1 Introduction
A small particle in a shear flow usually experiences a force perpendicular to the flow direction due to the flow velocity gradient (Stone Reference Stone2000). This phenomenon was first observed in capillaries where blood corpuscles tended to keep away from the walls (Poiseuille Reference Poiseuille1836). The force acting on the particle is termed ‘lift force’, which directly affects the particle migration and plays a crucial role in a variety of areas such as aerosol mechanics, chemical engineering, environmental science and biology (Herron, Davis & Bretherton Reference Herron, Davis and Bretherton1975; Gavze & Shapiro Reference Gavze and Shapiro1998; Loth Reference Loth2008; Zheng & Silber-Li Reference Zheng and Silber-Li2009). Therefore, it is of great importance to develop a quantitative description for the lift force in shear flows.
In a gaseous medium, the Knudsen number
$Kn$
(
$Kn=\unicode[STIX]{x1D706}/R$
, where
$\unicode[STIX]{x1D706}$
is the mean free path of the gas and
$R$
is the particle radius) plays a key role in the study of the lift force. In the continuum regime (
$Kn\ll 1$
), the linear analysis based on the Stokes solution predicts no lift force (Bretherton Reference Bretherton1962; Dandy & Dwyer Reference Dandy and Dwyer1990). Thus, the lift force can be attributed to the small inertia effects, and perturbation methods have been used to study the inertia terms in the Navier–Stokes equations (McLaughlin Reference McLaughlin1991; Asmolov Reference Asmolov1999). Saffman (Reference Saffman1965) gave an asymptotic solution for the lift force on a rigid sphere at low Reynolds numbers. The Saffman lift force in this regime is in the same direction as the flow velocity gradient, which is consistent with many experimental results (Merzkirch & Bracht Reference Merzkirch and Bracht1978; Busnaina, Taylor & Kashkoush Reference Busnaina, Taylor and Kashkoush1993). The particle in a shear flow may rotate in response to the torque induced by the shear stress (Bagchi & Balachandar Reference Bagchi and Balachandar2002). Nevertheless, there is very weak coupling between the shear effect and the rotation effect at low Reynolds numbers (Saffman Reference Saffman1965; Kurose & Komori Reference Kurose and Komori1999; Bagchi & Balachandar Reference Bagchi and Balachandar2002). In the free-molecule regime (
$Kn\gg 1$
), the velocity distribution of the gas molecules is not affected by the presence of the nanoparticles and the collisions between gas molecules and the particle dominate the gas–particle momentum transfer (Li Reference Li2009). The magnitude and direction of the lift force in this case can be determined by evaluating the momentum transfer upon collisions between gas molecules and the particle. Kröger & Hütter (Reference Kröger and Hütter2006) studied the lift force on a translationally moving particle in a shear flow at high Knudsen numbers. Their investigation suggests that the direction of the lift force is opposite to the flow velocity gradient, which is different from Saffman’s result. Several years ago, Liu & Bogy (Reference Liu and Bogy2008, Reference Liu and Bogy2009) derived an expression for the lift force on spherical particles in a gas shear flow based on the rigid-body collision model between gas molecules and the particle, which is given by
$$\begin{eqnarray}\boldsymbol{F}_{L}=-{\textstyle \frac{1}{3}}\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}\boldsymbol{G}R^{2}\unicode[STIX]{x1D706}V,\end{eqnarray}$$
where
$\unicode[STIX]{x1D70C}$
is the gas density,
$\boldsymbol{G}$
is the velocity gradient in the shear flow and
$V$
is the velocity of the particle relative to the gas. The lift force in (1.1) is also in the opposite direction to the velocity gradient, which confirms the prediction by Kröger & Hütter (Reference Kröger and Hütter2006). It should be noted that the rigid-body collision assumption for (1.1) is reasonable for relatively large particles. For nanoparticles, the molecular interactions between gas molecules and the particle may be significant and play an essential role (Li & Wang Reference Li and Wang2003a
,Reference Li and Wang
b
, Reference Li and Wang2004, Reference Li and Wang2005; Wang & Li Reference Wang and Li2011, Reference Wang and Li2012). Hence, (1.1) may not work well for nanoparticles. Recently, Luo et al. (Reference Luo, Wang, Xia and Li2016) considered the influence of gas–particle intermolecular interactions on the lift force in pure gases and obtained a lift force formula, which is given by
$$\begin{eqnarray}\boldsymbol{F}_{L}=\frac{1}{3}\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}\boldsymbol{G}R^{2}\unicode[STIX]{x1D706}V\left(5\frac{\unicode[STIX]{x1D6FA}_{gp}^{(1,1)\ast }}{\unicode[STIX]{x1D6FA}_{g}^{(2,2)\ast }}-6\frac{\unicode[STIX]{x1D6FA}_{gp}^{(1,2)\ast }}{\unicode[STIX]{x1D6FA}_{g}^{(2,2)\ast }}\right),\end{eqnarray}$$
where
$\unicode[STIX]{x1D6FA}_{gp}^{(1,1)\ast }$
and
$\unicode[STIX]{x1D6FA}_{gp}^{(1,2)\ast }$
are reduced collision integrals, which refer to gas–particle interactions, and
$\unicode[STIX]{x1D6FA}_{g}^{(2,2)\ast }$
is the reduced collision integral for gas–gas interactions. Equation (1.2) includes (1.1) as a special case as the reduced collision integrals are equal to 1 for rigid-body collisions. Moreover, it is found that the lift force on nanoparticles could be in the same direction as or in the opposite direction to the velocity gradient, depending on the gas–particle interaction potential and temperature.
In practical applications, the motion of particles in shear flows of gas mixtures is often encountered (Zou et al. Reference Zou, Cheng, Zhang and Zhao2007; Kleinstreuer, Zhang & Donohue Reference Kleinstreuer, Zhang and Donohue2008; Salmanzadeh et al. Reference Salmanzadeh, Zahedi, Ahmadi, Marr and Glauser2012). For binary gas mixtures, where the velocity distribution of gas molecules depends on the concentration distribution and the interactions between the two gas species, a nonlinear dependence of the lift force on the gas component concentration is expected according to the Chapman–Enskog theory (Chapman & Cowling Reference Chapman and Cowling1970; Wang & Li Reference Wang and Li2011). Unfortunately, the lift force on nanoparticles in shear flows of gas mixtures is poorly understood. This is primarily due to the complexity of mathematical analysis of the velocity distribution function in gas mixtures.
This work is devoted to generalize our previous work in pure gases to binary gas mixtures. The rest of the paper is organized as follows. Section 2 gives the assumptions and the velocity distribution function of the gas molecules in a binary gas mixture under a velocity gradient. In § 3, the formulae for the lift forces on nanoparticles in a binary gas mixture are derived on the basis of gas kinetic theory. It is shown that the lift force formulae can be simplified to those in pure gases. As an example, the lift forces on a silver nanoparticle in gas mixtures are illustrated in § 4. Finally, we conclude the paper in § 5.
