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Credit: [Notes on Curtis and "Collision Efficiency"]. Source: Wellcome Collection.
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![In order to solve the Stokes equations for this geometry, the coordinate system used by Stimson and Jeffrey 10 in their discussion of the axisymmetric Stokes flow past two spheres is required. These coordinates are defined by r= (A 2 — 1) J sin {/(cosh {—cos 0. (A8) z = (A 2 -1)1 sinh {/(cosh {-cos 0, (A9) and the spheres are the coordinate surfaces { = ±{ 0 , where cosh {o = h. (AIO) For the axisymmetric part of the solution, which satisfies the conditions (A5), a stream function can be used, defined by u\ = d\Iz/rdz, u\ = — di¡/¡rSr, and the value of í ¡/ which yields velocity components satisfying the Stokes equations (A2, A3) and the conditions at infinity is, as was shown by Stimson and Jeffrey, 1 ifr = (cosh { cos Q-ä- £ {A„smh(n-ì)è+B„smh(n + m}(Pr,- i -P n+l ), (All) where P„(cos{) is a Legendre polynomial. The conditions (A5) can now be applied with { = {o, and the conditions on { = — { 0 are automatically satisfied, since only the part of the complete solution which is odd in { has been included. The resulting equations can be solved for the coefficients A„ and B„. The force on the sphere { = { 0 is in the z-direction, and is given by (6nr¡a 2 k)F 3 , where F '-3ilb;| 1 (2 +1)( ' 1 - +i - ) - (A12) and is non-dimensional. When the values of the coefficients A n and 1 B„ are substituted, the value of this expression is found to be ^3 = -/iK 3 -/X sin «eos a (A13) h - * ,in { ° I, Tsr{áÍT) (2{ °'- 1+ 2§rs * A„ = 2 sinh (2/1 +1){ 0 —(2>i+1) sinh 2{ 0 . (A16) The force on the sphere { = — { 0 is of equal magnitude but in the opposite direction. The method of using these co-ordinates to deal with non-axisymmetric motions of two spheres was devised by Dean and O'Neill 11 and extended to deal with situations similar to the present one by O'Neill 12 and Wakiya. 13 Following these authors, we write the pressure and the components of u 2 as p 2 = r¡P cos 0/sinh { 0 , ? = iO^/sinh £ 0 +X+ Y) cos 0, u* = i(X-Y) sine, (A17) u] = Kzí/'sinh { 0 +2Z) cos 0, where P = (cosh {- cos 0* sin { £ C„ sinh (n + i)çP'„, X = (cosh {- cos 0* sin 2 { £ D „ sinh in+MP'l, . =2 ( A18 > Y = (cosh {- cos 0* £ E n sinh (n+£{P„, Z = (cosh {- cos 0* sin { £ H n cosh (h + l)çP' n , P' n (ß) = dPJdfi; PKji) = d 2 PJdß 2 . The boundary conditions (A6) then reduce to the sets of equations (cosh {„- cos Q 1 sin 2 { £ C„ sinh (n+i)Ç 0 P' n + sin 2 { £ D„ sinh (n+i)£ 0 P'¿+ £ E„ sinh (n + i){ 0 P„ = (cosh {„- cos 0~*(öi + fc 2 z), sin 2 { £ D„sinh(n + |){ 0 P;'- £ E„ sinh (n+i){ 0 P„ = - (cosh { 0 - cos 0 i (b L + b 2 z), (A 19) (cosh Co- cos 0 1 sinh {„ sin { £ C„ sinh (n+i)!; a P'„ + 2 sin { £ H„ cosh (« + i){ 0 P; = (cosh { 0 - cos 0~*b 3 r, where the coefficients b u b 2 and 63 are b 1 = 2(Ki—e> 2 Ä), b 2 = 2(a> 2 —k¡ sin 2 a), b 3 = 2(-a> 2 +k t cos 2 a). (A20) These equations also hold for u 3 if cos0 is replaced by sin0, sin0 by —cos0, and the ¿-coefficients have the values Ô! = 2( V 2 +w 1 /i), b 2 =-2(0!, Ô3 = 2(oj L +k 2 cosa). (A21) Eqn (A19) determine the coefficients in terms of the single set H n and the equation of con tinuity (div u 2 = 0) gives the set of conditions 5C„—(n— 1 )C„_ 1 +(n+ 2 )C n+1 + («- 2)(«- l)A>-i - 2(h- l)(n+ 2)Z>„+(«+ 2)(n+3)ö„+i - £■„_! + 2£„-£„ +1 -2(«- l)if I _ 1 + 2(2»+ l )H„-2(.n+2)H„ +l = 0. (A22) There results a set of linear equations each connecting three coefficients H n _ u H n and H n+1 , which can be solved numerically by truncating the infinite set of equations, but including enough equations to enable the forces to be determined as accurately as desired. The forces and couple derived from u 2 can then be found from the formulae F t = — (2*/3)sinh i« £ E„, = I (A23) G2 = -(2*/3) sinh 2 { 0 £ (2» + l- coth { 0 )£„, and the same expressions, when the values (A21) are given to b u b 2 and è 3 give the values of F 2 and — G u respectively, derived from u 3 . This completes the determination of the hydrodynamic forces.](https://iiif.wellcomecollection.org/image/b18182458_PP_CRI_H_4_21_0005.jp2/full/800%2C/0/default.jpg)
