Slow Motion of a Spherical Particle Perpendicular to Two Planar Walls with Slip Surfaces

Abstract

The quasi-steady creeping flow of a viscous fluid around a slip sphere translating perpendicular to one or two large slip planar walls at arbitrary relative positions is analyzed. To solve the axisymmetric Stokes equation for the fluid flow, we construct a general solution using fundamental solutions in spherical and cylindrical coordinate systems. Boundary conditions are first applied to the planar walls using the Hankel transform and then to the particle surface using a collocation method. Numerical results of the drag force exerted by the fluid on the particle are obtained for different values of the relevant stickiness/slip and configuration parameters. Our force results agree well with existing solutions for the motion of a slip sphere perpendicular to one or two nonslip planar walls. The hydrodynamic drag force acting on the particle is a monotonic increasing function of the stickiness of the planar walls and the ratio of its radius to distance from each planar wall. With other parameters remaining constant, this drag force generally increases with increasing stickiness of the particle surface. The influence of the slip planar walls on the axisymmetric translation of a slip sphere is significantly stronger than its axisymmetric rotation.

1. Introduction

Problems concerning the migration of a colloidal particle in a viscous fluid at low Reynolds numbers have received extensive attention from researchers in the fields of chemical, mechanical, biomedical, civil, and environmental engineering and science. Most such creeping motions are fundamental in nature, providing a basis for understanding numerous practical systems and industrial processes such as sedimentation, flocculation, flotation, aerosol technology, suspension rheology, locomotion of blood cells in arteries or veins, microfluidics, electrophoresis, and other phoretic motions. The theoretical study of this theme began with the classic work of Stokes [1] on the slow translation of a nonslip rigid sphere in an unbounded fluid.

In spite of that, a viscous fluid in contact with a solid surface is normally assumed to be nonslip in the Stokes problems; it can slip frictionally for the flows at lyophobic [2,3,4,5], permeable [6,7], airborne [8,9,10], and other [11,12] solid surfaces. Such slip velocities are presumably proportional to the shear stress of the fluid at the solid surfaces with the Navier slip coefficient ${\beta }^{-1}$ as a proportionality factor [13]. Basset [14] found that the force exerted by the surrounding unbounded fluid of viscosity $\eta$ on a migrating rigid sphere of radius $a$ with a slip-flow boundary condition at its surface is

$$F_0=-6\mathsf{\pi }\eta aU{\displaystyle \frac{\beta a+2\eta }{\beta a+3\eta }},$$

(1)

where $U$ is the translational velocity of the particle. In the particular case of $\beta \to \infty$, there is no slip at the particle surface and Equation (1) degenerates to Stokes’ law. When $\beta =0$, there is a perfect slip at the particle surface and Equation (1) is consistent with the translation of an inviscid gas bubble.

In most practical applications, particles move in bounded fluids. So, it is important to understand whether the proximity of confining walls significantly affects the migration of a particle in a fluid [15,16]. Through an exact representation in spherical bipolar coordinates, Reed and Morrison [17] and Chen and Keh [18] examined the quasi-steady creeping motion of a rigid sphere normal to an infinite planar wall, where the fluid may slip at the solid surfaces. Later, the migrations of a spherical particle inside a spherical cavity with slip surfaces at an arbitrary direction were analyzed for the concentric case [19,20] and semi-analytically studied with the use of a boundary collocation method [21,22,23,24] for the eccentric case [25,26]. The wall effects experienced by a slip spherical particle migrating along the axis of a slip circular tube [27,28], axisymmetric toward an orifice in a nonslip planar wall [29], as well as at arbitrary position and direction between two parallel nonslip planar walls [30,31] were also examined by using the boundary collocation method. In these investigations with confining walls, analytical and numerical results for the hydrodynamic drag force acting on the slip particle to correct Equation (1) were obtained for various cases.

The purpose of this article is to obtain a solution for the slow translational motion of a slip spherical particle perpendicular to two parallel slip planar walls at an arbitrary position between them. The creeping flow equations applicable to the systems are solved by using the boundary collocation method [23], and the wall-corrected hydrodynamic drag force acting on the particle is obtained with good convergence for various cases.