2 Assumptions and velocity distribution function
Consider a spherical particle moving at velocity
$\boldsymbol{V}_{p}$
in a linear shear flow, where
$\boldsymbol{V}_{p}$
is in the same direction as the velocity of the shear flow. The local mass velocity of the gas mixture is
$\boldsymbol{V}_{0}$
, and
$\boldsymbol{v}_{i}$
is the velocity of the gas molecule relative to
$\boldsymbol{V}_{0}$
, i.e.
$\boldsymbol{v}_{i}$
is the peculiar velocity and the subscript
$i$
denotes the gas species
$i$
. The particle velocity relative to the local mass velocity of the gas is
$\boldsymbol{V}$
(
$\boldsymbol{V}=\boldsymbol{V}_{p}-\boldsymbol{V}_{0}$
), and its direction is perpendicular to the velocity gradient of the shear flow
$\boldsymbol{G}$
. A reference frame is introduced with origin located at the mass centre of the particle, as shown in figure 1. The relative velocity
$\boldsymbol{V}$
is in the positive
$z$
-direction and the velocity gradient
$\boldsymbol{G}$
is in the negative
$y$
-direction.
Figure 1. (a) Collision model between a gas molecule and a small particle in a linear shear flow. (b) Relationship among various vectors.
Figure 2. Gas molecules travel in a cylindrical region.
Based on the gas kinetic theory (Chapman & Cowling Reference Chapman and Cowling1970; Bird Reference Bird1994; Li & Wang Reference Li and Wang2003a
,Reference Li and Wang
b
), the force acting on the particle can be obtained by finding the total momentum transfer from the gas molecules to the particle. The gas molecules travelling in a cylindrical shell with the same impact parameter
$l$
and the same relative velocity
$\boldsymbol{g}_{i}$
(
$\boldsymbol{g}_{i}=\boldsymbol{v}_{i}-\boldsymbol{V}$
) will undergo collisions with the particle during a short time
$\,\text{d}t$
. Then, the length of this cylinder is
$L=\boldsymbol{g}_{i}\,\text{d}t$
(see figure 2), and the number of gas molecules in the small sector of this cylindrical shell is
$f_{i}g_{i}l\,\text{d}l\,\text{d}\unicode[STIX]{x1D709}\,\text{d}t$
, where
$f_{i}$
is the velocity distribution of the gas molecules. The total momentum transfer of the gas molecules in the whole real space and velocity space can be obtained by integrating the momentum transfer of the gas molecules over
$l$
,
$\unicode[STIX]{x1D709}$
and
$\boldsymbol{v}_{i}$
. As a result, the force experienced by the particle upon collisions with gas molecules is given by
$$\begin{eqnarray}\boldsymbol{F}_{i}=m_{ip}\int _{\boldsymbol{v}_{i}}g_{i}\boldsymbol{g}_{i}\,f_{i}Q(g_{i})\,\text{d}\boldsymbol{v}_{i},\quad i=1,2,\end{eqnarray}$$
where
$m_{ip}=m_{i}m_{p}/(m_{i}+m_{p})$
is the reduced mass, with
$m_{p}$
and
$m_{i}$
being the particle mass and gas molecular mass of species
$i$
, respectively. In (2.1),
$Q(g_{i})$
represents the collision cross-section, which depends on the scattering scenario of the gas molecules upon collisions with the particle. Specular and diffuse scattering are the two limiting models of collision (Hirschfelder et al.
Reference Hirschfelder, Curtiss, Bird and Mayer1954; Chapman & Cowling Reference Chapman and Cowling1970). For specular scattering,
$$\begin{eqnarray}Q_{s}(g_{i})=2\unicode[STIX]{x03C0}\int _{0}^{\infty }(1-\cos \unicode[STIX]{x1D712})l\,\text{d}l,\end{eqnarray}$$
with
$\unicode[STIX]{x1D712}$
being the scattering angle given by
$$\begin{eqnarray}\unicode[STIX]{x1D712}=\unicode[STIX]{x03C0}-2l\int _{r_{m}}^{\infty }\frac{\text{d}r}{r^{2}\sqrt{1-{\displaystyle \frac{l^{2}}{r^{2}}}-{\displaystyle \frac{U(r)}{m_{ip}g_{i}^{2}/2}}}},\end{eqnarray}$$
where
$r$
is the separation between the particle and gas molecule,
$r_{m}$
is the distance of closest encounter and
$U(r)$
is the gas–particle interaction potential. The collision cross-section for diffuse scattering is defined as
$$\begin{eqnarray}Q_{d}(g_{i})=2\unicode[STIX]{x03C0}\left[\int _{0}^{l_{0}}\left(1+\frac{1}{g_{i}}\sqrt{\frac{\unicode[STIX]{x03C0}k_{B}T}{2m_{ip}}}\sin \frac{\unicode[STIX]{x1D712}}{2}\right)+\int _{l_{0}}^{\infty }(1-\cos \unicode[STIX]{x1D712})\right]l\,\text{d}l,\end{eqnarray}$$
where
$k_{B}$
is the Boltzmann constant,
$T$
is the temperature
$l_{0}$
is the critical impact parameter. Diffuse scattering can only occur if the impact parameter is less than
$l_{0}$
and the gas molecule collides with the particle physically; if the impact parameter is greater than
$l_{0}$
, then the gas molecule just grazes the particle and the scattering is considered to be specular.
As mentioned above, the velocity distribution of the gas molecules is required to calculate the force acting on the particle. In the free-molecule regime, whereby the mean free path of the gas is much larger than the radius of the spherical particle, the interactions between the incident gas molecules and those reflected by the particle can be ignored. Hence, the velocity distribution of gas molecules can be obtained by neglecting the presence of the particle (Liu & Bogy Reference Liu and Bogy2008). For a binary gas mixture, the velocity distribution function of the two gas species is more complicated than that of pure gases. According to the Chapman–Enskog theory (Chapman & Cowling Reference Chapman and Cowling1970), in the presence of a velocity gradient, the second-order approximation to the velocity distribution function is
$f_{i}=f_{i}^{(0)}+f_{i}^{(1)}$
, where the first term
$f_{i}^{(0)}$
is the molecular velocity distribution function in uniform state, and the second term
$f_{i}^{(1)}$
accounts for the flow velocity gradient. Thus, the integral in (2.1) consists of two terms, the first of which is the drag force and the second of which is the lift force. The present paper focuses on the lift force, which involves
$f_{i}^{(1)}$
(Chapman & Cowling Reference Chapman and Cowling1970; Bird Reference Bird1994)
$$\begin{eqnarray}f_{i}^{(1)}=-2f_{i}^{(0)}\unicode[STIX]{x1D63D}_{i}\boldsymbol{ : }\unicode[STIX]{x1D735}\boldsymbol{V}_{0},\quad i=1,2,\end{eqnarray}$$
where
$f_{i}^{(0)}$
is the velocity distribution function in uniform state given by
$$\begin{eqnarray}f_{i}^{(0)}=n_{i}\left(\frac{m_{i}}{2\unicode[STIX]{x03C0}k_{B}T}\right)^{3/2}\exp \!\left(\!-\frac{m_{i}v_{i}^{2}}{2k_{B}T}\right),\quad i=1,2,\end{eqnarray}$$
where
$n_{i}$
is the number density of gas
$i$
, and
$\unicode[STIX]{x1D63D}_{i}$
is a symmetrical and non-divergent tensor, which is obtained as a series of Sonine polynomials. Under the first-order approximation,
$$\begin{eqnarray}\unicode[STIX]{x1D63D}_{i}=b_{i}\frac{m_{i}}{2k_{B}T}\mathop{\circ }^{\boldsymbol{v}_{i}\boldsymbol{v}_{i}},\quad i=1,2,\end{eqnarray}$$
where
$b_{i}$
is a function of gas composition, temperature, gas molecular mass and the intermolecular potential among gas molecules. The superscript ‘
$\circ$
’ above
$\boldsymbol{v}_{i}\boldsymbol{v}_{i}$
denotes that the tensor is non-divergent. For the flow in figure 1,
$f_{i}^{(1)}$
can be written as
$$\begin{eqnarray}f_{i}^{(1)}=-f_{i}^{(0)}b_{i}\frac{m_{i}}{k_{B}T}v_{y,i}v_{z,i}G,\quad i=1,2,\end{eqnarray}$$
where
$v_{y,i}$
and
$v_{z,i}$
are the peculiar velocity components in
$y$
- and
$z$
-directions, respectively.
