Slow Rotation of a Soft Colloidal Sphere Normal to Two Plane Walls

Abstract

The creeping flow of a viscous fluid around a soft colloidal sphere rotating about a diameter normal to two planar walls at an arbitrary position between them is theoretically investigated in the steady limit of small Reynolds numbers. The fluid velocity outside the particle consists of the general solutions of the Stokes equation in circular cylindrical and spherical coordinates, while the fluid velocity inside the porous surface layer of the particle is expressed by the general solution of the Brinkman equation in spherical coordinates. The boundary conditions are implemented first on the planar walls by means of the Hankel transforms and then at the particle and hard-core surfaces by a collocation technique. The torque exerted on the particle by the fluid is calculated as a function of the ratio of the core-to-particle radii, ratio of the particle radius to the flow penetration length of the porous layer, and relative particle-to-wall spacings over the entire range. The wall effect on the rotating soft particle can be significant. The hydrodynamic torque exerted on the confined soft sphere increases as the relative particle-to-wall spacings decrease and stays finite even when the soft sphere contacts the plane walls. It is smaller than the torque on a hard sphere (or soft one with a reduced thickness or penetration length of the porous layer), holding the other parameters constant. For a given relative wall-to-wall spacing, this torque is minimal when the particle is situated midway between the walls and rises as it locates closer to either wall.

1. Introduction

The low-Reynolds-number translational and rotational motions of colloidal particles in incompressible Newtonian fluids have attracted wide attention from researchers in the fields of chemical, biomedical, mechanical, civil, and environmental engineering. These motions are practical and fundamental in numerous processes such as agglomeration, sedimentation, centrifugation, microfluidics, aerosol technology, and rheology of suspensions. The theoretical investigation of this topic began with Stokes’ studies [1,2] on the creeping motions of hard spherical particles in unbounded viscous fluids. Masliyah et al. [3] and Keh and Chou [4] extended this analysis to the translation and rotation, respectively, of a soft sphere.

A soft particle of radius $b$ has a hard core of radius $a$, covered by a permeable porous layer of thickness $b-a$. Polystyrene latices with surface layers [5] and biological cells with surface attachments [6] are examples of soft particles. To sterically stabilize colloidal dispersions, polymers are deliberately adsorbed by particles to form permeable layers [7]. When the porous layers of soft spheres disappear, the particles revert to hard spheres. When the hard cores of soft spheres vanish, they become fully porous spheres (like permeable colloid flocs and polymer coils) [8].

The hydrodynamic torque on a soft sphere of radius $b$ (a hard core of radius $a$ covered by a porous layer of thickness $b-a$) rotating with an angular velocity of $\Omega$ about a diameter in an unbounded fluid of viscosity $\eta$ at low Reynolds numbers is [4]

$$T_0=\frac{8\mathsf{\pi }\eta {\lambda }^{-2}b\Omega R}{\lambda a\cosh (\lambda b-\lambda a)+\sinh (\lambda b-\lambda a)},$$

(1)

where,

$$R=({\lambda }^{3}a{b}^2-3\lambda b+3\lambda a)\cosh (\lambda b-\lambda a)+({\lambda }^2{b}^2-3{\lambda }^{2}ab+3)\sinh (\lambda b-\lambda a),$$

(2)

and $1/\lambda$ is the penetration length (square root of permeability) of fluid flow within the surface layer of the soft particle ($T_0$ and $\Omega$ are in opposite directions). In the limiting case $\lambda b\to \infty$, Equation (1) degenerates to the Stokes result for a hard sphere of radius $b$.

In real situations of the rotation of particles, the surrounding fluid is bounded by solid walls [9,10,11,12]. Thus, it is necessary to know whether the proximity of boundary walls meaningfully affects particle rotation. The slow rotations of a hard sphere confined by adjacent boundaries, such as in a spherical cavity [13,14,15,16,17], in a circular cylinder [18,19,20], and near one or two planar walls [13,21,22,23], were analyzed. Alternatively, the low-Reynolds-number rotations of a soft or porous spherical particle in a spherical cavity [4,24,25,26,27] and in a cylinder [28] were also theoretically investigated. These studies show that the effect of boundaries on the rotation of particles can be very substantial and interesting.

