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Research Papers: Natural and Mixed Convection

# Laminar Natural Convection of Power-Law Fluids in a Square Enclosure With Differentially Heated Sidewalls Subjected to Constant Wall Heat Flux

[+] Author and Article Information
Osman Turan

School of Engineering,
University of Liverpool,
Brownlow Hill,
Liverpool, L69 3GH, UK;
Department of Mechanical Engineering,
Trabzon, 61080, Turkey
e-mail: osmanturan@ktu.edu.tr

Anuj Sachdeva

e-mail: a.sachdeva@liv.ac.uk

Robert J. Poole

e-mail: robpoole@liv.ac.uk
School of Engineering,
University of Liverpool,
Brownlow Hill,
Liverpool, L69 3GH, UK

Nilanjan Chakraborty

School of Mechanical and Systems Engineering,
Newcastle University,
Newcastle-Upon-Tyne, NE1 7RU, UK
e-mail: nilanjan.chakraborty@ncl.ac.uk

The definitions of $Ra$ and $Pr$ are provided later in Sec. 2.

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the Journal of Heat Transfer. Manuscript received July 24, 2011; final manuscript received April 3, 2012; published online October 8, 2012. Assoc. Editor: Sujoy Kumar Saha.

J. Heat Transfer 134(12), 122504 (Dec 08, 2012) (15 pages) doi:10.1115/1.4007123 History: Received July 24, 2011; Revised April 03, 2012

## Abstract

Two-dimensional steady-state laminar natural convection of inelastic power-law non-Newtonian fluids in square enclosures with differentially heated sidewalls subjected to constant wall heat flux (CHWF) are studied numerically. To complement the simulations, a scaling analysis is also performed to elucidate the anticipated effects of Rayleigh number (Ra), Prandtl number (Pr) and power-law index (n) on the Nusselt number. The effects of $n$ in the range 0.6 ≤ n ≤ 1.8 on heat and momentum transport are investigated for nominal values Ra in the range 103–106 and a Pr range of 10–105. In addition the results are compared with the constant wall temperature (CWT) configuration. It is found that the mean Nusselt number $Nu¯$ increases with increasing values of Ra for both Newtonian and power-law fluids in both configurations. However, the $Nu¯$ values for the vertical walls subjected to CWHF are smaller than the corresponding values in the same configuration with CWT (for identical values of nominal Ra, Pr and n). The $Nu¯$ values obtained for power-law fluids with $n<1$ ($n>1$) are greater (smaller) than that obtained in the case of Newtonian fluids with the same nominal value of Ra due to strengthening (weakening) of convective transport. With increasing shear-thickening (i.e., n > 1) the mean Nusselt number $Nu¯$ settles to unity ($Nu¯=1.0$) as heat transfer takes place principally due to thermal conduction. The effects of Pr are shown to be essentially negligible in the range 10–105. New correlations are proposed for the mean Nusselt number $Nu¯$ for both Newtonian and power-law fluids.

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## Figures

Fig. 1

Schematic diagram of the simulation domain (a) CWT configuration, (b) CWHF configuration

Fig. 2

(a) Different regimes of convection for both CWT and CWHF configurations for n = 0.6, (b) temporal evolution of Nu¯ with dimensionless time αt/L2 at Pr = 50, n = 0.6 for: (A) conduction regime RaCWHF (RaCWT) = 5, (B) laminar steady convection regime RaCWHF (RaCWT) = 1 × 106, (C) CWT case unsteady convection regime RaCWHF (RaCWT) = 5 × 106, (D) CWHF case unsteady convection regime RaCWHF (RaCWT) = 5 × 109

Fig. 3

Variations of nondimensional temperature θ and vertical velocity component V along the horizontal midplane at Pr = 100: (a) RaCWHF = 104, (b) RaCWHF = 105, and (c) RaCWHF = 106

Fig. 4

Contours of nondimensional stream functions (Ψ = ψ/α) for n = 0.6, 1.0, and 1.8 at Pr = 1000: (a) RaCWHF = 104, (b) RaCWHF = 105, and (c) RaCWHF = 106

Fig. 5

Contours of nondimensional temperature θ for n = 0.6, 1.0, and 1.8 at Pr = 1000: (a) RaCWHF = 104, (b) RaCWHF = 105, and (c) RaCWHF = 106

Fig. 6

The variation of the mean Nusselt number with Rayleigh number for both CWT (left column) and CWHF (right column) configurations for different values of power-law index n at (a) Pr = 100, (b) Pr = 1000, and (c) Pr = 10,000

Fig. 7

Variations of nondimensional temperature θ along the horizontal midplane for both (- - -) CWT, (—) CHWF configurations at different values of n for the same values of RaCWHF and RaCWT: RaCWHF = RaCWT = (a) 104, (b) 105, and (c) 106 at Pr = 1000

Fig. 8

Variations of nondimensional vertical velocity V along the horizontal midplane for both CWT (left column), CWHF (right column) configurations at different values of n for the same values of RaCWHF and RaCWT: RaCWHF = RaCWT = (a) 104, (b) 105, and (c) 106 at Pr = 1000

Fig. 9

Variations of Nu¯ with power-law index for both CWT (Δ) and CWHF (○) configurations for different values of Pr and RaCWT (RaCWHF) along with the predictions of Eqs. (34i)(34iii) (- - -) and Eqs. (35i)(35iv) (—)

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