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A Sphere Held Fixed in a Poiseuille Flow near a Rough Wall

We present here an analytical calculation of the hydrodynamic interactions between a smooth spherical particle held fixed in a Poiseuille flow and a rough wall. By the assumption of a low Reynolds number, the flow around a fixed spherical particle is described by the Stokes equations. The surface of the rigid wall has periodic corrugations, with small amplitude compared with the sphere radius. The asymptotic development coupled with the Lorentz reciprocal theorem are used to find the analytical solution of the couple, lift and drag forces exerted on the particle, generated by the second-order flow due to the wall roughness. These hydrodynamic effects are evaluated in terms of amplitude and period of roughness and also in terms of the distance between sphere and wall.