In the general theories of stirred vessels and rotational viscometers for highly viscous liquids, it is important to understand the variation of torque as the confinement boundary approaches. The objective of this paper is to analyze the rotation of a soft colloidal sphere (having a porous layer of arbitrary thickness and permeability) about its diameter normal to one or two plane walls at an arbitrary position between them at a low Reynolds number. The fluid velocity was found by solving the Stokes and Brinkman equations using the boundary collocation method, and semianalytical results were obtained for the hydrodynamic torque acting on the particle for various values of the relevant parameters (the core-to-particle radius ratio, shielding parameter of the porous surface layer, and relative separation distances from the walls), with excellent convergence over the entire range.

2. Analysis

As illustrated in Figure 1, we studied the creeping flow of a constant-property fluid around a soft spherical particle of radius $b$ rotating steadily with a constant angular velocity $\Omega$ about a diameter perpendicular to two large planar walls whose distances from the particle center are $c$ and $d$, respectively ($c\le d$ is taken without loss of generality), and $(r,\theta ,\phi )$ and $(\rho ,\phi ,z)$ represent the spherical and cylindrical coordinate systems, respectively, originating from the particle center. The soft sphere comprises a permeable porous surface layer of thickness $b-a$. Thus, the radius of its hard core is $a$. The fluid velocity inside the porous layer is finite, while the external fluid far from the particle is at rest. The objective is to find the correction to Equation (1) for the particle rotation caused by the confining plane walls.

The creeping flow is governed by the Stokes and Brinkman equations, yielding

$$[{\nabla }^2-{\rho }^{-2}-h(r){\lambda }^2]v_{\varphi }=\frac{1}{r^2}\frac{\partial }{\partial r}(r^2\frac{\partial v_{\varphi }}{\partial r})+\frac{1}{r^2}\frac{\partial }{\partial \theta }[\frac{1}{\sin \theta }\frac{\partial }{\partial \theta }(v_{\varphi }\sin \theta )]-h(r){\lambda }^2{v}_{\varphi }=0,$$

(3)

where $v_{\varphi }(\rho ,z)$ in cylindrical coordinates or $v_{\varphi }(r,\theta )$ in spherical coordinates is the $\varphi$ (only nontrivial) component of the fluid velocity distribution, the continuity equation is satisfied, the dynamic pressure is constant everywhere, ${\lambda }^{-1}$ is the penetration length (square root of permeability) of fluid flow within the surface layer, and $h(r)$ equals unity as $a<r<b$ and zero otherwise. The boundary conditions require that the fluid is no slip at the hard-core surface and plane walls, and that both velocity and stress are continuous at the particle surface. Thus,

$$r=a:\hspace{1em}{v}_{\varphi }=0,$$

(4)

$$r=b:\hspace{1em}{v}_{\varphi } \mathrm{and} {\tau }_{r\varphi } \mathrm{are} \mathrm{continuous},$$

(5)

$$z=-c, d:\hspace{1em}{v}_{\varphi }=-\Omega \rho ,$$

(6)

$$\rho \to \infty \mathrm{and}-c<z<d:\hspace{1em}{v}_{\varphi }=-\Omega \rho ,$$

(7)

where ${\tau }_{r\varphi }$ is the nontrivial shear stress at the particle surface. Equations (3)–(7) take the reference frame rotating with the particle.