3 Lift force in a binary gas mixture
3.1 Lift force formula on a spherical nanoparticle by gas
$i$
By substituting (2.8) into (2.1), the lift force
$F_{L,i}$
can be written as
$$\begin{eqnarray}\boldsymbol{F}_{L,i}=-\frac{m_{ip}m_{i}b_{i}G}{k_{B}T}\int _{\boldsymbol{v}_{i}}g_{i}\boldsymbol{g}_{i}\,f_{i}^{(0)}v_{y,i}v_{z,i}Q(g_{i})\,\text{d}\boldsymbol{v}_{i}.\end{eqnarray}$$
As shown in figure 1(b),
$\unicode[STIX]{x1D719}$
and
$\unicode[STIX]{x1D703}$
are the colatitude and azimuthal angles of
$\boldsymbol{g}_{i}$
in the frame of reference, respectively. The relative velocity vector
$\boldsymbol{g}_{i}$
is therefore given by
$$\begin{eqnarray}\boldsymbol{g}_{i}=g_{i}\sin \unicode[STIX]{x1D719}\cos \unicode[STIX]{x1D703}\,\boldsymbol{i}+g_{i}\sin \unicode[STIX]{x1D719}\sin \unicode[STIX]{x1D703}\,\boldsymbol{j}+g_{i}\cos \unicode[STIX]{x1D719}\,\boldsymbol{k}.\end{eqnarray}$$
Since the lift force is collinear with the velocity gradient, only the
$y$
-component of
$\boldsymbol{g}_{i}$
needs to be considered, and it is easy to check that the first and third terms in (3.2) vanish upon integration over the angles in spherical coordinates. This simplifies (3.1) to
$$\begin{eqnarray}\boldsymbol{F}_{L,i}=\frac{m_{ip}m_{i}b_{i}G}{k_{B}T}\int _{\boldsymbol{v}_{i}}g_{i}^{3}\sin ^{2}\unicode[STIX]{x1D719}\sin ^{2}\unicode[STIX]{x1D703}\,(g_{i}\cos \unicode[STIX]{x1D719}+V)\,f_{i}^{(0)}Q(g_{i})\,\text{d}\boldsymbol{v}_{i}.\end{eqnarray}$$
Substituting (2.6) into (3.3), we obtain the lift force (a detailed derivation is given in appendix A)
$$\begin{eqnarray}\boldsymbol{F}_{L,i}={\textstyle \frac{8}{15}}m_{ip}\sqrt{2\unicode[STIX]{x03C0}k_{B}T/m_{i}}\,\boldsymbol{G}VR^{2}n_{i}b_{i}[5\unicode[STIX]{x1D6FA}_{ip}^{(1,1)\ast }-6\unicode[STIX]{x1D6FA}_{ip}^{(1,2)\ast }],\end{eqnarray}$$
where
$\unicode[STIX]{x1D6FA}_{ip}^{(k,q)\ast }$
is the reduced collision integral given by
$$\begin{eqnarray}\unicode[STIX]{x1D6FA}_{ip}^{(k,q)\ast }=\int _{0}^{\infty }\frac{\text{e}^{-{\unicode[STIX]{x1D6FE}_{i}}^{2}}{\unicode[STIX]{x1D6FE}_{i}}^{2q+3}Q^{k}(g_{i})\,\text{d}\unicode[STIX]{x1D6FE}_{i}}{[(q+k)!/2]\{1-[1+(-1)^{k}]/(2+2k)\}\unicode[STIX]{x03C0}R^{2}},\end{eqnarray}$$
and
$\unicode[STIX]{x1D6FE}_{i}=g_{i}\sqrt{m_{i}/(2k_{B}T)}$
. More details can be found in appendix A.
3.2 Total lift force on a spherical nanoparticle in binary gas mixtures
Since the mass of the particle is usually much larger than that of a gas molecule, i.e.
