Creeping flow around a spherical bubble. When .1 liquid flows around .1 gas bubb

Question

Creeping flow around a spherical bubble. When .1 liquid flows around .1 gas bubble, circular lion takes place within the bubble. This circulation lowers the interfacial shear stress. and, to a first approximation, we may assume that it is entirely eliminated Repeat the development of Ex. 4.2-1 for such a gas bubble. assuming it is spherical. Show that B.C. 2 of Ex. 4.2-1 is replaced by at r = R, d/dr (1/r^2 df/dr) + 2 f/r^4 = 0 and that the problem set-up is otherwise the same Obtain the following velocity components: Upsilon_r= Upsilon_Infinity [1 - (R/r)] cos theta Upsilon_ = -Upsilon__infinity [1 - 1.2 (R/r)] sin theta Next obtain the pressure distribution by using the equation of motion: p = p_0 - rho g h - (mu upsilon_infinity/R) (R/r)^2 cos theta

Explanation / Answer

stokes law is the name given to the formula describing the force F on a stationary sphere of radius a held in a fluid of viscosity moving with steady velocity V. this is usually expressed in the form

F=pieVa

by translation can be more conveniently be expressed in terms of a drag coefficient and a reynolds number defined as follows

Cd=F/1/2rowV2piea2

Re=2aVp/n

these variables, takes the form

Cd=24/Re

however applicable for slow flows and should only be used for Re<1. the reason for this results for higher reynolds numbers are discussed below

F=4pienVa

Cd=16/Re

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