# Numerical Solutions of Dissipative Natural Convective Flow from a Vertical Cone with Heat Absorption, Generation, MHD and Radiated Surface Heat Flux

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## Abstract

The laminar natural convective hydromagnetic viscous fluid flow induced by a cone under aspect of radiated heat flux with thermal radiation, heat absorption and generation is addressed here. The basic equations of conservation of momentum, mass and energy are utilized for the modeling of physical problem. The consequential expressions are worked out by using Crank–Nicholson approach. The implementation of this method leads to conversion of non-dimensional expressions into system of tri-diagonal expressions. The obtain numerical data is visualized for momentum, local-average shear stresses, rate of heat transportation and temperature for various constraints *Pr*, Δ, *M*, ε and *Rd* with the help of graphical sketches. It is reported that the temperature of liquid is boost up with an enhancement in heat generation constraint. The larger Prandtl number corresponds to weaker temperature profiles. The average shear stress coefficient increase for higher radiation constraints and Prandtl number.

## Keywords

Finite difference method MHD Thermal radiation Viscous dissipation Vertical cone## List of Symbols

- \( F_{0}^{''} (0) \)
Shear-stress co-efficient in Ref: [13]

*Gr*_{L}Grashof number

- g
Rate of change of velocity due to gravity

- k
Thermal conductivity

- k
^{*} Mean sink co-efficient

- L
Reference span

- M
Magnetic constraint

*Nu*_{x}Local Nusselt number

*Nu*_{X}Dimensionless Local Nusselt numeral

- \( \overline{Nu} \)
Dimensionless average Nusselt numeral

*Pr*Prandtl number

*q*_{w}Uniform wall heat flux per unit area

- R
Non-dimensional local radius of the cone

- r
Local radius of the cone

*T*′Temperature

- T
Non-dimensional temperature

*t*′Time

- t
Non-dimensional time

- U
Non-dimensional velocity in X-direction

- u
Velocity component in x-direction

- V
Non-dimensional velocity in Y-direction

- v
Rate component in y-direction

- X
Non-dimensional spatial co-ordinate

- x
Spatial coefficient along cone generator

- Y
Non-dimensional spatial coefficient along the normal to the cone generator

- y
Spatial coefficient along the normal to the cone generator

## Greek Symbols

*α*Thermal diffusivity

*β*Volumetric thermal expansion

*σ*Electrical conductivity

*σ*^{*}Stefan–Boltzmann constant

- Δ
Non-dimensional heat source/sink constraint

- Δ
*t* Non-dimensional time step

- Δ
*X* Non-dimensional finite difference grid size in X-direction

- Δ
*Y* Non-dimensional finite difference grid size in Y-direction

*ε*Viscous dissipation parameter

*ϕ*Semi vertical angle of the cone

*μ*Dynamic viscosity

*γ*Kinematic viscosity

*ρ*Density

*τ*_{x}Non-dimensional local skin friction

*τ*_{X}Non-dimensional local skin friction

- \( \bar{\tau } \)
Non-dimensional average skin friction

## Subscripts

- w
Condition on the wall

- ∞
Free stream condition

## Notes

### Compliance with Ethical Standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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