$m_{i}/m_{ip}\approx 1$
, then the total lift force can be expressed as
$$\begin{eqnarray}\displaystyle \boldsymbol{F}_{L}=\boldsymbol{F}_{L1}+\boldsymbol{F}_{L2} & = & \displaystyle \frac{8\boldsymbol{G}VR^{2}\sqrt{2\unicode[STIX]{x03C0}k_{B}T}}{15} [n_{1}b_{1}\sqrt{m_{1}}(5\unicode[STIX]{x1D6FA}_{1p}^{(1,1)\ast }-6\unicode[STIX]{x1D6FA}_{1p}^{(1,2)\ast })\nonumber\\ \displaystyle & & \displaystyle +\,n_{2}b_{2}\sqrt{m_{2}}(5\unicode[STIX]{x1D6FA}_{2p}^{(1,1)\ast }-6\unicode[STIX]{x1D6FA}_{2p}^{(1,2)\ast })\!]\!.\end{eqnarray}$$
In (3.6), the coefficients
$b_{1}$
and
$b_{2}$
are given by (Chapman & Cowling Reference Chapman and Cowling1970)
$$\begin{eqnarray}b_{1}=\frac{\unicode[STIX]{x1D6FD}_{1}b_{22}-\unicode[STIX]{x1D6FD}_{2}b_{12}}{{\mathcal{B}}},\quad b_{2}=\frac{\unicode[STIX]{x1D6FD}_{2}b_{11}-\unicode[STIX]{x1D6FD}_{1}b_{12}}{{\mathcal{B}}},\end{eqnarray}$$
where
$\unicode[STIX]{x1D6FD}_{1}=5n_{1}/(2n^{2})$
,
$\unicode[STIX]{x1D6FD}_{2}=5n_{2}/(2n^{2})$
and
${\mathcal{B}}$
is the symmetric determinant,
$$\begin{eqnarray}{\mathcal{B}}=\left|\begin{array}{@{}cc@{}}b_{11} & b_{12}\\ b_{12} & b_{22}\end{array}\right|.\end{eqnarray}$$
For binary gas mixtures, the expressions for the elements of determinant
${\mathcal{B}}$
are
$$\begin{eqnarray}\displaystyle & b_{11}=\sqrt{{\displaystyle \frac{k_{B}T}{2\unicode[STIX]{x03C0}m_{0}}}}\left\{{\displaystyle \frac{4n_{1}^{2}}{n^{2}}}\sqrt{{\displaystyle \frac{2}{M_{1}}}}\unicode[STIX]{x1D6FA}_{11}^{(2,2)}+{\displaystyle \frac{16n_{1}n_{2}}{3n^{2}}}M_{2}\sqrt{{\displaystyle \frac{1}{M_{1}M_{2}}}}\left[5M_{1}\unicode[STIX]{x1D6FA}_{12}^{(1,1)}+{\displaystyle \frac{3}{2}}M_{2}\unicode[STIX]{x1D6FA}_{12}^{(2,2)}\right]\right\}, & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & b_{22}=\sqrt{{\displaystyle \frac{k_{B}T}{2\unicode[STIX]{x03C0}m_{0}}}}\left\{{\displaystyle \frac{4n_{2}^{2}}{n^{2}}}\sqrt{{\displaystyle \frac{2}{M_{2}}}}\unicode[STIX]{x1D6FA}_{22}^{(2,2)}+{\displaystyle \frac{16n_{1}n_{2}}{3n^{2}}}M_{1}\sqrt{{\displaystyle \frac{1}{M_{1}M_{2}}}}\left[5M_{2}\unicode[STIX]{x1D6FA}_{12}^{(1,1)}+{\displaystyle \frac{3}{2}}M_{1}\unicode[STIX]{x1D6FA}_{12}^{(2,2)}\right]\right\}, & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & b_{12}=\sqrt{{\displaystyle \frac{k_{B}T}{2\unicode[STIX]{x03C0}m_{0}}}}\left\{\sqrt{{\displaystyle \frac{1}{M_{1}M_{2}}}}\left(-{\displaystyle \frac{16n_{1}n_{2}}{3n^{2}}}\right)M_{1}M_{2}\left[5\unicode[STIX]{x1D6FA}_{12}^{(1,1)}-{\displaystyle \frac{3}{2}}\unicode[STIX]{x1D6FA}_{12}^{(2,2)}\right]\right\}, & \displaystyle\end{eqnarray}$$
where
$M_{1}=m_{1}/m_{0}$
,
$M_{2}=m_{2}/m_{0}$
and
$m_{0}=m_{1}+m_{2}$
. Here,
$\unicode[STIX]{x1D6FA}_{ij}^{(k,q)}$
is defined as (Hirschfelder et al.
Reference Hirschfelder, Curtiss, Bird and Mayer1954)
$$\begin{eqnarray}\unicode[STIX]{x1D6FA}_{ij}^{(k,q)}=\int _{0}^{\infty }\text{e}^{-\unicode[STIX]{x1D6FE}_{ij}^{2}}\unicode[STIX]{x1D6FE}_{ij}^{2q+3}Q^{k}(g_{ij})\,\text{d}\unicode[STIX]{x1D6FE}_{ij},\end{eqnarray}$$
which depends on the intermolecular interactions between gas molecules. It should be noted that the mass in the dimensionless quantity
$\unicode[STIX]{x1D6FE}_{ij}$
is the reduced mass between two gas molecules.
For simplicity, hereafter, (3.9)–(3.11) will be represented by
$$\begin{eqnarray}b_{11}=\sqrt{\frac{k_{B}T}{2\unicode[STIX]{x03C0}m_{0}}}\tilde{b}_{11},\quad b_{22}=\sqrt{\frac{k_{B}T}{2\unicode[STIX]{x03C0}m_{0}}}\tilde{b}_{22},\quad b_{12}=\sqrt{\frac{k_{B}T}{2\unicode[STIX]{x03C0}m_{0}}}\tilde{b}_{12},\end{eqnarray}$$
where
$\tilde{b}_{11}$
,
$\tilde{b}_{22}$
and
$\tilde{b}_{12}$
denote the terms in the curly braces in (3.9), (3.10) and (3.11), respectively. Then (3.7) can be rewritten as
$$\begin{eqnarray}\displaystyle b_{1} & = & \displaystyle \frac{\unicode[STIX]{x1D6FD}_{1}b_{22}-\unicode[STIX]{x1D6FD}_{2}b_{12}}{{\mathcal{B}}}=\frac{\unicode[STIX]{x1D6FD}_{1}\tilde{b}_{22}-\unicode[STIX]{x1D6FD}_{2}\tilde{b}_{12}}{\sqrt{{\displaystyle \frac{k_{B}T}{2\unicode[STIX]{x03C0}m_{0}}}}(\tilde{b}_{11}\tilde{b}_{22}-{\tilde{b}_{12}}^{2})}=\sqrt{\frac{2\unicode[STIX]{x03C0}m_{0}}{k_{B}T}}\tilde{b}_{1},\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle b_{2} & = & \displaystyle \frac{\unicode[STIX]{x1D6FD}_{2}b_{11}-\unicode[STIX]{x1D6FD}_{1}b_{12}}{{\mathcal{B}}}=\frac{\unicode[STIX]{x1D6FD}_{2}\tilde{b}_{11}-\unicode[STIX]{x1D6FD}_{1}\tilde{b}_{12}}{\sqrt{{\displaystyle \frac{k_{B}T}{2\unicode[STIX]{x03C0}m_{0}}}}(\tilde{b}_{11}\tilde{b}_{22}-{\tilde{b}_{12}}^{2})}=\sqrt{\frac{2\unicode[STIX]{x03C0}m_{0}}{k_{B}T}}\tilde{b}_{2}.\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \boldsymbol{F}_{L}=\frac{16\unicode[STIX]{x03C0}m_{0}\boldsymbol{G}VR^{2}}{15}[n_{1}\tilde{b}_{1}\sqrt{M_{1}}(5\unicode[STIX]{x1D6FA}_{1p}^{(1,1)\ast }-6\unicode[STIX]{x1D6FA}_{1p}^{(1,2)\ast })+n_{2}\tilde{b}_{2}\sqrt{M_{2}}(5\unicode[STIX]{x1D6FA}_{2p}^{(1,1)\ast }-6\unicode[STIX]{x1D6FA}_{2p}^{(1,2)\ast })]. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$
It should be noted that
$M_{i}$
and
$\unicode[STIX]{x1D6FA}_{ip}^{(k,q)\ast }$
in (3.15) are dimensionless, whereas
$n_{i}\tilde{b}_{i}$
has the same unit as
$1/R^{2}$
. Letting
$\unicode[STIX]{x1D6F1}_{i}^{\ast }=R^{2}n_{i}\tilde{b}_{i}\sqrt{M_{i}}(5\unicode[STIX]{x1D6FA}_{ip}^{(1,1)\ast }-6\unicode[STIX]{x1D6FA}_{ip}^{(1,2)\ast })$
, the lift force can be expressed as
$$\begin{eqnarray}\boldsymbol{F}_{L}=\frac{16\unicode[STIX]{x03C0}m_{0}\boldsymbol{G}V}{15}(\unicode[STIX]{x1D6F1}_{1}^{\ast }+\unicode[STIX]{x1D6F1}_{2}^{\ast }),\end{eqnarray}$$
where
$\unicode[STIX]{x1D6F1}_{1}^{\ast }$
and
$\unicode[STIX]{x1D6F1}_{2}^{\ast }$
are dimensionless quantities, which contain the interactions between gas species and the particle. It is worth mentioning that
$\boldsymbol{F}_{L1}$
and
$\boldsymbol{F}_{L2}$
in (3.6) are coupled with each other via the coefficients
$b_{1}$
and
$b_{2}$
, which depend on the interactions between the two gas species. Thus, the total lift force
$\boldsymbol{F}_{L}$
is not simply the linear summation of the lift force exerted by each gas component.