2. Analysis

As illustrated in Figure 1, we consider the quasi-steady creeping flow of a Newtonian fluid around a spherical particle of radius $a$ translating with velocity $U$ (in the $z$ direction) normal to two large planar walls whose distances from the center of the particle are $b$ and $c$, respectively (setting $b\le c$ without loss of generality). The circular cylindrical coordinates $(\rho ,\varphi ,z)$ and spherical coordinates $(r,\theta ,\varphi )$ with their origin at the particle center are used. The fluid may slip frictionally to any degree at the particle and wall surfaces and is at rest away from the particle. The purpose is to determine the correction to Equation (1) for the axisymmetric motion of the slip sphere due to the presence of the slip planar walls.

The fluid flow is governed by the equation of motion

$$E_S^2(E_S^2\Psi )=0,$$

(2)

where $\Psi$ is the Stokes stream function (automatically satisfying the continuity equation of the incompressible fluid) related to the nonvanishing velocity components in cylindrical coordinates by

$$v_{\rho }={\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial \Psi }{\partial z}}, v_z=-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial \Psi }{\partial \rho }},$$

(3)

and $E_S^2$ is the Stokes operator given by

$$E_{\mathrm{S}}^2=\rho {\displaystyle \frac{\partial }{\partial \rho }}({\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial }{\partial \rho }})+{\displaystyle \frac{{\partial }^2}{\partial z^2}}.$$

(4)

Since the slip velocities of the fluid at the solid surfaces are proportional to the local shear stresses (equal to $(I-nn)n:\tau /\beta$, where $\tau$ is the stress tensor, $n$ is the unit vector outwardly normal to the solid surfaces, and $I$ is the unit tensor) [13,14], the boundary conditions for the fluid flow at the particle surface, planar walls, and infinity are

$$r=a: v_{\rho }={\displaystyle \frac{1}{\beta }}{\tau }_{r\theta }\cos \theta , v_z=U-{\displaystyle \frac{1}{\beta }}{\tau }_{r\theta }\sin \theta ,$$

(5)

$$z=-b: v_{\rho }={\displaystyle \frac{1}{{\beta }_{\mathrm{w}1}}}{\tau }_{z\rho }, v_z=0,$$

(6)

$$z=c: v_{\rho }=-{\displaystyle \frac{1}{{\beta }_{\mathrm{w}2}}}{\tau }_{z\rho }, v_z=0,$$

(7)

$$\rho \to \infty \mathrm{and} -b<z<c: v_{\rho }=v_z=0,$$

(8)

where $1/\beta$ and $1/{\beta }_{\mathrm{w}i}$ (with $i=1$ and 2) are the Navier slip coefficients about the particle surface and related planar walls, respectively, and ${\tau }_{r\theta }$ and ${\tau }_{z\rho }$ are the matching shear stresses of the viscous fluid flow.

The stream function can be expressed as [23]

$$\Psi ={\displaystyle \sum _{n=2}^{\infty }(B_n{r}^{-n+1}+D_n{r}^{-n+3})G_n^{-1/2}(\cos \theta )}+{\displaystyle {\int }_0^{\infty }F(\omega ,z)\rho J_1(\omega \rho )\mathrm{d}\omega },$$

(9)

where the transformed function

$$F(\omega ,z)=A(\omega ){\mathrm{e}}^{\omega z}+B(\omega ){\mathrm{e}}^{-\omega z}+C(\omega )\omega z{\mathrm{e}}^{\omega z}+D(\omega )\omega z{\mathrm{e}}^{-\omega z},$$

(10)

$G_n^{-1/2}$ is the Gegenbauer polynomial of the first kind of order n and degree $-1/2$, $J_n$ is the Bessel function of the first kind of order $n$, boundary condition (8) is satisfied immediately, $B_n$, $D_n$, $A(\omega )$, $B(\omega )$, $C(\omega )$, and $D(\omega )$ are the unknown coefficients to be determined from the remaining boundary conditions, and $\omega$ is a separation (transform) variable. The infinite-series term in Equation (9) is a general solution to Equation (2) in spherical coordinates denoting the disturbance caused by the spherical particle and the Fourier-Bessel integral term is a general solution to Equation (2) in cylindrical coordinates representing the disturbance shaped by the planar walls. The substitution of Equation (9) into Equation (3) leads to

$$v_{\rho }={\displaystyle \sum _{n=2}^{\infty }[B_n{B}_n^{\prime }(\rho ,z)+D_n{D}_n^{\prime }(\rho ,z)]}+{\displaystyle {\int }_0^{\infty }E(\omega ,z)J_1(\omega \rho )\omega \mathrm{d}\omega },$$

