# Unsteady mixed convection flow from a rotating vertical cone with a magnetic field

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## Abstract.

An analysis is developed to study the unsteady mixed convection flow over a vertical cone rotating in an ambient fluid with a time-dependent angular velocity in the presence of a magnetic field. The coupled nonlinear partial differential equations governing the flow have been solved numerically using an implicit finite-difference scheme. The local skin friction coefficients in the tangential and azimuthal directions and the local Nusselt number increase with the time when the angular velocity of the cone increases, but the reverse trend is observed for decreasing angular velocity. However, these are not mirror reflection of each other. The magnetic field reduces the skin friction coefficient in the tangential direction and also the Nusselt number, but it increases the skin friction coefficient in the azimuthal direction. The skin friction coefficients and the Nusselt number increase with the buoyancy force.

## Keywords.

Unsteady mixed convection Magnetic effects## Nomenclature

## Roman letters

- C
_{fx} skin friction coefficient in the x-direction

- C
_{fy} skin friction coefficient in the y-direction

- C
_{p} specific heat at constant pressure, kJ·kg

^{–1}·K- Ec
Eckert number

**E**electric field

- f
dimensionless stream function

- G, H
similarity velocity functions, m·s

^{–1}- Gr
Grashof number

- H, h
total enthalpy and static enthalpy, respectively

- Ha
Hartman number

- k
thermal conductivity, W·m

^{–1}·K- M
magnetic parameter

- Ma
Mach number

- L
characteristic length, m

- Nu
Nusselt number

- Pr
Prandtl number

- Re
Reynolds number

- St
Stanton number

- t
dimensional time

- t*
dimensionless time

- T
temperature, K

- u, v, w
velocity components, m·s

^{–1}- V
characteristic velocity, m·s

^{–1}- x, y, z
curvilinear coordinates system

## Greek letters

- C
_{fx} skin friction coefficient in the x-direction

- α
semi-angle of the cone

- β
pressure gradient parameter

- γ
ratio of specific heats

- η, ξ
transformed co-ordinates

- μ
dynamic viscosity, kg·m

^{–1}·s^{–1}- ν
konematic viscosity, m

^{2}·s^{–1}- ρ
density, kg·m

^{–3}- ψ
dimensional stream function, m

^{2}·s^{–1}- Ω
angular velocity of the cone

- Ω
_{0} angular velocity of the cone at t=0

- ϕ(t)
a continuous function of time

## Subscripts

- i
initial conditions at the wall

- w
condition at the wall

- ∞
condition in the free stream

## References

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