# MHD Couette two-fluid flow and heat transfer in presence of uniform inclined magnetic field

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## Abstract

The MHD Couette flow of two immiscible fluids in a parallel plate channel in the presence of an applied electric and inclined magnetic field is investigated in the paper. One of the fluids is assumed to be electrically conducting, while the other fluid and the channel plates are assumed to be electrically insulating. Separate solutions with appropriate boundary conditions for each fluid are obtained and these solutions are matched at the interface using suitable matching conditions. The partial differential equations governing the flow and heat transfer are transformed to ordinary differential equations and closed-form solutions are obtained in both fluid regions of the channel. The results for various values of the Hartmann number, the angle of magnetic field inclination, the loading parameter and the ratio of the heights of the fluids are presented graphically to show their effect on the flow and heat transfer characteristics.

## Keywords

Viscous Dissipation Joule Heating Hartmann Number Micropolar Fluid Immiscible Fluid## List of symbols

**B**Magnetic field vector

*B*_{0}Applied magnetic field strength

*B*_{x}Strength of induced magnetic field

*b*Dimensionless ratio of magnetic fields

*C*Constant

*c*_{p}Specific heat capacity

*D*_{i}Coefficients

**E**Electric field vector

*F*Constant

*H*_{a}Hartmann number

*h*_{i}Height of the region

*i***J**Current density

*K*Loading parameter

*k*Fluid thermal conductivity

*p*Pressure

*Q*Constant

*R*_{m}Magnetic Reynolds number

*S*Constant

*T*Temperature

*t*Time

*U*_{0}Upper plate velocity

*u*Velocity in

*x*-direction**v**Velocity vector

*x*Longitudinal coordinate

*y*Transversal coordinate

## Greek symbols

- α
Fluid viscosities ratio

*β*Ratio of heights of two regions

*Φ*Dissipative function

*λ*Cosine of inclination angle

*μ*Dynamic viscosity

*μ*_{e}Magnetic permeability

- ν
Kinematic viscosity

*θ*Applied magnetic field inclination angle

- Θ
Dimensionless temperature

- ρ
Fluid density

- σ
Electrical conductivity

- ξ
Ratio of thermal conductivities

## Subscripts

- 1
Fluid in region I

- 2
Fluid in region II

*w*Plate

- *
Dimensionless quantities

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