Generating, from scratch, the near-field asymptotic forms of scalar resistance functions for two unequal rigid spheres in low-reynolds-number flow

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1 Generating, from scratch, the near-field asymptotic forms of scalar resistance functions for two unequal rigid spheres in low-reynolds-number flow Adam K. Townsend arxiv:8.86v3 [physics.flu-dyn] 5 Jan 9 Department of Mathematics, UCL, Gower Street, London WCE 6BT, UK 8 January 9 When investigating the behaviour of particle suspensions, for example in the Stokesian dynamics simulation technique, it is sometimes necessary to use near-field asymptotic forms of scalar resistance functions for two unequal rigid spheres, commonly notated X, A X, A Y, A Y, A Y, B Y, B X, C X, C Y, C Y, C X, G X, G Y, G Y, G Y H, Y H, X M, X M, Y M, Y M, Z M, Z M. () The required expressions for generating these scalars were initially published in Jeffrey & Onishi (984) and Jeffrey (99). These important papers suffer from a number of small errors, and furthermore, the reader may find it difficult to generate the required expressions (and therefore the value of these functions) independently, given the omission of intermediate formulae. A partial list of errata has been published by Kengo Ichiki ( net/twobody/errata.html), and some of these errors appear to have been noticed by authors using these papers in their extensions. However, I have not found a comprehensive description of how to fully generate, from scratch, expressions for these functions. This short article is a compilation of the relevant equations, with those originally omitted now added, and with any errors fixed. Equations from Jeffrey & Onishi (984) are labelled (JO.), those from Jeffrey (99) are labelled (J ), and those from the helpful Ichiki et al. (3) are labelled (I ). At the end of the article, the questions how do we know there are errors? and how do we know they are fixed? are addressed. Throughout this article, we use the same notation as these papers. For two spheres of a.townsend@ucl.ac.uk

2 radius a, a a distance s apart, we define the non-dimensional gap and size ratio λ as = The near-field forms are valid for and λ. s a + a, λ = a a. () In order to match the far-field forms, it is necessary to scale the terms by multiplying respectively by X A, Y A, Y B, X C, Y C, X G, Y G, Y H, X M, Y M, Z M (3),, 3, 4 3, 4 3, 3, 3, 4 3, 9, 9, 9. (4) X A terms Here the X A formulae are given in full, with changes from the source material when noted. The same directions for alteration, when required, will be given for the other terms in later sections. Set up the recurrence relations P n = δ n, V n = δ n, n V npq = P npq (n + )(n + 3) q ( n + s P npq = n s= q s= ( n + s n (5) (6) ) P s(q s)(p n ), (7) ) ( n(n + )(ns n s + ) P s(q s)(p n+) (n + )(s )(n + s) n(n ) (n + ) P s(q s)(p n ) n (4n ) (n + )(s + ) V s(q s )(p n+) (JO ). Then define the formulae f k (λ) = k ), (8) k P (k q)q λ q, (9) q= g (λ) = λ ( + λ) 3, () g (λ) = 5 λ( + 7λ + λ )( + λ) 3, () g 3 (λ) = 4 ( + 8λ 9λ + 8λ 3 + λ 4 )( + λ) 3, () m (m) = δ m + (m )( δ m ), (3)

3 (JO 3.5, 3.9); and A X = 4 g + m= m even [ m ( + λ) m f m g m g + 4m m g 3 ], (4) ( + λ)ax = 4 g + g log g 3 + m ( + λ) m f m g m g + 4m (m + ) g 3, (5) m= m odd (JO ), noting the correction from m to m + in A X. Then the resistance scalars are given by X A = g + g log( ) + A X + g 3 log( ), (6) X A = g + g log( ) ( + λ)ax + g 3 log( ), (7) from (JO ) up to O( log( )): note the different factor on X A. Y A terms The recurrence relations are (JO ), but with V npq corrected to V npq = P npq + n (n + )(n + 3) q ( ) n + s P n + s(q s)(p n ), (8) s= noticing the sign change on the in the last subscript. The required intermediate formulae for the f, g and m functions are eq. (9), (JO ), and eq. (3), respectively. Then the A Y formulae, terms are given by (JO ), leaving us with the resistance scalar Y A = g log( ) + A Y + g 3 log( ), (9) Y A = g log( ) ( + λ)ay + g 3 log( ), () from (JO ), with a different factor on Y A. Y B terms The recurrence relations are the same as those for the Y A terms. The required intermediate formulae for the f and g functions are f k (λ) = k+ and (JO between 5.6 and 5.7), respectively. k Q (k q)q λ q, () q= 3

