The force on a bubble, drop, or particle in arbitrary time-dependent ...
The force on a bubble, drop, or particle in arbitrary time-dependent ...
The force on a bubble, drop, or particle in arbitrary time-dependent ...
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unsteady p<strong>or</strong>ti<strong>on</strong> is small. A very similar derivati<strong>on</strong> can be<br />
perf<strong>or</strong>med here. <str<strong>on</strong>g>The</str<strong>on</strong>g> <strong>in</strong>terested reader is referred to the<br />
<strong>or</strong>ig<strong>in</strong>al w<strong>or</strong>k f<strong>or</strong> the details. <str<strong>on</strong>g>The</str<strong>on</strong>g> result is<br />
FH (t=7<br />
a(Z,+2Uj(t)+!Re[1+2U,(t)])<br />
± 2lrpa 3 U 1 (t) ±S8Tfa e vts/<br />
FA(t)<br />
-8ei"<br />
t (1/3 2/3A + 5/9<br />
6irftaU (A+l± (A+ 1)2<br />
( 1 +a) 2 f(a13)<br />
2Ag(af3) +(3±a)f(af3))<br />
-Re 5e-M- Z+2/3 ) 2<br />
- R<br />
e ~ e ~ ' )+<br />
with<br />
Xeric -T-- U(s) ds<br />
8-irjia ft U S ds<br />
TTJ_ 0 , T9v(t-s) Ia 2<br />
GWt =etU 2 /4v<br />
tU/ r X 3 /' 2 -xtU 2 /4v dx,<br />
J (l+ijI~ e<br />
(69)<br />
(70)<br />
where the imposed flow U' (t) = U[1 + U 1 (t)] has the c<strong>on</strong>diti<strong>on</strong><br />
U 1 (t) Re. At low frequency<br />
(e
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