(11)

$$v_z={\displaystyle \sum _{n=2}^{\infty }[B_n{B}_n^{″}(\rho ,z)+D_n{D}_n^{″}(\rho ,z)]}-{\displaystyle {\int }_0^{\infty }F(\omega ,z)J_0(\omega \rho )\omega \mathrm{d}\omega },$$

(12)

where

$$E(\omega ,z)=A(\omega ){\mathrm{e}}^{\omega z}-B(\omega ){\mathrm{e}}^{-\omega z}+C(\omega )(1+\omega z){\mathrm{e}}^{\omega z}+D(\omega )(1-\omega z){\mathrm{e}}^{-\omega z},$$

(13)

and the position functions $B_n^{\prime }(\rho ,z)$, $D_n^{\prime }(\rho ,z)$, $B_n^{″}(\rho ,z)$, and $D_n^{″}(\rho ,z)$ are given by Equations (A1)–(A4) in Appendix A.

The application of boundary conditions (6) and (7) to Equations (11) and (12) gives

$${\displaystyle {\int }_0^{\infty }[E(\omega ,z_i)+{(-1)}^{i}2\frac{\eta \omega }{{\beta }_{\mathrm{w}i}}{E}^{*}(\omega ,z_i)]J_1(\omega \rho )\omega \mathrm{d}\omega }=-{\displaystyle \sum _{n=2}^{\infty }[B_n{\gamma }_{in}(\rho ,z_i)+D_n{\delta }_{in}(\rho ,z_i)]},$$

(14)

$${\displaystyle {\int }_0^{\infty }F(\omega ,z_i)J_0(\omega \rho )\omega \mathrm{d}\omega }={\displaystyle \sum _{n=2}^{\infty }[B_n{B}_n^{″}(\rho ,z_i)+D_n{D}_n^{″}(\rho ,z_i)]},$$

(15)

where

$$E^{*}(\omega ,z)=A(\omega ){\mathrm{e}}^{\omega z}+B(\omega ){\mathrm{e}}^{-\omega z}+C(\omega )(1+\omega z){\mathrm{e}}^{\omega z}-D(\omega )(1-\omega z){\mathrm{e}}^{-\omega z},$$

(16)

the position functions ${\gamma }_{in}(\rho ,z)$ and ${\delta }_{in}(\rho ,z)$ are given by Equations (A5) and (A6) with $i=1$ and 2, $z_1=-b$, $z_2=c$, and $\eta$ is the viscosity of the fluid. Applying the Hankel transforms on the variable $\rho$ to Equations (14) and (15), we obtain

$$E(\omega ,z_i)+{(-1)}^{i}2{\displaystyle \frac{\eta \omega }{{\beta }_{\mathrm{w}i}}}{E}^{*}(\omega ,z_i)={\displaystyle \sum _{n=2}^{\infty }[B_n{S}_{1n}^{\prime }(\omega ,z_i)+D_n{S}_{2n}^{\prime }(\omega ,z_i)]},$$

(17)

$$F(\omega ,z_i)=-{\displaystyle \sum _{n=2}^{\infty }[B_n{S}_{1n}^{″}(\omega ,z_i)+D_n{S}_{2n}^{″}(\omega ,z_i)]},$$

(18)

where the transformed functions $S_{1n}^{\prime }(\omega ,z)$, $S_{1n}^{″}(\omega ,z)$, $S_{2n}^{\prime }(\omega ,z)$, and $S_{2n}^{″}(\omega ,z)$ are given by Equations (A12)–(A15).

After the substitution of Equations (10), (13) and (16), the four linear algebraic equations formed from Equations (17) and (18) with $i=1$ and 2 can be solved simultaneously to yield the four coefficients $A(\omega )$, $B(\omega )$, $C(\omega )$, and $D(\omega )$ in terms of the as yet unknown coefficients $B_n$ and $D_n$, with the result

$$\left[\begin{array}{l}A(\omega )\\ B(\omega )\\ C(\omega )\\ D(\omega )\end{array}\right]={\displaystyle \sum _{n=2}^{\infty }\{B_n}\left[\begin{array}{l}{A}_{1n}(\omega )\\ B_{1n}(\omega )\\ C_{1n}(\omega )\\ D_{1n}(\omega )\end{array}\right]+D_n\left[\begin{array}{l}{A}_{2n}(\omega )\\ B_{2n}(\omega )\\ C_{2n}(\omega )\\ D_{2n}(\omega )\end{array}\right]\},$$

(19)

where the functions $A_{in}(\omega )$, $B_{in}(\omega )$, $C_{in}(\omega )$, and $D_{in}(\omega )$ with $i=1$ and 2 are given by Equations (A40)–(A57).

