Coarse Particles in Homogeneous Non-Newtonian Slurries: Combined Effects of Shear-Thinning Viscosity and Fluid Yield Stress on Drag and Heat Transfer from Hemispherical Particles

  • Om Prakash
  • S. A. Patel
  • A. K. Gupta
  • R. P. ChhabraEmail author
Technical Paper


In this work, the momentum and energy equations have been solved numerically for predicting the hydrodynamic drag and heat transfer coefficient for a hemispherical particle submerged in a flow stream of yield-pseudoplastic fluids in order to elucidate the combined effects of shear-thinning viscosity and fluid yield stress. In this case, the momentum transfer aspects are influenced by the values of the Reynolds number (0.1 ≤ Re ≤ 100), Bingham number(0 ≤ Bn ≤ 100), shear-thinning index (0.2 ≤ n ≤ 1) and the orientation of the hemisphere. Similarly, the corresponding heat transfer results show additional dependence on the Prandtl number (0.7 ≤ Pr ≤ 100) and the type of thermal (isothermal or isoflux) boundary condition specified on the surface of the heated hemisphere. The numerical results are discussed in terms of the size and shape of the fluid-like yielded regions, wake lengths, hydrodynamic drag and heat transfer coefficients as functions of the preceding dimensionless parameters. Finally, the present values of the drag coefficient and Nusselt number have been fitted using simple expressions thereby enabling the interpolation of the present results for the intermediate values of the parameters and/or their prediction in a new application.


Hemisphere Bingham number Nusselt number Herschel–Bulkley fluid Reynolds number 

List of symbols


Surface area of hemisphere (m2)


Bingham number, Eq. (9), dimensionless


Total drag coefficient, dimensionless


Viscous (or friction) drag coefficient, dimensionless


Pressure (or form) drag coefficient, dimensionless


Specific heat of fluid (J kg−1 K−1)


Diameter of hemisphere (m)

\(D_{\infty }\)

Diameter of computational domain (m)


Colburn-j factor, dimensionless


Thermal conductivity of fluid (W m−1 K−1)


Fluid consistency index (Pa sn)


Recirculation length measured from the center of the hemisphere, dimensionless


Growth rate parameter used in Papanastasiou model, Eq. (5), dimensionless


Shear-thinning (or fluid behaviour) index, dimensionless


Prandtl number, Eq. (10), dimensionless


Modified Prandtl number (=Pr(1 + Bn)), dimensionless


Constant wall heat flux prescribed on the hemisphere (W m−2)


Reynolds number, Eq. (8) dimensionless


Modified Reynolds number (=Re/(1 + Bn)), dimensionless


Fluid temperature (K)


Fluid temperature in the free stream (K)


Constant wall temperature prescribed on the hemisphere (K)


Velocity vector, dimensionless


Free stream velocity (m s−1)


Cartesian coordinates (m)

Greek symbols

\(\dot{\gamma }\)

Rate of deformation tensor, dimensionless

\(\dot{\gamma }_{e}^{{\prime }}\)

Parameter used in Bercovier and Engelman model, Eq. (7), dimensionless


Plastic viscosity of Herschel–Bulkley fluid model (=K(V 0/d) n−1) (Pa s)


Yielding viscosity used in bi-viscosity model, Eq. (6) (Pa s)


Density of fluid (kg m−3)


Deviatoric stress tensor, dimensionless


Fluid yield stress (Pa)


Fluid temperature (=(T − T 0)/(T w  − T 0) for constant wall temperature); (=T − T 0/(q 0 d/k) for constant wall heat flux), dimensionless



RPC gratefully acknowledges the financial support of the Department of Science and Technology (Government of India) via the award of a J C Bose fellowship to him for the period 2015–2020. Funding was provided by Department of Science and Technology, Ministry of Science and Technology (SB/S2/JCB-06/2014).


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Copyright information

© The Indian Institute of Metals - IIM 2016

Authors and Affiliations

  • Om Prakash
    • 1
  • S. A. Patel
    • 1
    • 2
  • A. K. Gupta
    • 1
  • R. P. Chhabra
    • 1
    Email author
  1. 1.Department of Chemical EngineeringIndian Institute of Technology KanpurKanpurIndia
  2. 2.Department of Chemical, Biological, and Pharmaceutical EngineeringNew Jersey Institute of TechnologyNewarkUSA

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