An expresskm for ~he tk)rce on a sphere moving ,aith a ~ime-depv'ndent veiocily through an incompressible fldid m nonstationa
ON T i l E M O T I O N OF A S P H E R E WITH ARBITRARY SLIP IN A V I S C O U S I N C O M P R E S S I B L E FLUID A. ~,~.ALBANO*~. [). BEDEAUX and P. MAZLR l;~.~Hn~.l~t-Loremz, R(jkxm~k,e~'~'Refl Lekh~J~, Leh~).'m~ 77~.' ?~bg:eH(md',
f{ccei';~:d 24 Janua fie}J, P the pressure ter~sor, F~.. an external force exerted on the fluid, p the hydrostatic pressure, :? the shear viscosity and R(t) the position o f the sphere at time :. ~n ~he italy lh~ca.rized scheme, ~he time depe~dence o f R0) }n eq. (2,]) may be negiect.ed a e d R(I) may there%~e be replaced by zero. This poit~t b~ more fulty disct~ssed i~~ ref\ }. The motion o f Ibe sphere is ther~ governed by d m
--
d/ where o~ the vector The
u(O
=
K(:),: +
&,,d:,i ...... = :[P (~"+ ~) - n d.S' +
,~;:', ~,~'::+
= •q j
s
m is the mass ot the sphere~ u its velocity, K the ~k:.~rceexerted by */he fi¢~Jd splhere and K~,,t an cxmrnal force; S is the suri2qce o f the sph.ere a~d n a trait ,~ormai to the sur|3ace pointing out o f the sphere. set of" equations (2.t) and (2.4) must be supplemented by ~-::4mdary con~
* In refs. 3 and 4 Basset also derives t\~r the case of stick botmdary c, :',4i~ions the t))rmu!a . for the time-dependem drag force in terms of the ~ime-depender~t~,e~ocv.~ ~ "'° of the partic|e~ This l\-trmula was later derived also by Boussinesqs) to whom it is usually attributeE
< ......
{
(
(2J-}} wi~h
Eq,
.~;L i 7)
K~;., )......... ,{(-> i *#
~ /
+,£-, ' ~a ....
~.
{'7
J ,,j
ON T H E M O T I O N OF A SPHERE W I T H A R B I T R A R Y SLIP
97
For zero frequency this reduces to Bas~et s fo~mtua : (3.22)
:;'(0) = 6::.rfa (t + 3,%/a)- ~ (1 + 2 2 i a ) .
ii) If the sphere is at rest, u = 0, and the unperturbed fluid velocity is stationary, ~%(r, 0 = ~b(r), we have {?ore eq. (3.15)
/~ = -~6=~;~a (t, + ~.,~:>.-)
L1 4-
---a\ 2
..... a 8 a / . . ]
"
(3.~a)
wigch is the analogue of" ~!~e original Faxd'n theorem for arbitrary slip.
Acknowledgements ]~he au@ors are indebted u) Pvogessor L, W a l d m a n n and Dr. trl.Vestner fo; drawing the!; at{ention to Basset',; work.
References 1) 2) 3) 4) 5t 6} 7,~ ?/} 9) 10} II}
P. Mazur arid D, Bcdeaux, Physica 76 (t974) 235 D. Bedeaux artd P. Mazt;r, Physica 78 (11974) 505. A, B. Basset, Phil. Trans. Roy. Soc. London ! 79 ~: 888 ! 43. A,B. Basset, frea~ise orl Hydrodyrmmics, voW.1t (Cambridge I.%i~ersity Press. Londcr.,., 1888) oh,2I, J. Boussir~esq, Th~Sorie Analydque de la C'haieur T. I! {Paris, I9{)3). D.Bedeaux and P. Mazt~r~ Physica 76 (1974) 247. P.S,t£ps~ein, Phys. Rev, 23 (1924) 710, R.Zwa~zig and M.Bixon, Phys. Rev. A2 (1970) 2005. E° H,.Hauge, private communication. H, Fa×dm Arkiv for Matazmatik, Astr. och Fysik~ Bd, ! g (1924). R,k~erke,~, Encyclopedia of Physics, vol. VII[/2~ Fluid Dynam c:~ H (~963).
~" Basset uses the "coefficieat of sliding friction% fl = ~pL