Substituting Equation (19) back into Equations (10)–(13), we obtain the fluid velocity components as

$$v_{\rho }={\displaystyle \sum _{n=2}^{\infty }[B_n{\alpha }_{1n}^{(1)}(r,\theta )+D_n{\alpha }_{2n}^{(1)}(r,\theta )]},$$

(20)

$$v_z={\displaystyle \sum _{n=2}^{\infty }[B_n{\alpha }_{1n}^{(2)}(r,\theta )+D_n{\alpha }_{2n}^{(2)}(r,\theta )]},$$

(21)

where the position functions ${\alpha }_{in}^{(j)}(r,\theta )$ with $i$ and $j$ equal to 1 and 2 are given by Equations (A9) and (A10). The substitution of Equations (20) and (21) into Equation (5) for the boundary condition at the particle surface results in

$${\displaystyle \sum _{n=2}^{\infty }\{B_n[{\alpha }_{1n}^{(1)}(a,\theta )-\frac{\eta }{\beta }{\alpha }_{1n}^{*}(a,\theta )\cos \theta ]+D_n[{\alpha }_{2n}^{(1)}(a,\theta )-\frac{\eta }{\beta }{\alpha }_{2n}^{*}(a,\theta )\cos \theta ]\}}=0,$$

(22)

$${\displaystyle \sum _{n=2}^{\infty }\{B_n[{\alpha }_{1n}^{(2)}(a,\theta )+\frac{\eta }{\beta }{\alpha }_{1n}^{*}(a,\theta )\sin \theta ]+D_n[{\alpha }_{2n}^{(2)}(a,\theta )+\frac{\eta }{\beta }{\alpha }_{2n}^{*}(a,\theta )\sin \theta ]\}}=U,$$

(23)

where the position functions ${\alpha }_{in}^{*}(r,\theta )$ are given by Equation (A11).

To satisfy the conditions in Equations (22) and (23) along the surface of the slip sphere requires the solution of an infinite array of the coefficients $B_n$ and $D_n$. The boundary collocation method [23] enforces these conditions at a number of points on the longitudinal arc from $\theta =0$ to $\theta =\pi$ of the sphere and truncates their infinite series into finite series. If N points on the longitudinal arc are satisfied and the infinite series are truncated to N terms, we get a system of 2N simultaneous linear algebraic equations. These equations can be solved numerically to obtain the 2N coefficients $B_n$ and $D_n$ required for the truncated form of Equations (11) and (12) or (20) and (21). Once these coefficients are solved for sufficiently large values of N, the fluid velocity field can be fully obtained.

The drag force exerted by the fluid on the slip sphere can be determined according to [13]

$$F=4\pi \eta D_2.$$

(24)

This formula indicates that only the first coefficient $D_2$ contributes to the hydrodynamic force on the particle.

3. Results and Discussion

The numerical solutions obtained using the boundary collocation method for the hydrodynamic drag force acting on a slip spherical particle translating perpendicular to a single slip planar wall (viz., $c\to \infty$), two equally slipping planar walls with equal distances (viz., ${\beta }_{\mathrm{w}2}={\beta }_{\mathrm{w}1}$ and $c=b$), and one full-slip planar wall and another nonslip planar wall (viz., ${\beta }_{\mathrm{w}2}=0$ and ${\beta }_{\mathrm{w}1}\to \infty$) are listed in

Table 1. Normalized drag force $F/F_0$ on a slip sphere translating perpendicular to a single planar wall ($c\to \infty$) at different values of $\beta a/\eta$, ${\beta }_{\mathrm{w}1}b/\eta$, and $a/b$.