The fluid velocity can be expressed in the form [23,27]

$$v_{\varphi }={\displaystyle \sum _{n=1}^{\infty }{(\lambda r)}^{-1/2}[C_n{I}_{n+1/2}(\lambda r)+D_n{K}_{n+1/2}(\lambda r)]}{P}_n^1(\cos \theta ) \mathrm{if} a\le r\le b,$$

(8)

$$v_{\varphi }=-\Omega \rho +{\lambda }^{-2}{\displaystyle {\int }_0^{\infty }[X(\omega ){\mathrm{e}}^{\omega z}+Y(\omega ){\mathrm{e}}^{-\omega z}]\omega J_1(\omega \rho )\mathrm{d}\omega }+{\displaystyle \sum _{n=1}^{\infty }{A}_n{(\lambda r)}^{-n-1}{P}_n^1(\cos \theta )}$$

$$\mathrm{if} r\ge b \mathrm{and}-c\le z\le d,$$

(9)

where $P_n^1$ is the associated Legendre function of the first kind of order $n$ and degree 1, $J_n$ is the Bessel function of the first kind of order $n$, $I_n$ and $K_n$ are the modified Bessel functions of the first and second kinds, respectively, of order $n$, $X(\omega )$, $Y(\omega )$, $A_n$, $C_n$, and $D_n$ (all having the dimension of velocity) are the unknown functions and constants, respectively, to be determined. The parts of $v_{\varphi }$ involving $P_n^1$ in the previous equations are separable solutions to Equation (3) in spherical coordinates that represent the disturbance generated by the particle and the part of $v_{\varphi }$ involving $J_n$ in Equation (9) is a Fourier-Bessel integral solution to Equation (3) in cylindrical coordinates representing the disturbance produced by the planar walls. Note that Equation (9), which is a superposition of the general solutions in cylindrical and spherical coordinates due to the linearity of Equation (3), satisfies Equation (7) immediately.

Substitution of boundary condition (6) into Equation (9) leads to

$${\displaystyle {\int }_0^{\infty }[X(\omega ){\mathrm{e}}^{-\omega c}+Y(\omega ){\mathrm{e}}^{\omega c}]\omega J_1(\omega \rho )\mathrm{d}\omega }=-{\lambda }^2{\displaystyle \sum _{n=1}^{\infty }{A}_n{\alpha }_n(\rho ,-c)},$$

(10)

$${\displaystyle {\int }_0^{\infty }[X(\omega ){\mathrm{e}}^{\omega d}+Y(\omega ){\mathrm{e}}^{-\omega d}]\omega J_1(\omega \rho )\mathrm{d}\omega }=-{\lambda }^2{\displaystyle \sum _{n=1}^{\infty }{A}_n{\alpha }_n(\rho ,d)},$$

(11)

where

$${\alpha }_n(\rho ,z)={[{\lambda }^2({\rho }^2+z^2)]}^{-(n+1)/2}{P}_n^1[\frac{z}{{({\rho }^2+z^2)}^{1/2}}].$$

(12)

The application of the Hankel transform on the variable $\rho$ to Equations (10) and (11) yields

$$X(\omega ){\mathrm{e}}^{-\omega c}+Y(\omega ){\mathrm{e}}^{\omega c}=-{\lambda }^2{\displaystyle \sum _{n=1}^{\infty }{A}_n{\displaystyle {\int }_0^{\infty }{\alpha }_n(\rho ,-c)\rho J_1(\omega \rho )\mathrm{d}}\rho },$$

(13)

$$X(\omega ){\mathrm{e}}^{\omega d}+Y(\omega ){\mathrm{e}}^{-\omega d}=-{\lambda }^2{\displaystyle \sum _{n=1}^{\infty }{A}_n{\displaystyle {\int }_0^{\infty }{\alpha }_n(\rho ,d)\rho J_1(\omega \rho )\mathrm{d}}\rho }.$$

(14)

The solution of Equations (13) and (14) leads to

$$X(\omega )={\displaystyle \sum _{n=1}^{\infty }{A}_n}{X}_n(\omega ),$$

(15)

$$Y(\omega )={\displaystyle \sum _{n=1}^{\infty }{A}_n}{Y}_n(\omega ),$$

(16)

where

$$X_n(\omega )=\frac{{\mathrm{e}}^{\omega c}[-B_n(\omega ,-c)+{\mathrm{e}}^{\omega (c+d)}{B}_n(\omega ,d)]}{-1+{\mathrm{e}}^{2\omega (c+d)}},$$

