Faxen’s laws

This result should be checked against Faxen s law, which will be proven in Chap. 8. Faxen s law for a solid sphere holds that the force and torque that are due to an arbitrary undisturbed flow, Uoo = u(r), that satisfies the creeping-motion equations can be calculated from the following formula ... 

This important result is known as Faxen s law.22 According to this law, if we specify the undisturbed velocity u°°(x), then the force on a sphere can be calculated directly from the formula (8-220), without any need to actually solve the flow problem corresponding to the free-stream velocity u°°(x). 

Now, if the Reynolds number of the flow is sufficiently small for the creeping-motion approximation to apply, it can be shown by the arguments of Subsection B.3 in Chap. 7 that no lateral motion of the drop is possible unless the drop deforms. In other words, Us = Useiin this case, though, of course, Us is not generally equal to the undisturbed velocity of the fluid evaluated at the X3 position of the drop center. The drop may either lag or lead (in principle) because of a combination of interaction with the walls and the hydrodynamic effect of the quadratic form of the undisturbed velocity profile - see Faxen s law. Because the drop deforms, however, lateral migration can occur even in the complete absence of inertia (or non-Newtonian) effects. In this problem, our goal is to formulate two... 

The symbolic operators (87)-(88) for the sphere possess a greater degree of generality than do Faxen s laws. In particular, if we consider any Stokes flow v(r, r/r) vanishing at infinity and satisfying the arbitrary boundary condition V = f(r/r) s f(0, < ) at r = a, then the force on the sphere may be obtained directly from the prescribed velocity boundary condition via the expression... 

When the operators appearing in (96) and (97) are expanded via their infinite series representations the resulting expansions do not terminate after a finite number of terms, as do Faxen s laws for the sphere, Eqs. (89). 

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