$\mathit{\beta }\mathit{a}/\mathit{\eta }$	$\mathit{a}/\mathit{b}$	$\mathit{F}/{\mathit{F}}_0$
		${\mathit{\beta }}_{\mathbf{w}1}\mathit{b}/\mathit{\eta }=0$	${\mathit{\beta }}_{\mathbf{w}1}\mathit{b}/\mathit{\eta }=1$	${\mathit{\beta }}_{\mathbf{w}1}\mathit{b}/\mathit{\eta }=3$	${\mathit{\beta }}_{\mathbf{w}1}\mathit{b}/\mathit{\eta }=10$	${\mathit{\beta }}_{\mathbf{w}1}\mathit{b}/\mathit{\eta }\mathbf{\to }\mathbf{\infty }$
0	0.1	1.05264	1.06096	1.06776	1.07475	1.08110
	0.3	1.17719	1.20910	1.23616	1.26502	1.29224
	0.5	1.34106	1.41233	1.47662	1.54966	1.62382
	0.7	1.58556	1.73194	1.87850	2.06633	2.29122
	0.8	1.77839	1.99546	2.23133	2.56687	3.04339
	0.9	2.10931	2.46421	2.90022	3.63713	5.13001
	0.95	2.44379	2.95120	3.63751	4.98083	9.08847
	0.99	3.23330	4.12306	5.51052	8.91686	39.5699
	0.995	3.57695	4.63695	6.35356	10.8302	77.2772
	0.999	4.37868	5.83819	8.34279	15.4916	375.817
3	0.1	1.06661	1.07727	1.08599	1.09499	1.10318
	0.3	1.22971	1.27173	1.30759	1.34605	1.38253
	0.5	1.45436	1.54840	1.63355	1.73090	1.83047
	0.7	1.81040	1.99734	2.18340	2.42163	2.70754
	0.8	2.10904	2.37635	2.66233	3.06505	3.63259
	0.9	2.65566	3.06589	3.55523	4.36109	5.94513
	0.95	3.24477	3.79945	4.52184	5.88503	9.84840
	0.99	4.72957	5.61545	6.93895	10.0551	36.1757
	0.995	5.39933	6.42583	8.02021	12.0138	67.2352
	0.999	6.97895	8.33568	10.5539	16.7214	308.200
$\mathrm{\infty }$	0.1	1.08096	1.09408	1.10487	1.11602	1.12619
	0.3	1.28781	1.34205	1.38884	1.43964	1.48843
	0.5	1.59668	1.72363	1.84125	1.97937	2.12554
	0.7	2.17078	2.43602	2.71051	3.08005	3.55939
	0.8	2.77242	3.16366	3.60266	4.26209	5.30532
	0.9	4.33307	4.95945	5.74884	7.15332	10.4405
	0.95	7.14158	8.01670	9.22199	11.6810	20.5762
	0.99	27.8594	29.3309	31.6397	37.4262	100.892
	0.995	53.1698	54.9006	57.7131	65.1576	201.028
	0.999	252.546	254.881	258.888	269.711	995.829

,

Table 2. Normalized drag force $F/F_0$ on a slip sphere translating perpendicular to two equally slippery and distant planar walls (${\beta }_{\mathrm{w}2}={\beta }_{\mathrm{w}1}$ and $c=b$) at different values of $\beta a/\eta$, ${\beta }_{\mathrm{w}1}b/\eta$, and $a/b$.

$\mathit{\beta }\mathit{a}/\mathit{\eta }$	$\mathit{a}/\mathit{b}$	$\mathit{F}/{\mathit{F}}_0$
		${\mathit{\beta }}_{\mathbf{w}1}\mathit{b}/\mathit{\eta }=0$	${\mathit{\beta }}_{\mathbf{w}1}\mathit{b}/\mathit{\eta }=1$	${\mathit{\beta }}_{\mathbf{w}1}\mathit{b}/\mathit{\eta }=3$	${\mathit{\beta }}_{\mathbf{w}1}\mathit{b}/\mathit{\eta }=10$	${\mathit{\beta }}_{\mathbf{w}1}\mathit{b}/\mathit{\eta }\mathbf{\to }\mathbf{\infty }$
0	0.1	1.07448	1.08259	1.09027	1.09883	1.10714
	0.3	1.26260	1.29684	1.33053	1.36958	1.40911
	0.5	1.53229	1.61836	1.70793	1.81845	1.93887
	0.7	1.96739	2.17033	2.40129	2.72019	3.12597
	0.8	2.32772	2.65284	3.04895	3.65085	4.55048
	0.9	2.96524	3.54514	4.32675	5.71764	8.63955
	0.95	3.62238	4.49618	5.77069	8.36688	16.5126
	0.99	5.19212	6.82146	9.49163	16.2077	77.4399
	0.995	5.87829	7.84693	11.1697	20.0306	152.847
	0.999	7.48083	10.2390	15.1503	29.3489	749.987
3	0.1	1.09473	1.10520	1.11515	1.12627	1.13710
	0.3	1.34586	1.39171	1.43719	1.49035	1.54460
	0.5	1.72675	1.84159	1.96211	2.11220	2.27720
	0.7	2.37874	2.63651	2.93038	3.33719	3.85676
	0.8	2.94893	3.34311	3.82067	4.54221	5.61483
	0.9	4.01688	4.67066	5.53746	7.04987	10.1444
	0.95	5.18308	6.11195	7.43564	10.0545	17.9018
	0.99	8.14327	9.72311	12.2413	18.3596	70.5166
	0.995	9.48165	11.3360	14.3899	22.2729	132.628
	0.999	12.6405	15.1434	19.4922	31.6837	614.605
$\mathrm{\infty }$	0.1	1.11576	1.12878	1.14119	1.15511	1.16873
	0.3	1.44102	1.50180	1.56285	1.63513	1.70994
	0.5	1.98237	2.14363	2.31670	2.53798	2.78920
	0.7	3.06884	3.45148	3.90270	4.55654	5.45349
	0.8	4.24667	4.84996	5.61022	6.82698	8.84049
	0.9	7.34598	8.38578	9.82542	12.5142	19.0039
	0.95	12.9532	14.4727	16.7329	21.5216	39.2209
	0.99	54.3819	57.0794	61.5374	72.9733	199.811
	0.995	104.999	108.213	113.677	128.428	400.080
	0.999	503.860	508.290	516.140	539.033	1989.86