4 The B Y terms are given by B Y = g log g 3 + m ( + λ) m f m m g + 4m (m + ) g 3, () m= m odd 4 ( + λ) B Y = g 3 + m ( + λ) m f m m g + 4m (m + ) g 3, (3) m= m even having been corrected from (JO ). Then the resistance scalars are given by Y B = g log( ) + B Y + g 3 log( ), (4) Y B = g log( ) 4 ( + λ) B Y + g 3 log( ), (5) from (JO ), with a different factor on Y B. X C terms Expressions for the resistance scalars can be expressed directly as ( ) X C λ 3 = ( + λ) ζ λ λ 3, 3 + λ 4( + λ) log( ), (6) X C = λ3 ( + λ) ζ(3, ) + λ 3 4( + λ) log( ), (7) (JO ), where X C has been divided by 8/( + λ) 3, and where ζ(z, a) is the Hurwitz zeta function, ζ(z, a) = (k + a). (8) z k= Y C terms The recurrence relations are the same as those for the Y A terms except the initial conditions are replaced by (JO ). The intermediate formula for the f function is f k (λ) = k k Q (k q)q λ q+(k mod ), (9) q= with the g formula given by (JO between 7. and 7.), with the correction to g 5 of g 5 (λ) = 5 λ(43 4λ + 43λ )( + λ) 4. (3) 4

5 Then the C Y terms are C Y = g 3 + m ( + λ) m f m m g + 4m (m + ) g 3, (3) m= m even C Y = g 4 log g m ( + λ) 3 m f m m g 4 + 4m (m + ) g 5, (3) m= m odd noting the corrections to both of (JO 7.-7.). The resistance scalars are finally Y C = g log( ) + C Y + g 3 log( ), (33) 8 ( + λ) Y C 3 = g 4 log( ) + C Y + g 5 log( ), (34) noting the different factor on Y C from (JO ). X G terms The recurrence relations are the same as those for X A, and the f and g functions are (I 94) and (J between 9b and a). The G X terms are given by (J ), noting that in their notation, f(λ) = m f(λ). This gives us expressions for X G of X G = g + g log( ) + G X + g 3 log( ), (35) X G = g g log( ) + 4 ( + λ) G X g 3 log( ), (36) from (J 9) with a different factor on the X G. Y G terms The recurrence relations are the same as those for Y A, and the f and g functions are (I 5) and (J between 7b and 8a). The G Y terms are given by (J 9), giving us expressions for Y G of Y G = g log( ) + G Y + g 3 log( ), (37) Y G = g log( ) + 4 ( + λ) G Y g 3 log( ), (38) from (J 7) with a different factor on the Y G. Y H terms The recurrence relations are the same as those for Y C, and the f and g functions are (I ) and (J between 35b and 36a). The G Y terms are given by (J 37), giving us 5

6 expressions for Y H of Y H = g log( ) + H Y + g 3 log( ), (39) Y H = g 5 log( ) + 8 ( + λ)3 H Y + g 6 log( ), (4) from (J 35) with a different factor on the Y H. X M terms The recurrence relations are the same as those for X A, but with the different initial conditions (J 44). The f and g functions are given by (I 5) and (J between 48b and 49a). The M X terms are given by (J 5), giving us expressions for X M of X M = g + g log( ) + M X + g 3 log( ), (4) X M = g 4 + g 5 log( ) + 8 ( + λ)3 M X + g 6 log( ), (4) from (J 48) with a different factor on the X M. Y M terms The recurrence relations are the same as those for Y A, but with the different initial conditions (J 58). The f and g functions are given by (I 5) and (J between 64b and 65a). The M Y terms are given by (J 66), giving us expressions for Y M of Y M = g log( ) + M Y + g 3 log( ), (43) Y M = g 5 log( ) + 8 ( + λ)3 M Y + g 6 log( ), (44) from (J 64) with a different factor on the Y M. Z M terms The recurrence relations are (J 73 76). The f and g functions are given by (I 3) and (J between 79b and 8a). The M Z terms are given by (J 8), giving us expressions for Z M of from (J 79) with a different factor on the Z M. Z M = M Z + g 3 log( ), (45) Z M = 8 ( + λ)3 M Z g 3 log( ), (46) How do we know there are mistakes? The original articles provide tabulated values of the intermediate scalars A X, etc. The easiest way for the reader to confirm mistakes in the formulae is to confirm that the values computed from these formulae do not match those tabulated. For example: 6

