On The Modeling Of The Hydrodynamic Force Acting On A Spherical Particle

Abstract

A new model of the hydrodynamic force acting on a particle under an oscillating viscous fluid flow, at finite Reynolds and Strouhal numbers in the range of 0.01 to 25 is developed. This drag model is based on the novel concept of variable-order calculus, where the order of derivative can vary with the dynamics of the flow. A numerical simulation, based on a fourth-order compact scheme in space and a implicit third-order time-marching scheme has been used to compute the time-dependent, axisymmetric viscous flow past the rigid sphere. The solution is used to determine the appropriate differential operator (constant or variable) in the drag model for the several cases simulated. Also, it is determined: (i) the region of validity of the creeping flow equation for an oscillatory flow, (ii) the region where the order of the derivative is fractional, but constant and (iii) the region where the strong nonlinearity of the flow requires a variable-order derivative to account for the increased complexity of the drag force behavior. The proposed drag model is able to accurately predict the hydrodynamic force acting on the sphere within the parameter range investigated. For moderate values of the Reynolds numbers, the constant-order model provides an accurate description of the drag. Increasing the Reynolds number or decreasing the product of the Reynolds and Strouhal numbers leads to non-linear effects, which require a variable-order differential operator to model accurately the history effects on the drag.