and

Table 3. Normalized drag force $F/F_0$ on a slip sphere translating perpendicular to two planar walls with ${\beta }_{\mathrm{w}1}\to \infty$ and ${\beta }_{\mathrm{w}2}=0$ at different values of $\beta a/\eta$, $a/b$, and $b/(b+c)$.

$\mathit{b}/(\mathit{b}+\mathit{c})$	$\mathit{a}/\mathit{b}$	$\mathit{F}/{\mathit{F}}_0$
		$\mathit{\beta }\mathit{a}/\mathit{\eta }=0$	$\mathit{\beta }\mathit{a}/\mathit{\eta }=1$	$\mathit{\beta }\mathit{a}/\mathit{\eta }=3$	$\mathit{\beta }\mathit{a}/\mathit{\eta }=10$	$\mathit{\beta }\mathit{a}/\mathit{\eta }\mathbf{\to }\mathbf{\infty }$
0.25	0.1	1.08158	1.09257	1.10380	1.11616	1.12697
	0.3	1.29424	1.33811	1.38539	1.44067	1.49232
	0.5	1.62840	1.72493	1.83757	1.98287	2.13595
	0.7	2.29990	2.48092	2.72188	3.08955	3.58165
	0.8	3.05486	3.29394	3.65204	4.28056	5.33622
	0.9	5.14486	5.42003	5.97088	7.21721	10.4823
	0.95	9.10527	9.23415	9.87781	11.8890	20.6243
	0.99	39.5885	37.0743	36.2084	39.2119	100.947
	0.995	77.2932	70.9251	67.2661	69.0240	201.097
	0.999	339.308	304.622	279.012	264.561	890.644
0.5	0.1	1.09335	1.10609	1.11914	1.13353	1.14617
	0.3	1.34485	1.39842	1.45660	1.52514	1.58953
	0.5	1.75332	1.87767	2.02393	2.21356	2.41307
	0.7	2.57623	2.82433	3.15333	3.65150	4.30551
	0.8	3.47593	3.82082	4.32559	5.19233	6.59805
	0.9	5.84750	6.30701	7.13339	8.89744	13.2406
	0.95	10.1171	10.5210	11.6000	14.5364	26.1588
	0.99	41.3694	39.3699	39.3916	44.6814	127.172
	0.995	79.4179	73.6746	71.1181	75.8685	252.616
	0.999	378.755	340.898	313.664	301.139	1247.08

, respectively, as functions of the dimensionless stickiness/slip parameter $\beta a/\eta$ (ranging from 0 to $\infty$) of the particle, spacing parameter $a/b$ (ranging from 0 to 1), and parameter ${\beta }_{\mathrm{w}1}b/\eta$ (ranging from 0 to $\infty$) or $b/(b+c)$ (ranging from 0 to 1/2). The drag force $F_0$ acting on a particle in an unbounded fluid is given by Equation (1), which normalizes the wall-corrected drag force $F$. Therefore, in the limiting case of $a/b=0$ (the plane walls are at infinity), $F/F_0=1$, regardless of other parameters. Our results for $F/F_0$, which are always greater than unity, converge to at least the significant figures shown in the tables, and the number of collocation points $N=200$ is sufficiently large to achieve this convergence (the convergence data for the force at different N values are very similar to those given by Ganatos et al. [23]). In the limit ${\beta }_{\mathrm{w}i}b/\eta \to \infty$, these results agree excellently (to the same significant figures) with those obtained for the axisymmetric translation of a slip particle between two parallel nonslip planar walls [31]. For the cases of a slip sphere translating perpendicular to a no-slip or slip plane wall, our results of $F/F_0$ are also in good agreement with other analytical and numerical solutions [32,33,34,35].

