# Locally similar solutions for hydromagnetic and thermal slip flow boundary layers over a flat plate with variable fluid properties and convective surface boundary condition

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DOI: 10.1007/s11012-010-9372-2

- Cite this article as:
- Rahman, M.M. Meccanica (2011) 46: 1127. doi:10.1007/s11012-010-9372-2

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## Abstract

This paper presents heat transfer process in a two-dimensional steady hydromagnetic convective flow of an electrically conducting fluid over a flat plate with partial slip at the surface of the boundary subjected to the convective surface heat flux at the boundary. The analysis accounts for both temperature-dependent viscosity and temperature dependent thermal conductivity. The local similarity equations are derived and solved numerically using the Nachtsheim-Swigert iteration procedure. Results for the dimensionless velocity, temperature and ambient Prandtl number within the boundary layer are displayed graphically delineating the effect of various parameters characterizing the flow. The results show that momentum boundary layer thickness significantly depends on the surface convection parameter, Hartmann number and on the sign of the variable viscosity parameter. The results also show that plate surface temperature is higher when there is no slip at the plate compared to its presence. For both slip and no-slip cases surface temperature of the plate can be controlled by controlling the strength of the applied magnetic field. In modelling the thermal boundary layer flow with variable viscosity and variable thermal conductivity, the Prandtl number must be treated as a variable irrespective of flow conditions whether there is slip or no-slip at the boundary to obtain realistic results.

### Keywords

Convective flowHeat transferSimilar solutionSlip flowVariable thermal conductivityVariable viscosity### Nomenclature

### Roman

*A*constant appears in (9)

*a*surface convection parameter

*B*magnetic induction [Wb m

^{−2}]*B*_{0}constant

*C*_{f}local skin-friction coefficient

*c*constant

*c*_{p}specific heat at constant pressure [kJ kg

^{−1}K^{−1}]*f*dimensionless stream function

- Ha
Hartmann number

*h*_{w}convective heat transfer coefficient [W m

^{−2}K^{−1}]*L*slip length [m]

- Kn
_{x,L} local Knudsen number based on slip length

- Kn
_{x,δ} local Knudsen number based on mean free path

- Nu
_{x} local Nusselt number

- Pr
variable Prandtl number

- Pr
_{∞} ambient Prandtl number

- Re
_{x} local Reynolds number

*T*_{r}constant appears in (9)

*T*_{w}temperature at the surface of the plate [K]

*T*temperature of the fluid within the boundary layer [K]

*T*_{∞}temperature of the ambient fluid [K]

*U*_{∞}free stream velocity [m s

^{−1}]*u*,*v*the

*x*- and*y*-components of the velocity field [m s^{−1}]*x*,*y*distance along and normal to the plate [m]

### Greek

*ρ*fluid density [kg m

^{−3}]*ε*thermal conductivity parameter

*μ*dynamic viscosity [Pa s]

*μ*_{∞}dynamic viscosity at ambient temperature [Pa s]

*υ*kinematic viscosity [m

^{2}s^{−1}]*δ*slip parameter

*σ*tangential momentum accommodation coefficient

*σ*_{0}magnetic permeability [N A

^{−2}]*λ*mean free path [m]

*ψ*stream function [m

^{2}s^{−1}]*η*similarity variable

*θ*dimensionless temperature

*θ*_{r}variable viscosity parameter

*κ*thermal conductivity [W m

^{−1}K^{−1}]*κ*_{∞}thermal conductivity at ambient temperature [W m

^{−1}K^{−1}]*γ*constant appears in (8)

### Subscripts

*w*;∞surface condition; ambient condition