(17)

$$Y_n(\omega )=\frac{{\mathrm{e}}^{\omega d}{[\mathrm{e}}^{\omega (c+d)}{B}_n(\omega ,-c)-B_n(\omega ,d)]}{-1+{\mathrm{e}}^{2\omega (c+d)}},$$

(18)

and

$$B_n\left(\omega ,z\right)=\frac{{\mathrm{e}}^{-\omega \left|z\right|}}{\left(n-1\right)!}{(\frac{\omega \left|z\right|}{\lambda z})}^{n-1}.$$

(19)

Substitution of Equations (15) and (16) back into Equation (9) results in

$$v_{\varphi }=-\Omega \rho +{\displaystyle \sum _{n=1}^{\infty }{A}_n{\gamma }_n(r,\theta )}\hspace{1em}\mathrm{if} r\ge b \mathrm{and}-c\le z\le d,$$

(20)

where

$${\gamma }_n(r,\theta )={\lambda }^{-2}{\displaystyle {\int }_0^{\infty }[X_n(\omega ){\mathrm{e}}^{\omega r\cos \theta }+Y_n(\omega ){\mathrm{e}}^{-\omega r\cos \theta }]\omega J_1(\omega r\sin \theta )\mathrm{d}\omega }+{(\lambda r)}^{-n-1}{P}_n^1(\cos \theta ),$$

(21)

in which the integral can be calculated numerically.

The remaining boundary conditions to be fulfilled are those at the particle and hard core surfaces. Substituting Equations (8) and (20) into Equations (4) and (5) yields

$${\displaystyle \sum _{n=1}^{\infty }[C_n{I}_{n+1/2}(\lambda a)+D_n{K}_{n+1/2}(\lambda a)]} (\lambda a{)}^{-1/2}{P}_n^1(\cos \theta )=0,$$

(22)

$${\displaystyle \sum _{n=1}^{\infty }\{[C_n{I}_{n+1/2}(\lambda b)+D_n{K}_{n+1/2}(\lambda b)]{(\lambda b)}^{-1/2}{P}_n^1(\cos \theta )}-A_n{\gamma }_n(b,\theta )\}=-\Omega b\sin \theta ,$$

(23)

$${\displaystyle \sum _{n=1}^{\infty }\{[C_n\{\lambda b{I}_{n-1/2}(\lambda b)+\lambda b{I}_{n+3/2}(\lambda b)-3I_{n+1/2}(\lambda b)\}}-D_n\{\lambda b{K}_{n-1/2}(\lambda b)$$

$$+\lambda b{K}_{n+3/2}(\lambda b)+3K_{n+1/2}(\lambda b)\}]{(\lambda b)}^{-1/2}{P}_n^1(\cos \theta )-2A_n{\gamma }_n^{*}(b,\theta )\}=0,$$

(24)

where

$${\gamma }_n^{*}(r,\theta )=r^2\frac{\partial }{\partial r}[\frac{{\gamma }_n(r,\theta )}{r}]={\lambda }^{-2}r{\displaystyle {\int }_0^{\infty }\{[X_n(\omega ){\mathrm{e}}^{\omega r\cos \theta }-Y_n(\omega ){\mathrm{e}}^{-\omega r\cos \theta }]}{J}_1(\omega r\sin \theta )\cos \theta$$

$$-[X_n(\omega ){\mathrm{e}}^{\omega r\cos \theta }+Y_n(\omega ){\mathrm{e}}^{-\omega r\cos \theta }]J_2(\omega r\sin \theta )\sin \theta \}{\omega }^2\mathrm{d}\omega -(\mathrm{n}+2\left)\right(\lambda r{)}^{-n-1}{P}_n^1(\cos \theta )$$

(25)

The satisfaction of boundary conditions (22)–(24) at the inner and outer surfaces of the porous layer of the soft sphere requires solutions of the constants $A_n$, $C_n$, and $D_n$. The collocation technique [29] permits these boundary conditions to be imposed at $N$ points on the meridian semicircle of each surface and the infinite series in Equations (8) and (20) to be truncated after $N$ terms, leading to $3N$ simultaneous linear algebraic equations. These algebraic equations can be numerically solved for sufficiently large $N$ to result in the $3N$ constants $A_n$, $C_n$, and $D_n$. The details of the boundary collocation scheme were given in an early paper [30] for a hard sphere translating perpendicular to two parallel plane walls.

