient temperature

*u, v*Velocity components

*x, y*Cartesian coordinates

*U*(*x*)Free stream velocity

*f*(*η*)Dimensionless stream function

## Greek symbols

*α*Thermal diffusivity

*β*Thermal expansion coefficient

*ϕ*Nanoparticle volume fraction

*η*Similarity variable

*θ*(*η*)Dimensionless temperature

*λ*Buoyancy or mixed convection parameter

*μ*Dynamic viscosity

*υ*Kinematic viscosity

*ρ*Fluid density

- \( \tau_{w} \)
Wall shear stress

*Ψ*Stream function

## Subscripts

*w*Condition at the surface of the plate

- \( \infty \)
Ambient condition

*f*Fluid

*nf*Nanofluid

*s*Solid

## Superscript

- \( ^{'} \)
Differentiation with respect to

*η*

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