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ABSTRACT
The laminar forced convection momentum and heat transfer aspects of a circular disk oriented normal to the flow and maintained at a constant flux or a constant temperature condition in a stream of a Bingham plastic fluid are studied over wide ranges of parameters as follows: Reynolds number, Re ≤ 150; Prandtl number, 1 ≤Pr ≤ 100; Bingham number, Bn ≤ 100, and thickness-to-diameter ratio, 0.01 ≤ (t/d) ≤ 0.075. The new results on hydrodynamics are analyzed in terms of streamline plots, recirculation length, morphology of yielded/unyielded regions, and drag coefficient, and on heat transfer aspects in terms of isotherm contours, local and average Nusselt number. The flow domain is spanned by the simultaneous existence of the yielded and unyielded sub-regions, depending upon the relative strengths of the fluid inertia (Re) and yield stress (Bn). All else being equal, the rate of heat transfer is higher for an isothermal disk than that for the isoflux condition. Both the drag and average Nusselt number bear a positive dependence on the Bingham number. The drag is influenced only slightly (∼5%) by thickness (t/d); however, the heat transfer can increase on this count by up to 15% under appropriate conditions. Finally, the present numerical results on drag and Nusselt number (in terms of jH-factor) have been correlated via simple empirical equations using the modified definitions of the Reynolds (Re*) and Prandtl number (Pr*), thereby enabling a priori estimation of drag and heat transfer in a new application.
Nomenclature
| Ap | = | projected area of the disk normal to the direction of flow (m2) |
| Bn | = | Bingham number |
| = | modified Bingham number (= Bn/(1 + Bn)) (dimensionless) | |
| Bncr | = | critical Bingham number for cessation of flow separation (dimensionless) |
| Br | = | Brinkman number |
| C | = | thermal heat capacity of fluid (J kg−1 K−1) |
| CD | = | total drag coefficient |
| = | drag coefficient | |
| CDF | = | friction (or viscous) drag coefficient |
| CDP | = | pressure (or form) drag coefficient |
| CDS | = | Stokes drag coefficient (dimensionless) |
| Cp | = | pressure coefficient |
| d | = | disk diameter (m) |
| FD | = | drag force (N) |
| FDF | = | friction (viscous) component of the drag force (N) |
| FDP | = | pressure (form) component of the drag force (N) |
| h | = | local heat transfer coefficient (W m−2 K−1) |
| jH | = | Colburn jH-factor for heat transfer |
| k | = | thermal conductivity of fluid (W m−1 K−1) |
| L | = | size of the computational domain (m) |
| Lr | = | recirculation length measured from the rear stagnation point (dimensionless) |
| M | = | parameter used in Papanastasiou model, Eq. (8) (dimensionless) |
| ns | = | unit vector normal to the surface of disk (dimensionless) |
| N | = | total number of cells in the half computational domain (dimensionless) |
| Np | = | number of grid points on the surface of disk (dimensionless) |
| Nu | = | average Nusselt number (dimensionless) |
| NuH | = | average Nusselt number for CHF case (dimensionless) |
| Nul | = | local Nusselt number on the surface of disk (dimensionless) |
| NuT | = | average Nusselt number for CWT case (dimensionless) |
| = | conduction limit of the Nusselt number (dimensionless) | |
| p | = | pressure (dimensionless) |
| pref | = | reference pressure (Pa) |
| Pe | = | Peclet number |
| Pr | = | Prandtl number (dimensionless) |
| = | modified Prandtl number (= Pr (1 + Bn)) (dimensionless) | |
| q0 | = | uniform heat flux on the surface of the disk (W m−2) |
| r, z | = | cylindrical coordinates |
| R | = | radius of the disk (m) |
| Re | = | Reynolds number |
| Recr | = | critical Reynolds number for the onset of flow separation (dimensionless) |
| = | modified Reynolds number | |
| S | = | surface area of disk (m2) |
| t | = | disk thickness (m) |
| T | = | temperature of fluid (K) |
| V | = | velocity vector (dimensionless) |
| V0 | = | free stream velocity (m s−1) |
| Vr, Vz | = | velocity components in r- and z-directions (dimensionless) |
| Greek symbols | = | |
| ∇ | = | del operator (dimensionless) |
| = | rate of deformation tensor (dimensionless) | |
