## Appendix A. Multipole potentials

We define in this appendix some concepts from the theory of multipole potentials and briefly review some of its properties relevant to the work presented in the main body of this paper. For a more detailed discussion of the fundamentals of the method and its application to a variety of problems, the reader is referred to the textbooks by Leal (Reference Leal2007, chap. 8) – who names it as the method of superposition of vector harmonic functions – and Hess (Reference Hess2015, chap. 10). The material included in this review is taken from the latter.

The multipole potentials can be defined as tensorial solutions of the Laplace equation. There are two classes of multipole potentials, namely, descending and ascending potentials. The descending multipole potentials tend to zero when
$r\rightarrow \infty$
and diverge when
$r\rightarrow 0$
, where
$r\equiv |\text{x}|$
and
$r^{2}=x_{m}x_{m}$
. The descending multipole potentials are defined by

The first two descending potentials are given by

and

with the second potential being known as the dipole potential. The quadrupole and octupole potential tensors are

and

respectively. The rank-four multipole potential tensor is

Multipole potentials satisfy

This property is helpful in solving differential equations with elliptic operators. Multipole potential tensors are symmetric in any pair of indexes and, because they solve Laplace equation, also vanish after contracting any pair of indexes.

Finally, for problems in interior domains, the so-called ascending multipoles are needed. They arise because in (A 7), the factor
$g^{\prime \prime }-2\ell r^{-1}g^{\prime }=r^{2\ell }(r^{-2\ell }g^{\prime })^{\prime }=0$
not only for
$g=1$
– leading to the descending multipoles – but also for
$g=r^{(2\ell +1)}$
. Therefore, Laplace equation is also solved by

which are known as ascending multipole potentials. They are zero at
$r=0$
and, for
$\ell >0$
, diverge when
$r\rightarrow \infty$
.

## Appendix B. Solution of the ordinary differential equations

Consider the real-valued function
$\unicode[STIX]{x1D711}(x)$
. The ordinary differential equations in (3.20*a*
) and (3.20*c*
) can be represented in the generic form

where
$n=0,1,2,\ldots$
and
$\unicode[STIX]{x1D706}$
is a given parameter. The exact solution of this differential equation can be extracted from the handbook of solutions of ordinary differential equations by Zaitsev & Polyanin (Reference Zaitsev and Polyanin2002) (page 219). It can be written as

where
$J_{n+1/2}$
and
$Y_{n+1/2}$
are the Bessel functions of half-integer order of the first and second kind, respectively,
$i$
is the imaginary number, and
${\mathcal{A}}_{0}$
and
${\mathcal{B}}_{0}$
are arbitrary constants. With the relations (e.g. Arfken *et al.*, Reference Arfken, Weber and Harris2012, chap. 14; or Abramowitz & Stegun, Reference Abramowitz and Stegun1972, chap. 9)

and

where
$I_{\unicode[STIX]{x1D708}}$
and
$K_{\unicode[STIX]{x1D708}}$
are the modified Bessel functions of the first and second kind, respectively, and
$\unicode[STIX]{x1D708}$
may be a complex number, and introducing the modified spherical Bessel functions (notice the different scaling factors in these definitions)

we can write solution (B 2) as

This result indicates that the substitution
$\unicode[STIX]{x1D711}(x)=x^{\unicode[STIX]{x1D700}}\tilde{\unicode[STIX]{x1D711}}(\unicode[STIX]{x1D706}x)$
in (B 1) with the choice
$\unicode[STIX]{x1D700}=n+1$
leads to the modified spherical Bessel differential equation for
$\tilde{\unicode[STIX]{x1D711}}$
, an expression that, unlike (B 1), is rather well-known. Because
$i_{n}(x)$
grows unbounded whereas
$k_{n}(x)$
tends to zero when
$x\rightarrow \infty$
, we set
${\mathcal{A}}_{0}=0$
in order to use (B 7) to represent the solutions of (3.20*a*
) and (3.20*c*
) in the main body of the document. Finally, we can write
$k_{n}$
in terms of elementary functions with the relations

which can be extended with the recurrence formula
$k_{n-1}(x)-k_{n+1}(x)=-(2n+1)k_{n}(x)/x$
(Arfken *et al.*
Reference Arfken, Weber and Harris2012). Using (B 9) and (B 10), we can obtain the expressions in (3.21*a*
) and (3.21*c*
), respectively.

## Appendix D. Thermophoretic force and drag from Grad’s 13-moment method and from other approaches

Following Young’s (Reference Young2011) work, the non-dimensional thermophoretic force and velocity drag resulting from Grad’s 13-moment method G13 are given by

and

respectively, where
$Kn^{\prime }=\sqrt{\unicode[STIX]{x03C0}/2}\,Kn$
. For the results presented in this paper, the thermal creep, velocity slip and temperature jump coefficients take, respectively, the Maxwell–Smoluchowski values, namely,
$K_{tc}~=~3/4$
,
$C_{m}~=~1$
and
$C_{e}~=~15/8$
(Young Reference Young2011). It should be said that the first closing parenthesis in the numerator of (D 1) and the factor
$Kn^{\prime }$
after coefficient
$C_{m}$
in the numerator of (D 2) as well as the second closing parenthesis in the denominators of both (D 1) and (D 2) are missing in Young’s (Reference Young2011) article (see his formulae (32a) and (32b)). When passing the limit
$\unicode[STIX]{x1D6EC}\rightarrow \infty$
in (D 2), we recover the factor obtained by Lockerby & Collyer (Reference Lockerby and Collyer2016) in their formula (5.5), who have already noted the typographical errors in Young’s paper for
$\unicode[STIX]{x1D6F9}$
.

The interpolation formula presented by Young (Reference Young2011) for the thermophoretic force is

with
$K_{tc}~=~1.10$
,
$C_{m}~=~1.13$
,
$C_{e}~=~2.17$
and
$C_{int}=0.5$
.

For completeness, we add the expressions used in this paper for the free-molecule regime (
$Kn^{\prime }\gg 1$
). For the non-dimensional thermophoretic force, Waldmann (Reference Waldmann1959) obtained

whereas, for the drag caused by a free stream past a sphere, Epstein (Reference Epstein1924) obtained

These expressions are extracted from table 1 of Young’s article.

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