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Research Papers: Natural and Mixed Convection

# Laminar Natural Convection of Bingham Fluids in Inclined Differentially Heated Square Enclosures Subjected to Uniform Wall Temperatures

[+] Author and Article Information
Şahin Yİğİt

School of Mechanical and Systems Engineering,
Newcastle University,
Newcastle-Upon-Tyne NE1 7RU, UK
Department of Mechanical Engineering,
Trabzon 61080, Turkey
e-mail: syigit@ktu.edu.tr

Robert J. Poole

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

Nilanjan Chakraborty

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

As will be discussed in Sec. 3, the findings based on the current numerical experiment for Pr = 500 are qualitatively valid for other values of nominal Prandtl number and the resulting correlations capture the effects of Pr from 10 to 500.

R2 value is greater than 0.99 for all Ra values for the correlation given by Eq. (11).

The R2 value is greater than 0.99 for the correlation given by Eqs. (14b) and (14c). The maximum percentage error for the predictions of Eqs. (14b) and (14c) with respect to simulation data is found to be 4.8%.

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received February 7, 2014; final manuscript received January 19, 2015; published online March 3, 2015. Assoc. Editor: Oronzio Manca.

J. Heat Transfer 137(5), 052504 (May 01, 2015) (12 pages) Paper No: HT-14-1068; doi: 10.1115/1.4029763 History: Received February 07, 2014; Revised January 19, 2015; Online March 03, 2015

## Abstract

The effects of inclination $180deg≥φ≥0deg$ on steady-state laminar natural convection of yield-stress fluids, modeled assuming a Bingham approach, have been numerically analyzed for nominal values of Rayleigh number Ra ranging from 103 to 105 in a square enclosure of infinite span lying horizontally at $φ=0deg$, then rotated about its axis for $φ>0deg$ cases. It has been found that the mean Nusselt number $Nu¯$ increases with increasing values of Rayleigh number but $Nu¯$ values for yield-stress fluids are smaller than that obtained in the case of Newtonian fluids with the same nominal value of Rayleigh number Ra due to the weakening of convective transport. For large values of Bingham number Bn (i.e., nondimensional yield stress), the mean Nusselt number $Nu¯$ value settles to unity ($Nu¯=1.0$) as heat transfer takes place principally due to thermal conduction. The mean Nusselt number $Nu¯$ for both Newtonian and Bingham fluids decreases with increasing $φ$ until reaching a local minimum at an angle $φ*$ before rising with increasing $φ$ until $φ=90deg$. For $φ>90deg$ the mean Nusselt number $Nu¯$ decreases with increasing $φ$ before assuming $Nu¯=1.0$ at $φ=180deg$ for all values of $Ra$. The Bingham number above which $Nu¯$ becomes unity (denoted $Bnmax$) has been found to decrease with increasing $φ$ until a local minimum is obtained at an angle $φ*$ before rising with increasing $φ$ until $φ=90deg$. However, $Bnmax$ decreases monotonically with increasing $φ$ for $90deg<φ<180deg$. A correlation has been proposed in terms of $φ$, Ra, and Bn, which has been shown to satisfactorily capture $Nu¯$ obtained from simulation data for the range of Ra and $φ$ considered here.

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## Figures

Fig. 1

Schematic diagram of the simulation domain

Fig. 2

Variations of nondimensional temperature θ (1st column) and U=u1L/α (2nd column) with x2/L at x1/L = 0.5 for (a) φ=45 deg, (b) φ=60 deg, (c) φ=120 deg, and (d) φ=135 deg for different values of Ra at Pr = 500 and Bn=0.05

Fig. 3

Contours of θ at φ=45,60,120,135 deg at Pr = 500 and Bn=0.05 for Ra = (a) 1 × 103, (b) 5 × 103, (c) 1 × 104, (d) 5 × 104, and (e) 1 × 105

Fig. 4

Contours of nondimensional stream function (Ψ = ψ/α) and unyielded zones (gray) for φ=45,60,120,135 deg at Pr = 500 and Bn=0.05 for Ra = (a) 1 × 103, (b) 5 × 103, (c) 1 × 104, (d) 5 × 104, and (e) 1 × 105

Fig. 5

Variations of θ (1st column) and U=u1L/α (2nd column) with x2/L at x1/L = 0.5 for different values of Bingham numbers for (a) φ=30 deg, (b) φ=45 deg, (c) φ=60 deg, (d) φ=120 deg, (e) φ=135 deg, and (f) φ=150 deg at Pr = 500 and Ra = 105

Fig. 6

Contours of θ and Ψ with unyielded zones (gray) for φ=45,135 deg at Ra=105 and Pr = 500 for different values of Bn

Fig. 7

Variation of Nu¯ with Bn for Ra = (a) 5 × 103, (b) 1 × 104, (c) 5 × 104, and (d) 1 × 105 at Pr = 500 and φ=30,60,120,150 deg along with the prediction of Eq. (16d).

Fig. 8

(a) Variation of Nu¯ with φ for different values of Ra for Newtonian fluids along with the predictions of Eqs. (10) and (11). Contours of (b) nondimensional stream functions (Ψ = ψ/α) and (c) nondimensional temperature θ for Newtonian fluid (Bn=0) for different values of φ around φ=φ* for Ra = 105 at Pr = 500. The inclination φ at which the minimum Nu¯ is obtained for a given set of Ra and Pr is highlighted with an asterisk * in Figs. 8(b) and 8(c). (d) Variation of Bnmax with φ for different values of Ra along with the prediction of Eq. (13).

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