The boundary collocation results of the normalized drag force $F/F_0$ on a slip sphere translating perpendicular to two equally slipping planar walls with equal distances (${\beta }_{\mathrm{w}2}={\beta }_{\mathrm{w}1}$ and $c=b$, in solid curves) and to a single planar wall ($c\to \infty$, in dashed curves) versus the parameters ${\beta }_{\mathrm{w}1}b/\eta$, $\beta a/\eta$, and $a/b$ are plotted in Figure 2, Figure 3 and Figure 4, respectively. For given values of the spacing parameter $a/b$ and stickiness parameter $\beta a/\eta$ of the particle, as shown in Figure 2, Figure 3a and Figure 4b, the normalized force $F/F_0$ is a monotonic increasing function of the stickiness parameter ${\beta }_{\mathrm{w}1}b/\eta$ of the planar walls. For specified values of ${\beta }_{\mathrm{w}1}b/\eta$ and $a/b$, the normalized force $F/F_0$ generally increases with increasing $\beta a/\eta$ (with some exceptions for large values of ${\beta }_{\mathrm{w}1}b/\eta$ and $a/b$, as shown in Table 1, Table 2 and Table 3), as illustrated in Figure 2b, Figure 3 and Figure 4a. For fixed values of $\beta a/\eta$ and ${\beta }_{\mathrm{w}1}b/\eta$, the normalized force $F/F_0$ increases monotonically with increasing $a/b$ and becomes infinite in the limiting case of $a/b=1$ (the particle contacts the plane walls), as shown in Figure 2a, Figure 3b and Figure 4.

For any fixed values of the parameters ${\beta }_{\mathrm{w}1}b/\eta$, $\beta a/\eta$, and $a/b$, consistent with the previous results of nonslip planar walls (${\beta }_{\mathrm{w}1}b/\eta \to \infty$) [31], the assumption that the hydrodynamic effect of two planar walls on a translating sphere is simply the sum of the effects of each single wall will overestimate the correction to the drag force acting on the particle. That is, the increase in $F/F_0$ from unity for the case of two planar walls (as given by Table 2 or the solid curves in Figure 2, Figure 3 and Figure 4) is less than twice as it is for the case of a single planar wall (as given by Table 1 or the dashed curves in Figure 2, Figure 3 and Figure 4). The influence of the slip planar walls on the axisymmetric translation of the slip particle is significantly stronger (can be by as much as several orders of magnitude, depending especially on the value of $a/b$) than (but not as complex as) their influence on the axisymmetric rotation of the slip particle [36]. Particularly, the hydrodynamic torque $T$ on the rotating sphere near the confining planes normalized by the torque $T_0$ of the sphere far from the planes equals unity (the planes behave as an unbounded fluid) at a critical value of the wall stickiness parameter ${\beta }_{\mathrm{w}1}b/\eta$ close to 1.3 and increases with an increase in ${\beta }_{\mathrm{w}1}b/\eta$ from a positive value less than unity at perfect slip (${\beta }_{\mathrm{w}1}b/\eta =0$) to greater than unity at no slip (${\beta }_{\mathrm{w}1}b/\eta \to \infty$). When ${\beta }_{\mathrm{w}1}b/\eta$ is larger than the critical value, the normalized torque $T/T_0$ rises with increases of $a/b$ and $\beta a/\eta$. When ${\beta }_{\mathrm{w}1}b/\eta$ is less than the critical value, oppositely, $T/T_0$ reduces with increases of $a/b$ and $\beta a/\eta$. Moreover, for fixed values of the parameters ${\beta }_{\mathrm{w}1}b/\eta$, $\beta a/\eta$, and $a/b$, the assumption that the hydrodynamic effect of two planes on a rotating sphere is simply the sum of the effects of each single plane will overestimates/underestimates the correction to the torque acting on the sphere if ${\beta }_{\mathrm{w}1}b/\eta$ is greater/less than the critical value. Even for the limit that the sphere touches the planes ($a/b=1$), $T/T_0$ is still finite.

