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ABSTRACT
In this work, laminar mixed convection from an isothermal spheroidal particle immersed in a Bingham plastic fluid is studied numerically in the buoyancy-assisted regime. The results reported herein encompass the following ranges of conditions: Reynolds number, 0.1 ≤ Re ≤ 100; Prandtl number, 10 ≤ Pr ≤ 100; Bingham number, 0 ≤ Bn ≤ 100; Richardson number, 0 ≤ Ri ≤ 8; and aspect ratio of the spheroid, 0.2 ≤ e ≤ 5. In particular, consideration is given to the effect of shape and orientation of the particle on the detailed flow and temperature fields (in terms of streamlines, iso-vorticity, and isotherm contours), morphology of the yielded–unyielded regions, and the local and surface-averaged Nusselt number. All else being equal, the propensity for flow separation is seen to be greater for oblates (e < 1) than that for prolates (e > 1). In both cases, this reduces with the increasing Bingham number and/or the Richardson number. Both drag coefficient and the Nusselt number show a positive dependence on the Bingham number as well as on the Richardson number. Overall, the drag coefficient increases as the particle shape changes from an oblate to prolate, whereas the reverse trend is obtained for the average Nusselt number, which is in line with the general inference that more drag corresponds to more heat transfer. Finally, the average Nusselt number is correlated with the pertinent dimensionless parameters (Re, Pr, Bn, Ri, e) via a simple correlation, thereby enabling its prediction for intermediate values of the parameters and/or in a new application.
Nomenclature
| a | = | spheroid semi-axis normal to flow, m |
| Ap | = | projected area of the spheroid normal to flow (= πa2), m2 |
| b | = | spheroid semi-axis parallel to flow, m |
| Bn | = | Bingham number, dimensionless |
| C | = | specific heat of fluid, J/kg · K |
| CD | = | total drag coefficient, dimensionless |
| CDP | = | pressure drag coefficient |
| D∞ | = | diameter of the computational domain, m |
| e | = | aspect ratio of the spheroid, (= b/a), dimensionless |
| FD | = | total drag force, N |
| FDP | = | pressure drag force, N |
| Gr | = | Grashof number, dimensionless |
| g | = | acceleration due to gravity, m · s−2 |
| h | = | local heat transfer coefficient, W · m−2 · K−1 |
| k | = | thermal conductivity of the fluid, W · m−1 · K−1 |
| m | = | regularization or growth rate parameter, dimensionless |
| Np | = | number of grid points on the surface of the spheroid, dimensionless |
| Nu | = | average Nusselt number, dimensionless |
| Nu∞ | = | conduction limit of the Nusselt number for a spheroid, dimensionless |
| Nuθ | = | local Nusselt number on the surface of the spheroid, dimensionless |
| P | = | pressure, dimensionless |
| Pr | = | Prandtl number, dimensionless |
| Re | = | Reynolds number, dimensionless |
| Ri | = | Richardson number (= Gr/Re2), dimensionless |
| T | = | fluid temperature, K |
| T0 | = | temperature of the fluid in free stream, K |
| Tw | = | temperature on the surface of the spheroid, K |
| U0 | = | free stream velocity, m · s−1 |
| V | = | velocity vector, dimensionless |
| x, y | = | cartesian coordinates, m |
| β | = | coefficient of volumetric expansion, K−1 |
| = | rate of strain tensor, dimensionless | |
| δ | = | minimum distance between two grid points on the surface of the spheroid, m |
| η | = | viscosity of the fluid, dimensionless |
| θ | = | position along the surface of the spheroid, degree |
| λ | = | growth rate parameter in the Bercovier and Engelman model, dimensionless |
| μB | = | plastic viscosity, Pa · s |
| μy | = | yielding viscosity, Pa · s |
| ξ | = | fluid temperature |
| ρ | = | density of fluid, kg · m−3 |
| ρ0 | = | density of fluid at the reference temperature T0, kg · m−3 |
| τ | = | deviatoric stress tensor, dimensionless |
| τ0 | = | fluid yield stress, Pa |
Nomenclature
| a | = | spheroid semi-axis normal to flow, m |
| Ap | = | projected area of the spheroid normal to flow (= πa2), m2 |
| b | = | spheroid semi-axis parallel to flow, m |
| Bn | = | Bingham number, dimensionless |
| C | = | specific heat of fluid, J/kg · K |
| CD | = | total drag coefficient, dimensionless |
| CDP | = | pressure drag coefficient |
| D∞ | = | diameter of the computational domain, m |
| e | = | aspect ratio of the spheroid, (= b/a), dimensionless |
| FD | = | total drag force, N |
| FDP | = | pressure drag force, N |
| Gr | = | Grashof number, dimensionless |
| g | = | acceleration due to gravity, m · s−2 |
| h | = | local heat transfer coefficient, W · m−2 · K−1 |
| k | = | thermal conductivity of the fluid, W · m−1 · K−1 |
| m | = | regularization or growth rate parameter, dimensionless |
| Np | = | number of grid points on the surface of the spheroid, dimensionless |
| Nu | = | average Nusselt number, dimensionless |
| Nu∞ | = | conduction limit of the Nusselt number for a spheroid, dimensionless |
| Nuθ | = | local Nusselt number on the surface of the spheroid, dimensionless |
| P | = | pressure, dimensionless |
| Pr | = | Prandtl number, dimensionless |
| Re | = | Reynolds number, dimensionless |
| Ri | = | Richardson number (= Gr/Re2), dimensionless |
| T | = | fluid temperature, K |
| T0 | = | temperature of the fluid in free stream, K |
| Tw | = | temperature on the surface of the spheroid, K |
| U0 | = | free stream velocity, m · s−1 |
| V | = | velocity vector, dimensionless |
| x, y | = | cartesian coordinates, m |
| β | = | coefficient of volumetric expansion, K−1 |
| = | rate of strain tensor, dimensionless | |
| δ | = | minimum distance between two grid points on the surface of the spheroid, m |
| η | = | viscosity of the fluid, dimensionless |
| θ | = | position along the surface of the spheroid, degree |
| λ | = | growth rate parameter in the Bercovier and Engelman model, dimensionless |
| μB | = | plastic viscosity, Pa · s |
| μy | = | yielding viscosity, Pa · s |
| ξ | = | fluid temperature |
| ρ | = | density of fluid, kg · m−3 |
| ρ0 | = | density of fluid at the reference temperature T0, kg · m−3 |
| τ | = | deviatoric stress tensor, dimensionless |
| τ0 | = | fluid yield stress, Pa |
Acknowledgements
R.P. Chhabra is grateful to the Department of Science & Technology (Government of India) for the award of JC Bose fellowship.