The force on a bubble, drop, or particle in arbitrary time-dependent ...

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motion at small but finite Reynolds number. <strong>The</strong> general expression is given by ?fdU=F+Fet (75) where md is the mass of the body. Here, we have assumed that the only <strong>force</strong>s acting on the body are hydrodynamic <strong>force</strong>s and external body <strong>force</strong>s, Fext, such as the buoyancy <strong>force</strong> due to gravity. In the absence of velocity gradients in the imposed flow (the requirement here is that the Oseen time scale v/Ut is much less than the characteristic shear rate in the imposed flow), the general expression for the hydrodynamic <strong>force</strong> is given by (51), where one can take into account the influence of the deformation of the body by including the last term of (26). <strong>The</strong> two volume integrals in (51) are absent for spherical bodies. For long time scale motion ( > a 2 /v) of nonspheres they can be approximated using the solution of the steady Stokes flow field for the translating body. Under this condition, it can be shown that they contribute only a side <strong>force</strong> perpendicular to the direction of motion. I For a spherical body the hydrodynamic <strong>force</strong> to O(Re) simplifies to Fd- a~p t) + FH (St_ f - 4, _7_Sd WAe
where mf is the mass of the exterior fluid displaced by the sphere and g is the acceleration due to gravity. Here, we have assumed the only external <strong>force</strong> is the buoyancy <strong>force</strong>, Fext= (md-Mn)g. Apart from the two additional history integrals in (81), the governing equations for the two bodies are essentially the same, in that they have the same terms with only a difference in numerical coefficients. For very short time scale motion (
Page 1 and 2: The force<
Page 3 and 4: =fSd L -(r' d ,) J d d Tn-(,-o* Pd-
Page 5 and 6: = fV", (-Vp*).- (M*..)dV = .fVh -V
Page 7 and 8: govern the flow when the time scale
Page 9 and 10: sion for a bubble, (60), and with t
Page 11: 1/3 1 3 (A23 2 1 1 (74a) 4 Afla 3 R

The force on a bubble, drop, or particle in arbitrary time-dependent ...

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