3.3 Lift force in a pure gas
It is expected that (3.6) can be simplified to (1.2) for pure gases if the two species are identical. Letting
$\unicode[STIX]{x1D6FA}_{1p}^{(1,1)\ast }=\unicode[STIX]{x1D6FA}_{2p}^{(1,1)\ast }=\unicode[STIX]{x1D6FA}_{gp}^{(1,1)\ast }$
,
$\unicode[STIX]{x1D6FA}_{1p}^{(1,2)\ast }=\unicode[STIX]{x1D6FA}_{2p}^{(1,2)\ast }=\unicode[STIX]{x1D6FA}_{gp}^{(1,2)\ast }$
and
$m_{1}=m_{2}=m_{g}$
, equation (3.6) becomes
$$\begin{eqnarray}\boldsymbol{F}_{L}=\frac{8\boldsymbol{G}VR^{2}\sqrt{2\unicode[STIX]{x03C0}k_{B}Tm_{g}}}{15}(5\unicode[STIX]{x1D6FA}_{gp}^{(1,1)\ast }-6\unicode[STIX]{x1D6FA}_{gp}^{(1,2)\ast })(n_{1}b_{1}+n_{2}b_{2}),\end{eqnarray}$$
and (3.9)–(3.11) can be rewritten as
$$\begin{eqnarray}\displaystyle & \displaystyle b_{11}=\sqrt{\frac{k_{B}T}{\unicode[STIX]{x03C0}m_{g}}}\left\{\frac{4n_{1}^{2}}{n^{2}}\unicode[STIX]{x1D6FA}_{g}^{(2,2)}+\frac{8n_{1}n_{2}}{3n^{2}}\left[\frac{5}{2}\unicode[STIX]{x1D6FA}_{g}^{(1,1)}+\frac{3}{4}\unicode[STIX]{x1D6FA}_{g}^{(2,2)}\right]\right\}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle b_{22}=\sqrt{\frac{k_{B}T}{\unicode[STIX]{x03C0}m_{g}}}\left\{\frac{4n_{2}^{2}}{n^{2}}\unicode[STIX]{x1D6FA}_{g}^{(2,2)}+\frac{8n_{1}n_{2}}{3n^{2}}\left[\frac{5}{2}\unicode[STIX]{x1D6FA}_{g}^{(1,1)}+\frac{3}{4}\unicode[STIX]{x1D6FA}_{g}^{(2,2)}\right]\right\}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle b_{12}=\sqrt{\frac{k_{B}T}{\unicode[STIX]{x03C0}m_{g}}}\left(\frac{8n_{1}n_{2}}{3n^{2}}\right)\left[\frac{3}{4}\unicode[STIX]{x1D6FA}_{g}^{(2,2)}-\frac{5}{2}\unicode[STIX]{x1D6FA}_{g}^{(1,1)}\right]. & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}n_{1}b_{1}+n_{2}b_{2}=\frac{5}{2}\left[\frac{n_{1}^{2}b_{22}-2n_{1}n_{2}b_{12}+n_{2}^{2}b_{11}}{n^{2}(b_{11}b_{22}-b_{_{12}}^{2})}\right].\end{eqnarray}$$
Then, by substituting (3.18)–(3.20) into (3.21), we obtain
$$\begin{eqnarray}n_{1}b_{1}+n_{2}b_{2}=\frac{5}{8\sqrt{{\displaystyle \frac{k_{B}T}{\unicode[STIX]{x03C0}m_{g}}}}\unicode[STIX]{x1D6FA}_{g}^{(2,2)}}.\end{eqnarray}$$
As a result, (3.17) is simplified to
$$\begin{eqnarray}\boldsymbol{F}_{L}=\frac{\sqrt{2}\unicode[STIX]{x03C0}m_{g}}{3\unicode[STIX]{x1D6FA}_{g}^{(2,2)}}\boldsymbol{G}R^{2}V[5\unicode[STIX]{x1D6FA}_{1p}^{(1,1)\ast }-6\unicode[STIX]{x1D6FA}_{1p}^{(1,2)\ast }],\end{eqnarray}$$
which is the same as (1.2) by using
$\unicode[STIX]{x1D70C}=m_{g}n$
,
$\unicode[STIX]{x1D6FA}_{g}^{(2,2)\ast }=\unicode[STIX]{x1D6FA}_{g}^{(2,2)}/(2\unicode[STIX]{x03C0}\unicode[STIX]{x1D70E}_{g}^{2})$
and
$\unicode[STIX]{x1D706}=1/(\sqrt{2}\unicode[STIX]{x03C0}\unicode[STIX]{x1D70E}_{g}^{2}n)$
, where
$\unicode[STIX]{x1D70E}_{g}$
is the collision diameter of the gas molecules.
As another test of consistency, we consider the case in which the concentration of one gas component tends to zero. For instance, in the limit of
$n_{2}\rightarrow 0$
and
$n_{1}\rightarrow n$
,
$b_{1}n_{1}=5\sqrt{\unicode[STIX]{x03C0}m_{1}}/(8\unicode[STIX]{x1D6FA}_{11}^{(2,2)}\sqrt{k_{B}T})$
,
$b_{2}n_{2}=0$
and (3.6) converges to that in a pure gas.
For large particles, the gas–particle interactions can be neglected and the rigid-body collision model is reasonable. In this case, (3.23) can be easily reduced to (1.1) by assuming rigid-body collisions. However, as the particle size decreases to the nanoscale, the influence of the van der Waals interactions between the gas molecules and particle on lift force becomes notable. This influence is manifested in the term
$\unicode[STIX]{x1D6FA}_{ip}^{(k,q)\ast }$
.
