Time-dependent lift force acting on a particle moving arbitrarily in a pure shear flow, at small Reynolds number

In this paper we seek an approximate expression for the time-dependent lift force, initially derived by Saffman in the steady case, by using the Asmolov and McLaughlin frequential results. The single-frequency expression derived by these authors can indeed be used to obtain a temporal expression of the lift force acting on a sphere with arbitrary motion, since the problem is linear. The major difficulty lies in the fact that their frequential results are given under a very complicated algebraic expression. Therefore, no analytical inverse Fourier transform can be carried out and the expression of the time-dependent lift force acting on a particle cannot be obtained in the general case. In the present paper, however, it is shown that a closed temporal expression for this force can be found, provided one makes additional approximations. This expression takes the form of a convolution product involving an empirical kernel as well as the slip velocity of the particle.