The hydrodynamic torque acting on the soft sphere is [27]

$$T=8\mathsf{\pi }\eta {\lambda }^{-2}{A}_1,$$

(26)

where $\eta$ is the viscosity of the fluid and only the lowest-order constant $A_1$ makes a contribution.

When the surface layer of the soft sphere vanishes, it degenerates to a hard sphere of radius $b=a$, the constants $C_n=D_n=0$, Equations (5), (8), (22), (24), and (25) are trivial, and just Equation (23) is required to be solved for the $N$ constants $A_n$. When the hard core vanishes, the soft sphere reduces to an entirely porous sphere of radius $b$, the constants $D_n=0$, Equations (4) and (22) become trivial, and only Equations (23) and (24) need to be solved for the $2N$ constants $A_n$ and $C_n$.

3. Results and Discussion

The numerical solutions for the hydrodynamic torque $T$ acting on a soft spherical particle rotating about its diameter perpendicular to two plane walls as a function of the ratio of the particle radius to the porous layer penetration length $\lambda b$, ratio of the core-to-particle radii $a/b$, particle-wall spacing parameter $b/c$, and relative position parameter $c/(c+d)$ obtained from the boundary collocation method, are provided in

Table A1. Normalized torque $T/T_0$ for a porous sphere ($a=0$) rotating about a diameter perpendicular to two parallel planar walls.

$\mathit{c}/(\mathit{c}+\mathit{d})$	$\mathit{b}/\mathit{c}$			$\mathit{T}/{\mathit{T}}_0$
		$\mathit{\lambda }\mathit{b}=1$	$\mathit{\lambda }\mathit{b}=10$	$\mathit{\lambda }\mathit{b}=100$	$\mathit{\lambda }\mathit{b}=600$
0	0.1	1.00001	1.00009	1.00012	1.00012
	0.3	1.00021	1.00247	1.00329	1.00337
	0.5	1.00095	1.01156	1.01544	1.01585
	0.6	1.00165	1.02021	1.02714	1.02786
	0.7	1.00262	1.03270	1.04432	1.04554
	0.8	1.00392	1.05031	1.06937	1.07144
	0.9	1.00559	1.07526	1.10766	1.11143
	0.95	1.00659	1.09183	1.13641	1.14210
	0.99	1.00746	1.10804	1.17065	1.18075
	0.995	1.00758	1.11031	1.17647	1.18807
	0.999	1.00767	1.11218	1.18165	1.19516
1/4	0.1	1.00001	1.00009	1.00012	1.00013
	0.3	1.00021	1.00251	1.00333	1.00342
	0.5	1.00097	1.01173	1.01567	1.01607
	0.6	1.00167	1.02050	1.02753	1.02826
	0.7	1.00266	1.03317	1.04495	1.04619
	0.8	1.00397	1.05103	1.07034	1.07243
	0.9	1.00567	1.07631	1.10907	1.11289
	0.95	1.00668	1.09307	1.13811	1.14386
	0.99	1.00757	1.10946	1.17261	1.18280
	0.995	1.00769	1.11175	1.17849	1.19017
	0.999	1.00778	1.11363	1.18371	1.19734
1/2	0.1	1.00001	1.00016	1.00022	1.00022
	0.3	1.00037	1.00446	1.00594	1.00609
	0.5	1.00172	1.02101	1.02813	1.02886
	0.6	1.00297	1.03690	1.04972	1.05106
	0.7	1.00473	1.06007	1.08180	1.08410
	0.8	1.00708	1.09308	1.12918	1.13312
	0.9	1.01011	1.14040	1.20263	1.20992
	0.95	1.01192	1.17212	1.25847	1.26959
	0.99	1.01352	1.20337	1.32558	1.34554
	0.995	1.01373	1.20775	1.33710	1.36004
	0.999	1.01389	1.21135	1.34733	1.37420

