Lateral Migration of a Sphere in Poiseuille Flow

Resistance Matrices for the Force and Torque on a Body in Creeping Flow 

Finally, linearity of the creeping-flow equations and boundary conditions allows a great a priori simplification in calculations of the force or torque on a body of fixed shape that moves in a Newtonian fluid. To illustrate this assertion, we consider a solid particle of arbitrary shape moving with translational velocity U(t) and angular velocity il(t) through an unbounded, quiescent viscous fluid in the creeping-flow limit Re 1 and Re/S 1. The problem of calculating the force or torque on the particle requires a solution of 

x is a position vector measured from the center of gravity of the particle. The force on the particle is 

The critical difficulty with this problem is that the solution depends on the orientations of U and f2 relative to axes fixed in the particle, as well as on the relative magnitudes of U and f2. Thus, for every possible orientation of U and/or f2, a new solution appears to be required to calculate u, p, F, or G. Fortunately, however, the possibility of constructing solutions of a problem as a sum (or superposition) of solutions to a set of simpler problems means that this is not actually necessary in the creeping-flow limit. Rather, to evaluate u, p, F, or G for any arbitrary choice of U and f2, we will show that it is sufficient to obtain detailed solutions for translation in three mutually orthogonal directions (relative to axes fixed in the particle) with unit velocity U = e, and il = 0, and for rotation about three mutually orthogonal axes with unit angular velocity il = e, and U = 0. 

We see that the problem is linear in U. Thus the solution (ui, p ) can depend only linearly on U, and in view of the relationships (7 18a) and (7-18b), this means that the force and 

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