| Δ | = | minimum grid spacing on the surface of disk (m) |
| η | = | scalar viscosity function (dimensionless) |
| λ | = | parameter used in Bercovier and Engelman model, Eq. (10) (dimensionless) |
| = | fluid viscosity (Pa s) | |
| = | yielding viscosity used in bi-viscosity model, Eq. (9) (Pa s) | |
| ξ | = | temperature of fluid (dimensionless) |
| ρ | = | density of fluid (kg m3) |
| τ | = | deviatoric stress tensor (dimensionless) |
| = | fluid yield stress (Pa) | |
| Subscripts | = | |
| = | dimensional variable | |
| 0 | = | corresponds to faraway (free stream) condition |
| w | = | condition at disk surface |
| Abbreviations | = | |
| CHF | = | constant heat flux |
| CWT | = | constant wall temperature |
Nomenclature
| Ap | = | projected area of the disk normal to the direction of flow (m2) |
| Bn | = | Bingham number |
| = | modified Bingham number (= Bn/(1 + Bn)) (dimensionless) | |
| Bncr | = | critical Bingham number for cessation of flow separation (dimensionless) |
| Br | = | Brinkman number |
| C | = | thermal heat capacity of fluid (J kg−1 K−1) |
| CD | = | total drag coefficient |
| = | drag coefficient | |
| CDF | = | friction (or viscous) drag coefficient |
| CDP | = | pressure (or form) drag coefficient |
| CDS | = | Stokes drag coefficient (dimensionless) |
| Cp | = | pressure coefficient |
| d | = | disk diameter (m) |
| FD | = | drag force (N) |
| FDF | = | friction (viscous) component of the drag force (N) |
| FDP | = | pressure (form) component of the drag force (N) |
| h | = | local heat transfer coefficient (W m−2 K−1) |
| jH | = | Colburn jH-factor for heat transfer |
| k | = | thermal conductivity of fluid (W m−1 K−1) |
| L | = | size of the computational domain (m) |
| Lr | = | recirculation length measured from the rear stagnation point (dimensionless) |
| M | = | parameter used in Papanastasiou model, Eq. (8) (dimensionless) |
| ns | = | unit vector normal to the surface of disk (dimensionless) |
| N | = | total number of cells in the half computational domain (dimensionless) |
| Np | = | number of grid points on the surface of disk (dimensionless) |
| Nu | = | average Nusselt number (dimensionless) |
| NuH | = | average Nusselt number for CHF case (dimensionless) |
| Nul | = | local Nusselt number on the surface of disk (dimensionless) |
| NuT | = | average Nusselt number for CWT case (dimensionless) |
| = | conduction limit of the Nusselt number (dimensionless) | |
| p | = | pressure (dimensionless) |
| pref | = | reference pressure (Pa) |
| Pe | = | Peclet number |
| Pr | = | Prandtl number (dimensionless) |
| = | modified Prandtl number (= Pr (1 + Bn)) (dimensionless) | |
| q0 | = | uniform heat flux on the surface of the disk (W m−2) |
| r, z | = | cylindrical coordinates |
| R | = | radius of the disk (m) |
| Re | = | Reynolds number |
| Recr | = | critical Reynolds number for the onset of flow separation (dimensionless) |
| = | modified Reynolds number | |
| S | = | surface area of disk (m2) |
| t | = | disk thickness (m) |
| T | = | temperature of fluid (K) |
| V | = | velocity vector (dimensionless) |
| V0 | = | free stream velocity (m s−1) |
| Vr, Vz | = | velocity components in r- and z-directions (dimensionless) |
| Greek symbols | = | |
| ∇ | = | del operator (dimensionless) |
| = | rate of deformation tensor (dimensionless) | |
| Δ | = | minimum grid spacing on the surface of disk (m) |
| η | = | scalar viscosity function (dimensionless) |
| λ | = | parameter used in Bercovier and Engelman model, Eq. (10) (dimensionless) |
| = | fluid viscosity (Pa s) | |
| = | yielding viscosity used in bi-viscosity model, Eq. (9) (Pa s) | |
| ξ | = | temperature of fluid (dimensionless) |
| ρ | = | density of fluid (kg m3) |
| τ | = | deviatoric stress tensor (dimensionless) |
| = | fluid yield stress (Pa) | |
| Subscripts | = | |
| = | dimensional variable | |
| 0 | = | corresponds to faraway (free stream) condition |
| w | = | condition at disk surface |
| Abbreviations | = | |
| CHF | = | constant heat flux |
| CWT | = | constant wall temperature |
Acknowledgments
R. P. Chhabra would like to thank the Department of Science and Technology, Government of India, New Delhi for the award of a J. C. Bose fellowship to him for the period 2015–2020.