and

Table A2. Normalized torque $T/T_0$ for a soft sphere with $\lambda b=1$ rotating about a diameter perpendicular to two parallel planar walls.

$\mathit{c}/(\mathit{c}+\mathit{d})$	$\mathit{b}/\mathit{c}$		$\mathit{T}/{\mathit{T}}_0$
		$\mathit{a}/\mathit{b}=0.5$	$\mathit{a}/\mathit{b}=0.8$	$\mathit{a}/\mathit{b}=0.95$
0	0.1	1.00002	1.00006	1.00011
	0.3	1.00055	1.00175	1.00290
	0.5	1.00253	1.00816	1.01362
	0.6	1.00438	1.01420	1.02387
	0.7	1.00698	1.02282	1.03880
	0.8	1.01046	1.03469	1.06020
	0.9	1.01498	1.05079	1.09159
	0.95	1.01769	1.06088	1.11354
	0.99	1.02009	1.07019	1.13645
	0.995	1.02041	1.07145	1.13982
	0.999	1.02066	1.07246	1.14263
1/4	0.1	1.00002	1.00007	1.00011
	0.3	1.00055	1.00177	1.00294
	0.5	1.00256	1.00827	1.01381
	0.6	1.00444	1.01440	1.02421
	0.7	1.00708	1.02315	1.03935
	0.8	1.01061	1.03519	1.06104
	0.9	1.01520	1.05151	1.09283
	0.95	1.01794	1.06173	1.11501
	0.99	1.02038	1.07117	1.13814
	0.995	1.02070	1.07244	1.14154
	0.999	1.02096	1.07347	1.14437
1/2	0.1	1.00004	1.00012	1.00019
	0.3	1.00098	1.00316	1.00525
	0.5	1.00457	1.01479	1.02477
	0.6	1.00792	1.02584	1.04365
	0.7	1.01264	1.04172	1.07145
	0.8	1.01899	1.06378	1.11175
	0.9	1.02727	1.09397	1.17167
	0.95	1.03225	1.11304	1.21400
	0.99	1.03667	1.13074	1.25853
	0.995	1.03725	1.13313	1.26511
	0.999	1.03772	1.13507	1.27059

in Appendix A for the distinct case of $a=0$ (a fully porous sphere) and the general case, respectively. The torque $T_0$, given by Equation (1) for the soft sphere in the unbounded fluid is used to normalize $T$. The accuracy and convergence behavior of the collocation technique depends upon the relevant parameters. All the results obtained converge to at least six significant figures. For the most difficult case, the number of collocation points, $N=46$, is sufficiently large to achieve this convergence. These results are the same as those obtained for a hard sphere [23] in the limiting case of $\lambda b\to \infty$ or $a=b$. Obviously, $T/T_0=1$ is the limit $b/c=0$, regardless of other parameters. The wall effects on the rotational motion of the soft sphere can be significant.

The normalized torque $T/T_0$ for a fully porous sphere rotating about its diameter perpendicular to two planar walls is plotted against the parameters $\lambda b$, $b/c$, $c/(c+d)$ in Figure 2, Figure 3 and Figure 4, respectively, over the entire range. For fixed values of $b/c$ and $c/(c+d)$, as expected and shown in Table A1 and Figure 2a,b, Figure 3a and Figure 4b, $T/T_0$ is a monotonically increasing function of the shielding parameter $\lambda b$ (decreasing function of the permeability) for the fluid in the porous particle from unity (with $T=T_0=0$) at $\lambda b=0$ to a larger finite value as $\lambda b\to \infty$. When $\lambda b$ is smaller than unity, the variation of $T/T_0$ with $b/c$ and $c/(c+d)$ is weak (<1.4%). $T/T_0$ of a porous sphere with low permeability (say, $\lambda b>100$) in general is close to that of a hard one (with $\lambda b\to \infty$), though their difference can be noticeable when the particle is very close to a wall ($b/c\to 1$).

