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Aiding and opposing mixed convection of water with density inversion about a wall of varying temperature

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Abstract

Mixed convection of water over a vertical surface of varying temperature with density inversion is studied for aiding and opposing convection configurations. The temperature of the surface is assumed to be an arbitrary function of vertical distance. The governing equations are transformed using dimensionless stream function and temperature. The temperature differentials of varying wall temperature are used as perturbation functions. The dimensionless stream function and temperature are expanded in power series of perturbation variables and coefficient functions. The obtained coefficient functions are valid for any arbitrary wall temperature function, and hence they are ’universal functions’. Power law wall temperature variation is chosen to show the usefulness of universal functions. The results are presented for velocity and temperature distributions in the boundary layer, velocity and thermal boundary layer thicknesses, skin friction coefficient and heat transfer rates for various values of governing parameter and wall temperature power law index for both aiding and opposing flows. It is found that for the range of \(Gr_y/Re_y^2\) values considered in the study, the skin friction coefficient and heat transfer rates vary almost linearly with wall temperature power law index value for a given \(Gr_y/Re_y^2\) value for both aiding and opposing flows. For special wall temperature cases, the present results are compared to benchmark solutions available in literature and good agreement is found.

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Abbreviations

A :

constant in power law wall temperature variation (Eq. (45))

\(C_{f}\) :

skin friction coefficient

f :

non-dimensional stream function

\(F,f_0,f_1,\dots ,\) :

universal functions for velocity in

\(f_{00},f_{11}(\eta ),\dots \) :

perturbation technique

g :

gravitational acceleration (m/s\(^2\))

Gr :

modified Grashof number

h :

heat transfer coefficient (W/m\(^2\) K)

\(H,\theta _0,\theta _1,\dots , \) :

universal functions for temperature

\(\theta _{00},\theta _{11}(\eta ),\dots \) :

in perturbation technique

k :

thermal conductivity of fluid (W/m K)

m :

power index of power law variation of wall temperature

Nu :

Nusselt number

Pr :

Prandtl number

Re :

Reynolds number

T :

temperature (\(^\circ \)C)

\(T_0\) :

temperature at maximum density of water (\(^\circ \)C)

u, v :

velocity components in x- and y-direction, respectively (m/s)

x, y :

Cartesian coordinates (m)

\(\alpha \) :

thermal diffusivity (m\(^2\)/s)

\(\delta _v\) :

velocity boundary layer thickness (m)

\(\delta _T\) :

thermal boundary layer thickness (m)

\(\eta \) :

non-dimensional independent variable

\(\gamma \) :

constant in function (Eq. (1)) of density variation of water with temperature (°C\(^{-2}\))

\(\lambda _0,\lambda _1,\dots ,\lambda _n \) :

a set of variables that are functions of y

\(\mu \) :

dynamic viscosity (Ns/m\(^2\))

\(\nu \) :

kinematic viscosity (m\(^2\)/s)

\(\psi \) :

dimensional stream function (m\(^2\)/s)

\(\rho \) :

density (1/specific volume) (kg/m\(^3\))

\(\rho _0\) :

maximum density of water (kg/m\(^3\))

\(\tau \) :

shear stress (N/m\(^2\))

\(\theta \) :

non-dimensional temperature

w:

wall

y:

local quantity

\(\infty \) :

free-stream quantity

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Authors and Affiliations

  1. Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati, 781039, India

    Rajendra Prasad Soni & Madhusudhana R Gavara

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  1. Rajendra Prasad Soni
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  2. Madhusudhana R Gavara
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Correspondence to Madhusudhana R Gavara.

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Soni, R.P., Gavara, M.R. Aiding and opposing mixed convection of water with density inversion about a wall of varying temperature. Sādhanā 44, 172 (2019). https://doi.org/10.1007/s12046-019-1152-7

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  • DOI: https://doi.org/10.1007/s12046-019-1152-7

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