The lift force given by (1.2) could be positive
$(5\unicode[STIX]{x1D6FA}_{gp}^{(1,1)\ast }>6\unicode[STIX]{x1D6FA}_{gp}^{(1,2)\ast })$
or negative
$(5\unicode[STIX]{x1D6FA}_{gp}^{(1,1)\ast }<6\unicode[STIX]{x1D6FA}_{gp}^{(1,2)\ast })$
, depending on the gas–particle interactions. In most cases, the direction of the lift force on nanoparticles in pure gases (1.2) is negative, i.e. opposite to the velocity gradient, which is consistent with (1.1). However, the positive lift force takes place in a certain temperature range for a given gas–particle interaction. The physical reason for the direction change is that the gas molecules on the low-velocity side transfer more momentum to the nanoparticle due to its larger scattering angles upon non-rigid-body collisions. As for binary gas mixtures, the total lift force is a nonlinear combination of individual lift forces contributed from species 1 and 2. A direction change of the lift force is expected if the lift forces of gas species 1 and 2 are in opposite directions, as will be demonstrated in the following section.
4 Lift force on a silver spherical nanoparticle in gas mixtures
As an example, a suspended silver nanoparticle in a Ne–Ar binary gas mixture is considered. A widely used model to describe the van der Waals interactions is the Lennard-Jones (LJ) 12–6 potential,
$$\begin{eqnarray}U_{12-6}(r)=4\unicode[STIX]{x1D700}\left[\left(\frac{\unicode[STIX]{x1D70E}}{r}\right)^{12}-\left(\frac{\unicode[STIX]{x1D70E}}{r}\right)^{6}\right],\end{eqnarray}$$
where
$\unicode[STIX]{x1D700}$
and
$\unicode[STIX]{x1D70E}$
represent the binding energy and collision diameter, respectively. The LJ potential parameters for Ne (
$\unicode[STIX]{x1D70E}_{1}=2.82$
Å and
$\unicode[STIX]{x1D700}_{1}/k_{B}=32.8~\text{K}$
), Ar (
$\unicode[STIX]{x1D70E}_{2}=3.47$
Å and
$\unicode[STIX]{x1D700}_{2}/k_{B}=114.0~\text{K}$
) and Ag (
$\unicode[STIX]{x1D70E}_{p}=2.57$
Å and
$\unicode[STIX]{x1D700}_{p}/k_{B}=4075.0~\text{K}$
) are adopted from the literature (Svehla Reference Svehla1962; Hippler, Troe & Wendelken Reference Hippler, Troe and Wendelken1983; Agrawal, Rice & Thompson Reference Agrawal, Rice and Thompson2002). The potential parameters for the interactions between different species 1 and 2 are obtained using the conventional Lorentz–Berthelot rules, i.e.
$\unicode[STIX]{x1D70E}_{12}=(\unicode[STIX]{x1D70E}_{1}+\unicode[STIX]{x1D70E}_{2})/2$
and
$\unicode[STIX]{x1D700}_{12}=\sqrt{\unicode[STIX]{x1D700}_{1}\unicode[STIX]{x1D700}_{2}}$
.
Figure 3 shows the dimensionless quantity
$\unicode[STIX]{x1D6F1}_{i}^{\ast }$
of a silver nanoparticle in a rarefied Ne–Ar gas mixture as a function of reduced Ne concentration
$n_{1}/n$
. It is seen that
$\boldsymbol{F}_{L}$
converges to
$\boldsymbol{F}_{L,1}$
(open circle) or
$\boldsymbol{F}_{L,2}$
(open triangle) as
$n_{1}\rightarrow n$
or
$n_{2}\rightarrow n$
, respectively. It is also found that
$\unicode[STIX]{x1D6F1}_{i}^{\ast }$
varies nonlinearly with Ne concentration. In (3.6), it seems that the total lift force is simply a linear superposition of the lift forces given by the gas components. However, both
$b_{1}$
and
$b_{2}$
depend on the composition ratio and the interactions between the two gas species, by which the dependence of the total lift force on the concentration is nonlinear.
Figure 3. Variation of
$\unicode[STIX]{x1D6F1}_{i}^{\ast }$
as a function of reduced Ne concentration
$n_{1}/n$
at
$T=100~\text{K}$
(a),
$T=300~\text{K}$
(b),
$T=500~\text{K}$
(c) and
$T=700~\text{K}$
(d). Open symbols corresponding to the results in pure gases.
As discussed by Luo et al. (Reference Luo, Wang, Xia and Li2016), if the LJ 12–6 potential function is employed for the gas–particle interaction, a positive lift force that directs towards the high-velocity side exists in a certain reduced temperature range,
$0.417<T^{\ast }=k_{B}T/\unicode[STIX]{x1D700}<0.951$
(Wang & Li Reference Wang and Li2012; Khrapak Reference Khrapak2014; Luo et al.
Reference Luo, Wang, Xia and Li2016). Therefore, the lift forces acting on the nanoparticle in pure neon gas (
$F_{L1}$
) and pure argon gas (
$F_{L2}$
) become positive when the temperature is roughly in the ranges of 154–347 K and 286–648 K, respectively. If the temperature is extremely high or low, as shown in figure 3(a) for
$T=100~\text{K}$
and figure 3(d) for
$T=700~\text{K}$
, since the lift forces in both gas species are negative, the total lift forces are definitely negative. At moderate temperatures, positive
$F_{L1}$
(Ne) and/or
$F_{L2}$
(Ar) can take place, as shown in figures 3(b) and 3(c). As previously mentioned, the total lift force consists of two contributions from species 1 and 2. If the lift forces of gas species 1 and 2 are in opposite directions, it is expected that the direction of the total lift force changes as the gas concentration varies. For example, in the case of
$T=500~\text{K}$
(see figure 3
c),
$F_{L1}$
(Ne) is negative and
$F_{L2}$
(Ar) is positive; as a result, the direction of
$F_{L}$
changes from positive to negative as the Ne concentration varies. This means that in a certain temperature range, there exists a critical value of
$n_{1}/n$
, where the total lift force changes its direction. However, the complex nonlinear dependence of the lift force on the gas concentrations makes it difficult to predict this critical value.
Figure 4 shows the dimensionless quantity
$\unicode[STIX]{x1D6F1}_{i}^{\ast }$
of a silver nanoparticle in other gas mixtures (He–Ar and He–Kr gas mixtures). The collision diameter
$\unicode[STIX]{x1D70E}$
and binding energy
$\unicode[STIX]{x1D700}/k_{B}$
for He and Kr are 2.55 Å, 10.22 K and 3.66 Å, 178.9 K, respectively (Svehla Reference Svehla1962). As can be seen from (3.16), the total molecular mass of the two species
$m_{0}$
appears in the coefficient of
$(\unicode[STIX]{x1D6F1}_{1}^{\ast }+\unicode[STIX]{x1D6F1}_{2}^{\ast })$
, and
$\unicode[STIX]{x1D6F1}_{1}^{\ast }$
depends on
$1/m_{0}$
as
$n_{1}/n$
converges to 1. Since the total molecular mass of He and Ar is different from that of He and Kr, the values of
$\unicode[STIX]{x1D6F1}_{1}^{\ast }$
in figures 4(a) and 4(b) are not equal to each other as
$n_{1}/n$
converges to 1 (pure He gas). It has been checked that the values of
$\unicode[STIX]{x1D6F1}_{1}^{\ast }$
in He–Ar and He–Kr gas mixtures are equal to each other if
$\unicode[STIX]{x1D6F1}_{1}^{\ast }$
is multiplied by
$m_{0}$
. By comparing figures 3(b) with 4, it is seen that the dependence of the total lift force on the concentration is still nonlinear and the degree of the nonlinearity between
$\unicode[STIX]{x1D6F1}_{i}^{\ast }$
and
$n_{1}/n$
is different for different gas mixtures. The nonlinearity in the case of figure 4(b) is the most distinct one because the differences between the potential parameters and the molecular masses of He and Kr are the largest. Similarly, for the case in figure 3(b), the nonlinearity is weak because the potential parameters and the molecular mass of Ne are close to those of Ar. This is consistent with the discussion in § 3.3: if the potential parameters and molecular mass of species 1 are identical to those of species 2, the formulae reduce to those in pure gases and the nonlinearity disappears.