Figure 5 presents the numerical results of the normalized drag force $F/F_0$ acting on a spherical particle with typical values of the stickiness parameter $\beta a/\eta$ translating perpendicular to two equally slipping planar walls (${\beta }_{\mathrm{w}2}={\beta }_{\mathrm{w}1}$) versus the ratio $b/(b+c)$ (denoting the relative position of the particle between the planar walls). The solid curves (with constant spacing parameter $a/b$) illustrate the effect of the position of the second wall (at $z=c$, where $c\ge b$) on the drag force. Obviously, the proximity of a second wall [an increase in $b/(b+c)$] will enhance the drag force experienced by the particle near the first wall. The dashed curves [with constant ratio of the particle diameter to the wall-to-wall distance $2a/(b+c)$] denote the variation in the drag force as a function of the particle position between the two planar walls. At a constant value of $2a/(b+c)$, consistent with the previous results of nonslip planar walls (${\beta }_{\mathrm{w}1}b/\eta \to \infty$) [23,31], when the particle is in the middle between the two planar walls [$b/(b+c)=1/2$], it will experience the minimum drag force, and as the particle approaches either wall, the force increases monotonically.

Figure 6a,b present the numerical results of the normalized drag force $F/F_0$ acting on a spherical particle translating perpendicular to a nonslip planar wall (the first wall with ${\beta }_{\mathrm{w}1}b/\eta \to \infty$) and another slip plane wall (the second wall) versus the ratio $b/(b+c)$ with $a/b=0.9$ and various values of the stickiness parameters $\beta a/\eta$ and ${\beta }_{\mathrm{w}2}c/\eta$, respectively. Again, the proximity of a second wall [an increase in $b/(b+c)$] will enhance the drag force experienced by the particle near the first wall (which increases with an increase in $\beta a/\eta$). Naturally, $F/F_0$ increases with an increase in ${\beta }_{\mathrm{w}2}c/\eta$, keeping other parameters unchanged.

4. Concluding Remarks

The slow translation of a slip spherical particle in a viscous fluid perpendicular to two large slip planar walls at any position between them is analyzed. A boundary collocation method is used to solve the axisymmetric Stokes equation for the fluid flow. Numerical results of the drag force $F$ exerted by the fluid on the confined particle normalized by the force $F_0$ of the particle far from the planar walls are obtained as a function of the scaled particle radius $a/b$, particle position parameter $b/(b+c)$, particle stickiness parameter $\beta a/\eta$, and planar wall stickiness parameters ${\beta }_{\mathrm{w}1}b/\eta$ and ${\beta }_{\mathrm{w}2}c/\eta$. The normalized drag force $F/F_0$ is a monotonic increasing function of ${\beta }_{\mathrm{w}i}b/\eta$ and $a/b$. With ${\beta }_{\mathrm{w}i}b/\eta$ and $a/b$ remaining constant, $F/F_0$ generally increases with increasing $\beta a/\eta$.

Section 3 presents results for a resistance problem, in which the drag force $F$ is calculated for a sphere translating perpendicular to two planar walls at a given velocity $U$ [equal to $(F_0/6\mathsf{\pi }\eta a)(\beta a+3\eta )/(\beta a+2\eta )$ according to Equation (1)]. In a mobility problem, on the other hand, the force $F$ [equal to $6\mathsf{\pi }\eta a{U}_0(\beta a+2\eta )/(\beta a+3\eta )$] applied to a sphere is given, and the wall-corrected velocity $U$ needs to be determined. For the axisymmetric translation of a sphere perpendicular to two planar walls considered here, the ratio $U/U_0$ for the mobility problem equals the reciprocal of the ratio $F/F_0$ for the corresponding resistance problem, as shown Table 1, Table 2 and Table 3 and Figure 2, Figure 3, Figure 4, Figure 5 and Figure 6.

The paper only focuses on the motion of a single particle perpendicular to dual slip walls. However, the similar boundary collocation method can be easily used for solving the Stokes problems of actual slip multi-particle systems [18,37,38] and slip particles near other applicable slip boundaries, such as in a spherical cavity [25,26] and circular tube [27]. Also, this research is expected to provide valuable insights for the design of hydrodynamic metamaterials that have emerged in recent years [39,40,41,42].