Hydrodynamic Force Expression with Couple Stress Drag Over Sphere at Small Reynolds Numbers

Abstract

Introduction: The flow past a sphere has been an important problem since Stokes and the Navier Stokes (NS) equation is the basic governing equation for it. The main reason for its importance is that the NS equation is not yet solvable analytically and has been solved in the absence of either or both unsteady inertia and convective inertia. The hydrodynamic force on a spherical particle at low Reynolds numbers has been meticulously derived by Lovalenti and Brady [2] and Stokes [1], has derived the same for a couple stress fluid.

Objectives: In this problem we extend the works of Stokes [1] and Lovalenti and Brady [2] to obtain an expression for the hydrodynamic force on a couple stress flow over a rigid sphere at low Reynolds numbers.

Methods: We consider both the expressions for the hydrodynamic forces and combine them to get a new expression for the effects of couple stress fluids at low Reynolds numbers on a rigid sphere. Then the expression is validated by reproducing the results of both the authors.

Results: We consider both the extremes and obtain the plots of Stokes [1] and Lovalenti and Brady [2] and also obtain the transition from Newtonian fluid to Couple stress fluid at small Reynolds numbers. We note that the hydrodynamic force on the rigid sphere is the effect of viscous force and pressure and at zero couple stress! (an ideal situation) the fluid behaves as a Newtonian fluid with zero or constant flow when fluid is quiescent and as the couple stress parameters are introduces the force tend to grow from constant to exponential flow.

Conclusions: This work fills the gap in the literature for the expression for force on a couple stress fluid on a rigid sphere at small finite Reynolds number.