Research Papers: Natural and Mixed Convection

Natural Convection of Viscoplastic Fluids in a Square Enclosure

[+] Author and Article Information
Manabendra Pathak

e-mail: mpathak@iitp.ac.in

Mohd. Kaleem Khan

Department of Mechanical Engineering,
Indian Institute of Technology Patna,
Patliputra Colony,
Patna-13, Bihar, India

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the Journal of Heat Transfer. Manuscript received September 11, 2012; final manuscript received June 5, 2013; published online October 14, 2013. Assoc. Editor: William P. Klinzing.

J. Heat Transfer 135(12), 122501 (Oct 14, 2013) (12 pages) Paper No: HT-12-1495; doi: 10.1115/1.4024896 History: Received September 11, 2012; Revised June 05, 2013

Viscoplastic fluids are special kind of non-Newtonian materials which deform or flow only when applied stresses are more than a critical value known as yield stress. In this work, a numerical investigation of natural convection in a square enclosure filled with viscoplastic fluids has been reported. The enclosure has been partially heated from the bottom wall by a heating source and symmetrically cooled from both the side walls. The rheology of the viscoplastic fluids has been modeled with Bingham fluid model. A scaling analysis has been presented to establish the gross dependence of heat transfer on different values of operating parameters of the problem. The effects of yield stress of the fluid on heat and fluid transport inside the enclosure have been investigated for different values of temperature difference, across the hot and cold surfaces and also for different fluids. The effects of different lengths of heated zone on the flow phenomena and heat transfer characteristics have been investigated for three different values of the heated zones. All the important results have been expressed in terms of Bingham number (Bn), Rayleigh number (Ra), and Prandtl number (Pr). It has been observed that with the increase in Bingham number, the buoyancy induced fluid circulation and convection effect decreases inside the enclosure. For each Rayleigh number, there correspond a critical Bingham number for which the heat transfer inside the enclosure takes place solely by conduction mode. This critical value increases with the increase in Rayleigh number. For fixed value of Bingham number, i.e., fixed value of yield stress, the effects of Rayleigh number and heated length on heat transfer have been observed similar to the case of natural convection in Newtonian fluid. It has been also observed that at high Bingham number the effect of increase in Rayleigh number on average Nusselt number is lesser compared to the effect of increasing Rayleigh number at low Bingham number. Using the present numerical results, a correlation of average Nusselt number as a function of other nondimensional numbers has been established.

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Grahic Jump Location
Fig. 1

Schematic diagram of the computational domain

Grahic Jump Location
Fig. 2

Contours plot of stream function and temperature for different values of Bn at Pr = 7, Ra = 106, and ε = 1/5

Grahic Jump Location
Fig. 3

Yielded (unshaded) and unyielded region (shaded) for various values of Bn at Pr = 7, Ra = 106, and ε = 1/5

Grahic Jump Location
Fig. 4

Vertical velocity component profile along the horizontal midplane for different Bn at Ra = 104, Pr = 7, and ε = 1/5

Grahic Jump Location
Fig. 5

The plot of average Nusselt number Nu¯ and Bn for different Ra at Pr = 7 and ε = 1/5

Grahic Jump Location
Fig. 6

Contours plot of stream function and temperature for different Ra at Bn = 0.5, Pr =0.71, and ε =1/5

Grahic Jump Location
Fig. 7

Vertical velocity component along the horizontal midplane for different Ra at Bn = 1, Pr = 7, and ε = 2.5/5

Grahic Jump Location
Fig. 8

Variation of local Nusselt number, Nu at the heated surface for Ra = 104 at Bn = 1, Pr = 7, and ε = 4/5

Grahic Jump Location
Fig. 9

Contours plot of stream function and temperature for various ε at Bn = 5, Ra = 105, and Pr = 7

Grahic Jump Location
Fig. 10

Variation of the average Nusselt number, Nu¯ at the heated surface with Ra for various dimensionless heating zone length ε and Pr = 0.71

Grahic Jump Location
Fig. 11

Contours plot of stream function and temperature for various Pr at Bn = 0.5, Ra = 106, and ε =4/5



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