Errata on the papers for two-body exact solutions in Stokes flows
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Here, some typos and errors in the publications
for the exact solutions of two particles in Stokes flows
by Jeffrey et al are summarized.
(Mathematical notations are done by
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-
D.J.Jeffrey and Y.Onishi,
J. Fluid Mech. 139 (1984) pp.261-290.
- (JO-1.6) and (JO-1.12):
those scalar functions are defined for the non-dimensional matrices
$\hat{A}, \hat{B}, \hat{C}, \hat{a}, \hat{b}, \hat{c}$,
not for the bare matrices $A, B, C, a, b, c$.
- (JO-3.11a) $\chi_{mn}^{(\alpha)} = (-1)^{3-\alpha} U \delta_{m0}\delta_{n1}$
- (JO-4.9) The index of the last P is $s (q-s) (p-n-1)$
- (JO-4.10) typo on the closing brace at the index of the second P.
- (JO-6.12) the index of $f$ is $2k$ (not $2j$)
- (JO-6.13) there are two corrections:
one term outside the summation (the third term in the right-hand side below)
and the factor 4 in the denominator at the second term in the summation
were missing. The correct form is
$X_{12}^{C} = \frac{4\lambda^2}{(1+\lambda)^4}\ln\frac{s+2}{s-2}
+\frac{8\lambda^2}{(1+\lambda)^4}s^{-1}\ln(1-4s^{-2})
-\frac{16\lambda^2}{(1+\lambda)^4}s^{-1}
-\frac{8}{(1+\lambda)^3}\sum_{k=1}^{\infty}
\left\{
(1+\lambda)^{-2k-1}f_{2k+1}
-2^{2k+2}k^{-1}(2k+1)^{-1}\frac{\lambda^2}{4(1+\lambda)}
\right\}s^{-2k-1}$
- p. 277 in JO,
$g_5 = \frac{1}{125}\lambda (43 - 24\lambda + 43\lambda^2)(1+\lambda)^{-4}$
,from
Ladd (1990) J.Chem.Phys. p.3487 (or Kim-Mifflin -- I do not check it).
- p.282 in JO (3rd line from the bottom),
the left-hand side is $P_{1pq}$ (not $p_{1pq}$).
- (JO-12.1) $Q_{1pq} = \delta_{0p} \delta_{0q}
- (JO-12.2) $x_{11}^{c} = \sum f_{n}(\lambda) (1+\lambda)^{-n} s^{-n}$
(drop the factor of $2^{n}$ from the right-hand side)
- (JO-12.3)
$x_{12}^{c} = -\frac{1}{8}(1+\lambda)^{3}\sum f_{n}(\lambda) (1+\lambda)^{-n} s^{-n}$
(drop the factor of $2^{n}$ from the right-hand side)
-
D.J.Jeffrey,
Phys. Fluids A 4 (1992) pp.16-29.
- Note: the differences of (J-60) and (J-61) from (JO-4.10) and (JO-4.11)
are coming from the different definition of $Q_{npq}$.
That is, $Q$ in [J-1992] is equal to $(2/3)$ times $Q$ in [JO-1984].
-
D.J.Jeffrey, J.F.Morris and J.F.Brady,
Phys. Fluids A 5 (1993) pp.2317-2325.
- at p. 2321, $f_8$ for $X^Q$ should be
- $f_8 = 432 \lambda^2 -1410 \lambda^3 +5400 \lambda^4 +6240 \lambda^5$.