For the given values of $\lambda b$ and $c/(c+d)$, as indicated in Table A1 and Figure 2b, Figure 3a,b and Figure 4a, the normalized torque $T/T_0$, acting on the confined porous sphere, is an increasing function of the particle-to-wall spacing parameter $b/c$ from unity at $b/c=0$ to a greater finite value at $b/c=1$ (note that $T/T_0$ is still finite even for the limit that the particle touches the plane walls), since the hydrodynamic hindrance caused by the plane walls is stronger when they locate closer to the particle. The dependence of $T/T_0$ on $b/c$ is robust when $\lambda b$ is large but vanishes in the limit $\lambda b=0$. The supposition that the two-wall effect on the rotation of a particle can be viewed as a sum of single-wall effects will overestimate the hydrodynamic torque exerted on the particle. That is, the increase in $T/T_0$ from unity for the two-equidistant-wall case $c/(c+d)=1/2$ is less than twice that for the corresponding single-wall case $c/(c+d)=0$, which can be seen in Table A1 and Figure 2a, Figure 3b and Figure 4a,b.

For specified values of $\lambda b$ and $b/c$, the normalized torque $T/T_0$ of the porous sphere increases with an increase in the parameter $c/(c+d)$ (denoting the relative position of the porous sphere between the walls) from a finite value at $c/(c+d)=0$ (the case of a single wall) to a greater one at $c/(c+d)=1/2$ (the case of two equally distant walls). Namely, the nearness of a second wall will enhance the torque acting on the particle close to the first wall. The variation of $T/T_0$ with $c/(c+d)$ can be significant when the value of $\lambda b$ is large, though it disappears in the limits $\lambda b=0$ and $b/c=0$. For a given value of $2b/(c+d)$ (the ratio of the particle diameter to the wall-to-wall distance), as revealed by the dashed curves in Figure 4a, the torque is minimum when the particle locates in the middle between the two walls [$c/(c+d)=1/2$] and increases monotonically as the particle approaches either wall.

After understanding the hydrodynamic effect of two parallel plane walls on the axially symmetric rotation of a porous particle, we study the general case of that on a rotating soft particle. The results of the normalized torque $T/T_0$ on a soft sphere rotating about its diameter perpendicular to two planar walls for different values of the core-to-particle radius ratio $a/b$, shielding parameter in the porous layer $\lambda b$, dimensionless spacing parameter $b/c$, and relative position parameter $c/(c+d)$ are presented in Figure 5, Figure 6, Figure 7 and Figure 8 (together with Table A2), respectively, over the entire ranges. Again, $T/T_0$ increases as $b/c$ increases from unity at $b/c=0$ to a finite value at $b/c=1$ and increases as $c/(c+d)$ increases from a finite value at $c/(c+d)=0$ to another at $c/(c+d)=1/2$, keeping the other parameters unchanged. Also, $T/T_0$ is a monotonic increasing function of $\lambda b$ from a constant (equal to zero for the entirely porous limit $a/b=0$) at $\lambda b=0$ (the porous surface layer is completely permeable) to a great one as $\lambda b\to \infty$ (the surface layer is impermeable).