7 value from (JO) formulae value in (JO) tables correct value A X (λ = ) B(λ Y = ) Alternatively, these can be spotted by either deriving the equations independently, or observing that the values provided do not match those in the mid-field; both methods are described below. How do we know they are fixed? The reader is invited to derive and confirm the above formulae themselves, should they wish. The method is perhaps best explained in Jeffrey (99), sections II (starting at the paragraph containing the definition of ), III B & III C, where X G is used as an example. We can also confirm that the formulae produce values of the resistance scalars which match those in the mid-field. Figures to 3 demonstrate the near-field values matching to the mid-field values, which have been computed independently for.4 using the two-sphere method of Wilson (3), based on the solution to Stokes flow given by Lamb (93). Recall that the near-field equations are valid only for λ. Acknowledgements With thanks to Anubhab Roy (IIT Madras), Donald L. Koch (Cornell), Pankaj Rohilla (Texas Tech), Yixiang Luo and Aleksandar Donev (NYU) for correcting my own typos. References Ichiki, K., Kobryn, A. E. & Kovalenko, A. 3 Resistance functions for two unequal spheres in linear flow at low Reynolds number with the Navier slip boundary condition. arxiv:3.46 [cond-mat, physics:physics]. Jeffrey, D. J. 99 The calculation of the low Reynolds number resistance functions for two unequal spheres. Physics of Fluids A: Fluid Dynamics 4 (), 6 9. Jeffrey, D. J. & Onishi, Y. 984 Calculation of the resistance and mobility functions for two unequal rigid spheres in low-reynolds-number flow. Journal of Fluid Mechanics 39, 6 9. Lamb, H. 93 Hydrodynamics. Cambridge University Press. Wilson, H. J. 3 Stokes flow past three spheres. Journal of Computational Physics 45,

8 XA 4 3 ZM.4. XA 4. YM YA YB YC XG 4 XM YM ZM. YH.5 4 YG..5. XC YH YG XM 4 XC.4.35 XG 4 4 YB YC λ= Resistance scalars for YA Figure : Values of the scalar resistance functions over non-dimensional gap,, for size ratio λ =. Those generated from the near-field formulae are represented by hollow circles (#), and those generated from Lamb s solution (Wilson, 3) are filled circles ( ). The dashed vertical line appears at = λ, recalling that the near-field formulae are only valid for λ. XA 3.5 λ =. Resistance scalars for YA YB. XC YC XG 3 YG YH.5 XM 3 YM YA 3 YC XM XC XG 3 YG YH.5 YB XA ZM YM. 3. ZM Figure : Values of the scalar resistance functions over non-dimensional gap,, for size ratio λ =.. Those generated from the near-field formulae are represented by hollow circles (#), and those generated from Lamb s solution (Wilson, 3) are filled circles ( ). The dashed vertical line appears at = λ, recalling that the near-field formulae are only valid for λ. 8

9 Resistance scalars for λ =. X A.5. Y A. Y B X C Y C X G Y G Y H.35 X M Y M 6 +. Z M X A Y A.5 Y B 6 X C Y C X G Y G.5 5 Y H. X M 4 Y M 6 Z M Figure 3: Values of the scalar resistance functions over non-dimensional gap,, for size ratio λ =.. Those generated from the near-field formulae are represented by hollow circles ( ), and those generated from Lamb s solution (Wilson, 3) are filled circles ( ). The dashed vertical line appears at = λ, recalling that the near-field formulae are only valid for λ. 9

Anomalous effect of turning off long-range mobility interactions in Stokesian Dynamics

Anomalous effect of turning off long-range mobility interactions in Stokesian Dynamics Adam K. Townsend, a), b) and Helen J. Wilson Department of Mathematics, University College London, Gower Street, London