Approximate solutions for drag coefficient of bubbles moving in shear-thinning elastic fluids

The drag coefficient for bubbles with mobile or immobile interface rising in shear-thinning elastic fluids described by an Ellis or a Carreau model is discussed. Approximate solutions based on linearization of the equations of motion are presented for the highly elastic region of flow. These solutions are in reasonably good agreement with the theoretical predictions based on variational principles and with published experimental data.C_D Drag coefficient - E^* Differential operator [E^{* 2} =²/² + (sin/²)/(1/sin /)] - El Ellis number - F_D Drag force - K Consistency index in the power-law model for non-Newtonian fluid - n Flow behaviour index in the Carreau and power-law models - P Dimensionless pressure [=(p – p₀)/₀ (U/R)] - p Pressure - R Bubble radius - Re₀ Reynolds number [= 2R U/₀] - Re Reynolds number defined for the power-law fluid [= (2R)ⁿU^2–n/K] - r Spherical coordinate - t Time - U Terminal velocity of a bubble - u Velocity - Wi Weissenberg number - Ellis model parameter - Rate of deformation - Apparent viscosity - ₀ Zero shear rate viscosity - Infinite shear rate viscosity - Spherical coordinate - Parameter in the Carreau model - ^* Dimensionless time [=/(U/R)] - Dimensionless length [=r/R] - Second invariant of rate of deformation tensors - ^* Dimensionless second invariant of rate of deformation tensors [=/(U/R)²] - Second invariant of stress tensors - ^* Dimensionless second invariant of second invariant of stress tensor [=/₀²(U/R)²] - Fluid density - Shear stress - ^* Dimensionless shear stress [=/₀ (U/R)] - _1/2 Ellis model parameter - ₁^2/* Dimensionless Ellis model parameter [=_1/2/₀(U/R)] - Stream function - ^* Dimensionless stream function [=/UR²]