The steady axisymmetric flow internal and external to a deformable viscous liquid drop falling through a quiescent gas under the action of gravity is computed by solving the nonlinear Navier–Stokes equations using a Galerkin finite-element method with a boundary-fitted quadrilateral mesh. Considering typical values of the density and viscosity for common liquids and gases, numerical solutions are first computed for the liquid-to-gas density ratio ρ = 1000 and viscosity ratio μ from 50 to 1000. Visually noticeable drop deformation is shown to occur when the Weber number We ~ 5. For μ ≥ 100, drops of Reynolds number Re < 200 tend to have a rounded front and flattened or even dimpled rear, whereas those at Re > 200 a flattened front and somewhat rounded rear, with that at Re = 200 exhibiting an almost fore–aft symmetric shape. As an indicator of drop deformation, the axis ratio (defined as drop width versus height) increases with increasing We and μ, but decreases with increasing Re. By tracking the solution branches around turning points using an arclength continuation algorithm, critical values of We for the ‘shape instability’ are determined typically within the range of 10 to 20, depending on the value of Re (for Re ≥ 100). The drop shape can change drastically from prolate- to oblate-like when μ < 80 (for 100 ≤ Re ≤ 500). For example, for μ = 50 a drop at Re ≥ 200 exhibits a prolate shape when We < 10 and an upside-down button mushroom shape when We > 10. The various solutions computed at ρ = 1000 with the associated values of drag coefficient and drop shapes are found to be almost invariant at other values of ρ (e.g. from 500 to 1500) as long as the value of ρ/μ2 is fixed, despite the fact that the internal circulation intensity changes according to the value of μ. The computed values of drag coefficient are shown to agree quite well with an empirical formula for rigid spheres with the radius of the sphere replaced by the radius of the cross-sectional area.
A novel predictive model of drag coefficient and settling velocity of drill cuttings in non-Newtonian drilling fluids