Figure 4. Variation of
$\unicode[STIX]{x1D6F1}_{i}^{\ast }$
as a function of reduced He concentration
$n_{1}/n$
in He–Ar gas mixtures at
$T=300~\text{K}$
(a) and in He–Kr gas mixtures at
$T=300~\text{K}$
(b).
It should be noted that the LJ 12–6 potential might not be an adequate model to describe the interactions between gas molecules and a particle. A rigorous description of the gas–particle interaction is unavailable. As a coarse approach, the potential for gas–particle interactions can be determined as a sum of the LJ interactions of gas molecules with all atoms in the particle. If the distribution of atoms in the particle is assumed to be continuous, integration can be carried out in place of summation (Rouquerol et al. Reference Rouquerol, Rouquerol, Llewellyn, Maurin and Sing2013). In this way, the interaction between gas molecules and a single solid lattice plane is given by (Everett & Powl Reference Everett and Powl1976)
$$\begin{eqnarray}U_{10-4}(r)=\frac{10}{3}\unicode[STIX]{x1D700}\left[\frac{1}{5}\left(\frac{\unicode[STIX]{x1D70E}}{r}\right)^{10}-\frac{1}{2}\left(\frac{\unicode[STIX]{x1D70E}}{r}\right)^{4}\right],\end{eqnarray}$$
and the interaction between gas molecules and a semi-infinite slab of solid is given by (Everett & Powl Reference Everett and Powl1976)
$$\begin{eqnarray}U_{9-3}(r)=\frac{3}{\sqrt{10}}\unicode[STIX]{x1D700}\left[\frac{2}{15}\left(\frac{\unicode[STIX]{x1D70E}}{r}\right)^{9}-\left(\frac{\unicode[STIX]{x1D70E}}{r}\right)^{3}\right].\end{eqnarray}$$
Although there is no evidence that these two potential functions are suitable for gas–particle interactions, in the present paper, we consider them as a possible choice for illustrative purposes. It is expected that the
$r$
dependence for the gas–particle interaction potential is covered by these three cases (12–6, 10–4 and 9–3) or their combinations (Luo et al.
Reference Luo, Wang, Xia and Li2016).
In figure 5,
$\unicode[STIX]{x1D6F1}_{1}^{\ast }+\unicode[STIX]{x1D6F1}_{2}^{\ast }$
is depicted as a function of reduced Ne concentration
$n_{1}/n$
in a Ne–Ar gas mixture at various temperatures, wherein (4.1)–(4.3) are employed to model the interactions between the particle and gas molecules. Being similar to the results in figure 3, for interaction potentials given by (4.1)–(4.3), the dependence of
$\unicode[STIX]{x1D6F1}_{1}^{\ast }+\unicode[STIX]{x1D6F1}_{2}^{\ast }$
on the single-species concentration ratio is also nonlinear. A positive lift force can be observed at certain temperature associated with the direction change from positive to negative with increasing Ne concentration. Furthermore, at the same concentration of Ne, the temperature value corresponding to the lift force direction change (if there is a change in lift force direction) increases from 12–6, to 10–4, to 9–3 potentials. This trend may be due to the increasing attractive interactions between gas molecules and the particle. Since we can observe the direction change of the lift force with all three types of interaction potentials, it is expected that the direction change can be observed experimentally. Owing to the computational complexity of the reduced collision integrals, it is not easy to offer a rigorous treatment of the gas–particle interactions. Further intensive experimental studies and theoretical analyses are needed on this issue.
Figure 5. Variation of
$\unicode[STIX]{x1D6F1}_{1}^{\ast }+\unicode[STIX]{x1D6F1}_{2}^{\ast }$
(Ne–Ar) as a function of reduced Ne concentration
$n_{1}/n$
for the different gas–particle interaction potential functions at
$T=400~\text{K}$
(a),
$T=600~\text{K}$
(b),
$T=800~\text{K}$
(c) and
$T=1000~\text{K}$
(d).
5 Conclusions
To summarize, the lift force on spherical nanoparticles in shear flows of rarefied binary gas mixtures is investigated. In the light of the importance of the molecular interactions between the gas molecules and a nanoparticle, we take into account the non-rigid-body collision effect in our model. On the basis of gas kinetic theory, we derive an analytical lift force formula by calculating the momentum transfer upon collisions between the nanoparticle and surrounding gas molecules. The expression involves terms of the collision integrals, which relies on the intermolecular interactions. The present formula is consistent with its counterpart in pure gases and can be simplified to that in the case of rigid-body collisions. For a particle suspended in a binary gas mixture, the total lift force acting on the particle is considered as the combination of the individual lift forces contributed by the two species, and the dependence of the total lift force on the gas concentration is nonlinear, which can contribute to the molecular interactions between two gas species.
Furthermore, it is illustrated that the positive lift force, which is in the opposite direction of the lift force in the case of rigid-body collisions and drives nanoparticles towards high-velocity regions, may occur in binary gas mixtures. The emergence of a positive lift force is a result of the higher gas–particle momentum transfer in the low-velocity region compared with that in the high-velocity region. Three types of gas–particle interaction potential models are employed to evaluate the lift force. It is found that the direction of the lift force acting on nanoparticles depends on temperature, gas concentrations and potential parameters and the direction change of the lift force can be observed with varying gas concentration.
Acknowledgements
This work is supported by the National Natural Science Foundation of China (grant no. 51306004) and the Science and Technology Project of Beijing Education Committee (grant no. KM201610005016).