For fixed values of $\lambda b$, $b/c$, and $c/(c+d)$, Table A2 and Figure 5, Figure 6, Figure 7 and Figure 8 show that the normalized torque $T/T_0$ for the confined soft spherical particle undergoing rotation increases monotonically with an increase in the ratio of the core-to-particle radii $a/b$, where the limits $a/b=0$ and $a/b=1$ denote a porous sphere and an impermeable sphere, respectively. That is, if the porous layer is thicker for specified permeability, particle size, and separation from walls, the torque exerted on the particle will be less. All results for the soft spherical particle fall between the upper and lower bounds of $a/b=1$ and $a/b=0$, respectively. For the circumstance where the surface layer has low to mediate permeability (e.g., $\lambda b\ge 10$), as shown in Figure 5a and Figure 8, $T/T_0$ on the particle with $a/b$ smaller than about 0.8 can be well approximated by that for a fully porous particle of the same size, permeability, and distances from walls. In this case, the relative motion of the fluid is barely felt by the hard core of the soft sphere, and its hindrance to the flow is negligible. However, this approximation is not valid for the porous layer with high permeability.

Recently, collocation results were obtained for the normalized hydrodynamic torque $T/T_0$ of a soft sphere of radius $b$ rotating about a diameter on the axis of a circular cylinder of radius $c$ [28]. Similar to the currently considered case of axisymmetric rotation of the particle perpendicular to two equidistant plane walls (i.e., at the center of a slit), $T/T_0$ is a monotonically growing function of the shielding parameter $\lambda b$ (from a value at $\lambda b=0$ to a higher one at $\lambda b\to \infty$), particle-wall spacing parameter $b/c$ (from unity at $b/c=0$ to a greater constant at $b/c=1$), and core-to-particle radius ratio $a/b$, holding other parameters constant. The particle in the circular cylinder bears much more torque than the particle in the slit does. This result manifests that the retardation to the particle rotation caused by the confinement walls is freed in both principal lateral directions of the slit, though only in an axial direction of the cylinder.

4. Concluding Remarks

The slow rotational motion of a soft spherical particle in a viscous fluid about its diameter perpendicular to one or two planar walls is semianalytically studied using the method of boundary collocation. Convergent numerical results for the torque exerted on the particle by the fluid were obtained for various values of the ratio of the particle radius to the flow penetration length of the porous layer $\lambda b$, the ratio of the core-to-particle radii $a/b$, particle-wall spacing parameter $b/c$, and relative position parameter $c/(c+d)$. The wall effect on the rotating soft particle can be significant. The normalized torque, $T/T_0$, acting on the confined particle increases with an increase in $b/c$ from unity as $b/c=0$ (the particle is far from the walls) and remains finite even at the contact limit $b/c=1$. This torque is smaller than that on a hard sphere (or soft one with larger $a/b$ or $\lambda b$), keeping the other parameters’ constant. For a given ratio of the particle diameter to the wall-to-wall distance $2b/(c+d)$, $T/T_0$ is minimal when the particle is midway between the two walls [$c/(c+d)=1/2$] and increases as it locates closer to either wall [the value of $c/(c+d)$ decreases]. Experimental data of the normalized torque for the slow rotation of a soft particle near one or two plane walls would be needed to confirm the validity of our semianalytical collocation results at various ranges of $\lambda b$, $a/b$, $b/c$, and $c/(c+d)$.

Section three provides results for a resistance problem, where the hydrodynamic torque $T$ on a particle rotating normal to two planar walls is considered for a given angular velocity $\Omega$ [equal to $(T_0/8\mathsf{\pi }\eta {\lambda }^{-2}bR)[\lambda a\cosh (\lambda b-\lambda a)+\sinh (\lambda b-\lambda a)]$ according to Equation (1)]. In a mobility problem, the torque, $T$ (equal to $8\mathsf{\pi }\eta {\lambda }^{-2}b{\Omega }_{0}R/[\lambda a\cosh (\lambda b-\lambda a)+\sinh (\lambda b-\lambda a)]$), imposed to the particle is assumed and the boundary-corrected angular velocity, $\Omega$, is considered. For a soft sphere rotating normal to two plane walls dealt with here, the normalized angular velocity $\Omega /{\Omega }_0$ for the mobility problem is equal to ${(T/T_0)}^{-1}$, as given in Table A1 and Table A2 and Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8 for the resistance problem.