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Heatline and Massline Analysis Due to Magnetohydrodynamic Double Diffusive Natural Convection in a Trapezoidal Enclosure with Various Aspect Ratios

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Abstract

As streamlines are the utmost visualization tools of fluid flow and not the isobars, heatlines and masslines are the sufficient visualization tools of heat and mass transfers and not the isotherms and isoconcentrations respectively. Motivated by various applications of conjugate heat and mass transfer in drying, cooling, sterilization process etc., the present work aims to find out the detailed analysis of conjugate heat and mass transfer using heatlines and masslines in a trapezoidal enclosure with different heating and various massive walls in the presence of magnetic field. The main impact of the paper is the analysis of heatlines and masslines with different aspect ratios of the cavity, which has not been considered by any earlier researcher. The second order accurate finite difference approximation streamfunction—velocity formulation of full Navier Stokes equations has been used to find out the numerical solutions. The discretised non-homogeneous system of linear equations are solved by Biconjugate Gradient Stabilized method.

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Abbreviations

A :

Aspect ratio

B :

Magnetic induction (\(\mathrm{Tesla} = \mathrm{N}/\mathrm{Am}^{2}\))

C :

Concentration

D :

Mass diffusivity

g :

Acceleration due to gravity, (ms\(^{-2}\))

Ha :

Hartmann number

h :

Step length in both \(\xi \) and \(\eta \) coordinates

L :

Length of the bottom wall of the enclosure, m

Le :

Lewis number, \(Le = \alpha /D\)

N :

Buoyancy ratio

Nu :

Nusselt number

P :

Dimensional pressure (Pa)

p :

Dimensionless pressure

Pr :

Prandtl number

Ra :

Rayleigh number

S :

Dimensionless concentration

Sh :

Sherwood number

T :

Temperature (K)

U :

x component of velocity, (ms\(^{-1}\))

u :

Dimensionless x component of velocity

V :

y component of velocity, (ms\(^{-1}\))

v :

Dimensionless y component of velocity

X :

Distance along x axis (m)

x :

Dimensionless distance along x axis

Y :

Distance along y axis (m)

y :

Dimensionless distance along y axis

\(C_{c}\) :

Low concentration at the wall of the cavity

\(C_{h}\) :

High concentration at the wall of the cavity

\(S_{1}\) :

Length of left wall (m)

\(S_{2}\) :

Length of right wall (m)

\(T_{h}\) :

Temperature of hot wall (K)

\(T_{c}\) :

Temperature of cold wall (K)

\({\overline{Nu}}\) :

Average Nusselt number

\({\overline{Sh}}\) :

Average Sherwood number

\(\alpha \) :

Thermal diffusivity (m\(^{2}\hbox {s}^{-1}\))

\(\xi \) :

Horizontal coordinate in a unit square

\(\eta \) :

Vertical coordinate in a unit square

\(\nu \) :

Kinematic viscosity (\(\hbox {m}^{2}\hbox {s}^{-1}\))

\(\psi _{ij}\) :

\(\psi (\xi +ih, \eta +jh)\)

\(\theta \) :

Dimensionless temperature

\(\rho \) :

Density, (kg \(\hbox {m}^{-3}\))

\(\phi \) :

Angle of inclination

\(\psi \) :

Streamfunction

\(\tau \) :

Massfunction

\(\pi \) :

Heatfunction

\(\omega \) :

Vorticityfunction

\(\beta _{T}\) :

Coefficient of thermal expansion

\(\beta _{S}\) :

Coefficient of solutal expansion

b :

Bottom wall

l :

Left wall

r :

Right wall

t :

Top wall

max :

Maximum

\(\xi \) :

Partial derivative w.r.t. \(\xi \)

\(\eta \) :

Partial derivative w.r.t. \(\eta \)

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Acknowledgements

The authors thank the referees for their valuable comments, which enable to produce an improved presentation of their paper. The work of one of the authors (T.R.M.) is supported under SAP (DRS Phase-III, Letter No. F.510/3/DRS-III/2015(SAPI)) under UGC, New Delhi, India.

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  1. Department of Mathematics, Visva-Bharati (A Central University), Santiniketan, West-Bengal, 731 235, India

    T. R. Mahapatra & Priyajit Mondal

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  1. T. R. Mahapatra
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Correspondence to T. R. Mahapatra.

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Mahapatra, T.R., Mondal, P. Heatline and Massline Analysis Due to Magnetohydrodynamic Double Diffusive Natural Convection in a Trapezoidal Enclosure with Various Aspect Ratios. Int. J. Appl. Comput. Math 5, 82 (2019). https://doi.org/10.1007/s40819-019-0657-4

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