Appendix A. Derivation of lift force on a spherical nanoparticle by gas
$i$
By substituting (2.6) into (3.3), one obtains
$$\begin{eqnarray}\boldsymbol{F}_{L,i}=\frac{\sqrt{2\unicode[STIX]{x03C0}}m_{ip}n_{i}b_{i}\boldsymbol{G}}{4\unicode[STIX]{x03C0}^{2}}\left(\frac{m_{i}}{k_{B}T}\right)^{5/2}\int _{\boldsymbol{v}_{i}}g_{i}^{3}\sin ^{2}\unicode[STIX]{x1D719}\sin ^{2}\unicode[STIX]{x1D703}\,(g_{i}\cos \unicode[STIX]{x1D719}+V)\exp \!\left(\!-\frac{m_{i}v_{i}^{2}}{2k_{B}T}\right)Q(g_{i})\,\text{d}\boldsymbol{v}_{i}.\end{eqnarray}$$
Considering the fact that the particle relative velocity
$V$
is much smaller than the peculiar velocity of the gas molecules
$v_{i}$
, the exponential term in (A 1) can be simplified to
$$\begin{eqnarray}\exp \!\left(\!-\frac{m_{i}v_{i}^{2}}{2k_{B}T}\right)=\exp \!\left(\!-\frac{g_{i}^{2}+2g_{i}V\cos \unicode[STIX]{x1D719}+V^{2}}{2k_{B}T/m_{i}}\right)\approx \exp \!\left(\!-\frac{m_{i}g_{i}^{2}}{2k_{B}T}\right)\exp \!\left(\!-\frac{m_{i}g_{i}V\cos \unicode[STIX]{x1D719}}{k_{B}T}\right),\end{eqnarray}$$
where
$\exp [-m_{i}g_{i}V\cos \unicode[STIX]{x1D719}/(k_{B}T)]$
can be expanded to
$$\begin{eqnarray}\exp \!\left(\!-\frac{m_{i}g_{i}V\cos \unicode[STIX]{x1D719}}{k_{B}T}\right)=1-\frac{m_{i}g_{i}V\cos \unicode[STIX]{x1D719}}{k_{B}T}+\frac{1}{2}\left(\!\frac{m_{i}g_{i}V\cos \unicode[STIX]{x1D719}}{k_{B}T}\right)^{2}-\frac{1}{6}\left(\!\frac{m_{i}g_{i}V\cos \unicode[STIX]{x1D719}}{k_{B}T}\right)^{3}+\cdots \,.\end{eqnarray}$$
Since
$k_{B}T\sim m_{i}g_{i}^{2}\gg m_{i}g_{i}V$
and
$m_{i}g_{i}V/(k_{B}T)\ll 1$
, the second- and higher-order Taylor expansion terms in (A 3) can be neglected. Therefore, (A 2) can be simplified to
$$\begin{eqnarray}\exp \!\left(\!-\frac{m_{i}v_{i}^{2}}{2k_{B}T}\right)=\exp \!\left(\!-\frac{m_{i}g_{i}^{2}}{2k_{B}T}\right)\left(1-\frac{m_{i}g_{i}V\cos \unicode[STIX]{x1D719}}{k_{B}T}\right).\end{eqnarray}$$
Moreover, in the short time
$\text{d}t$
,
$\boldsymbol{V}$
can be treated as constant, thus
$$\begin{eqnarray}\text{d}\boldsymbol{v}_{i}=\text{d}\boldsymbol{g}_{i}=g_{i}^{2}\sin \unicode[STIX]{x1D719}\,\text{d}\unicode[STIX]{x1D719}\,\text{d}\unicode[STIX]{x1D703}\,\text{d}g_{i}.\end{eqnarray}$$
Substituting (A 4) and (A 5) into (A 1), we obtain
$$\begin{eqnarray}\displaystyle \boldsymbol{F}_{L,i} & = & \displaystyle \frac{\sqrt{2\unicode[STIX]{x03C0}}m_{ip}n_{i}b_{i}\boldsymbol{G}}{4\unicode[STIX]{x03C0}^{2}}\left(\frac{m_{i}}{k_{B}T}\right)^{5/2}\int _{\boldsymbol{v}_{i}}g_{i}^{3}\sin ^{2}\unicode[STIX]{x1D719}\sin ^{2}\unicode[STIX]{x1D703}\,(g_{i}\cos \unicode[STIX]{x1D719}+V)\nonumber\\ \displaystyle & & \displaystyle \times \exp \!\left(\!-\frac{m_{i}g_{i}^{2}}{2k_{B}T}\right)\left(1-\frac{m_{i}g_{i}V\cos \unicode[STIX]{x1D719}}{k_{B}T}\right)Q(g_{i})\,\text{d}\boldsymbol{v}_{i}\nonumber\\ \displaystyle & = & \displaystyle \frac{\sqrt{2\unicode[STIX]{x03C0}}m_{ip}n_{i}b_{i}\boldsymbol{G}}{4\unicode[STIX]{x03C0}}\left(\frac{m_{i}}{k_{B}T}\right)^{5/2}\int _{0}^{\infty }\int _{0}^{\unicode[STIX]{x03C0}}g_{i}^{5}\sin ^{3}\unicode[STIX]{x1D719}\,(g_{i}\cos \unicode[STIX]{x1D719}+V)\nonumber\\ \displaystyle & & \displaystyle \times \exp \!\left(\!-\frac{m_{i}g_{i}^{2}}{2k_{B}T}\right)\left(1-\frac{m_{i}g_{i}V\cos \unicode[STIX]{x1D719}}{k_{B}T}\right)Q(g_{i})\,\text{d}\unicode[STIX]{x1D719}\,\text{d}g_{i}.\end{eqnarray}$$
Integrating over
$\unicode[STIX]{x1D719}$
, we have
$$\begin{eqnarray}\boldsymbol{F}_{L,i}=\frac{\sqrt{2\unicode[STIX]{x03C0}}m_{ip}n_{i}b_{i}\boldsymbol{G}V}{\unicode[STIX]{x03C0}}\left(\frac{m_{i}}{k_{B}T}\right)^{5/2}\int _{0}^{\infty }\left(\frac{1}{3}g_{i}^{5}-\frac{1}{15}\frac{m_{i}g_{i}^{7}}{k_{B}T}\right)\exp \!\left(\!-\frac{m_{i}g_{i}^{2}}{2k_{B}T}\right)Q(g_{i})\,\text{d}g_{i}.\end{eqnarray}$$
By using the definition of the reduced collision integrals (3.5) and
$\unicode[STIX]{x1D6FE}_{i}=g_{i}\sqrt{m_{i}/(2k_{B}T)}$
, equation (A 7) can be rewritten as
$$\begin{eqnarray}\displaystyle \boldsymbol{F}_{L,i} & = & \displaystyle \frac{8m_{ip}n_{i}b_{i}\boldsymbol{G}V}{15\unicode[STIX]{x03C0}}\sqrt{2\unicode[STIX]{x03C0}k_{B}T/m_{i}}\int _{0}^{\infty }(5\unicode[STIX]{x1D6FE}_{i}^{5}-2\unicode[STIX]{x1D6FE}_{i}^{7})\exp (-\unicode[STIX]{x1D6FE}_{i}^{2})Q(g_{i})\,\text{d}\unicode[STIX]{x1D6FE}_{i}\nonumber\\ \displaystyle & = & \displaystyle \frac{8}{15}m_{ip}\sqrt{2\unicode[STIX]{x03C0}k_{B}T/m_{i}}\,\boldsymbol{G}VR^{2}n_{i}b_{i}[5\unicode[STIX]{x1D6FA}_{ip}^{(1,1)\ast }-6\unicode[STIX]{x1D6FA}_{ip}^{(1,2)\ast }],\end{eqnarray}$$
which is (3.4).
