1. Introduction
Particle dynamics in low-Reynolds-number fluids under confinement is fundamental to various applications, including particle sorting and drug delivery (Hamilton et al. Reference Hamilton, Gilbert, Petrov and Ogrin2018; Kabacaoğlu & Biros Reference Kabacaoğlu and Biros2019; Shi & Schwartz Reference Shi and Schwartz2024). In confined geometries, particle–wall hydrodynamic interaction plays a key role in the structure and transport properties of particles. For example, hydrodynamic interaction of treadmilling swimmers or microrollers with confining walls can affect translation and rotation of the particles, and also lead to hydrodynamic bound states similar to the dancing Volvox states (Drescher et al. Reference Drescher, Leptos, Tuval, Ishikawa, Pedley and Goldstein2009; Crowdy Reference Crowdy2011; Crowdy & Samson Reference Crowdy and Samson2011; Davis & Crowdy Reference Davis and Crowdy2012; Martinez-Pedrero et al. Reference Martinez-Pedrero, Navarro-Argemí, Ortiz-Ambriz, Pagonabarraga and Tierno2018; Delmotte Reference Delmotte2019; Guo, Man & Zhu Reference Guo, Man and Zhu2024). It was also observed that persistent motile structures can form spontaneously from hydrodynamic interactions in a system of colloidal rollers suspended and translating above a planar wall (Driscoll et al. Reference Driscoll, Delmotte, Youssef, Sacanna, Donev and Chaikin2017). Microrotors in a confined circular domain were found to induce chaotic advection and mixing, which can act as controllable mobile micromixers to produce fast mixing of the fluid in microfluidic devices (Tallapragada & Sudarsanam Reference Tallapragada and Sudarsanam2019).
In recent times, particle motion under total confinement has drawn considerable attention due to its pivotal role in intracellular transport and microfluidic encapsulation technology (Aponte-Rivera & Zia Reference Aponte-Rivera and Zia2016; Aponte-Rivera, Su & Zia Reference Aponte-Rivera, Su and Zia2018; Maheshwari et al. Reference Maheshwari, Sunol, Gonzalez, Endy and Zia2019; Gonzalez, Aponte-Rivera & Zia Reference Gonzalez, Aponte-Rivera and Zia2021; Sunol & Zia Reference Sunol and Zia2023). Intracellular processes such as transcription, signal transmission and metabolism are closely related to restricted motion of biological particles in the cytoplasm, which is confined by the cell membrane (Xiang et al. Reference Xiang, Chen, Yan, Li and Xu2020). In microfluidic encapsulation technology or droplet-based microfluidics, cells and particles are encapsulated in droplets, which are then used for a wide variety of applications such as single cell/molecule analysis and nanoparticle synthesis (Nan et al. Reference Nan, Zhang, Weitz and Shum2024; Tang et al. Reference Tang, Zhao, Shen, Yang and Lim2024). Particulate transport within the droplets is therefore crucial for suspension stability, reaction rates, and ultimately the performance of droplet-based microfluidic devices (Vollenbroek et al. Reference Vollenbroek, Nieuwelink, Bomer, Tiggelaar, van den Berg, Weckhuysen and Odijk2023).
To better understand and manipulate particle motion, characteristics and fundamental mechanisms of particulate transport in low-Reynolds-number fluids under total confinement have been studied using experiments, theories and numerical simulations (Sun et al. Reference Sun, Chen, de Pablo and Jiang2024). Many experimental studies have focused on dynamics of biological particles in cells. By using advanced experimental methods, including magnetic resonance velocimetry, single-particle tracking, and fluorescence recovery after photobleaching, intriguing phenomena such as enhanced transport by cytoplasmic streaming, and hindered and heterogeneous diffusion due to confinement and crowding, were observed and analysed (Konopka et al. Reference Konopka, Shkel, Cayley, Record and Weisshaar2006; van de Meent et al. Reference van de Meent, Gladden and Goldstein2010; Prindle et al. Reference Prindle, de Cuba and Gahlmann2023; Luo et al. Reference Luo, Jiang, Dou, Wang and Li2024). For non-biological systems, experiments have focused on particle dynamics inside droplets, in which diffusion of Brownian particle is measured and also used to probe interaction between particles and mechanical properties of lipid vesicles (Cervantes-Martínez et al. Reference Cervantes-Martínez, Ramírez-Saito, Armenta-Calderón, Ojeda-López and Arauz-Lara2011; Goto et al. Reference Goto, Sakata, Moritani and Inui2017; Sandoval-Jiménez et al. Reference Sandoval-Jiménez, Jacinto-Méndez, Toscano-Flores and Carbajal-Tinoco2020).
In theoretical studies, the totally confined system is usually modelled as a spherical cavity containing particles and the incompressible Newtonian fluid. Under the low-Reynolds-number assumption, viscous forces are much more important than inertial forces, such that the Navier–Stokes equation is reduced to the Stokes equation. Analytical solutions to the Stokes problems could then be obtained for simple geometries. Nearly a century ago, the Green’s function describing motion of a spherical particle in the spherical cavity was derived (Oseen Reference Oseen1927). Later, translational and rotational motion of the spherical particle was analysed by combining theoretical and numerical methods (O’Neill & Majumdar Reference O’Neill and Majumdar1970a , Reference O’Neill and Majumdarb ). The Green’s function obtained by Oseen has also been used to calculate the mobility matrix, diffusion tensor and escape rate of a spherical particle in the cavity (Felderhof & Sellier Reference Felderhof and Sellier2012). More recently, a theoretical framework was developed to describe the motion of spherical particles with an arbitrary number in the cavity (Aponte-Rivera & Zia Reference Aponte-Rivera and Zia2016), and the framework has been used for studying equilibrium structure and diffusion of the particles (Aponte-Rivera et al. Reference Aponte-Rivera, Su and Zia2018; Maheshwari et al. Reference Maheshwari, Sunol, Gonzalez, Endy and Zia2019; Gonzalez et al. Reference Gonzalez, Aponte-Rivera and Zia2021; Sunol & Zia Reference Sunol and Zia2023). Effects of surface slip and wall elasticity on particle motion in the cavity were also analysed in detail using different theoretical approaches (Keh & Lee Reference Keh and Lee2010; Faltas & Saad Reference Faltas and Saad2011; Lee & Keh Reference Lee and Keh2013; Daddi-Moussa-Ider, Löwen & Gekle Reference Daddi-Moussa-Ider, Löwen and Gekle2018; Hoell et al. Reference Hoell, Löwen, Menzel and Daddi-Moussa-Ider2019).
Numerical simulations were also employed for studying particle dynamics under total confinements. By using the boundary element method, gravity-driven motions of spherical and ellipsoidal particles in the spherical cavity were studied, and it was found that particle settling migration deeply depends on the gravity and both its shape and location in the cavity (Sellier Reference Sellier2008). Dynamics of concentrated intracellular macromolecules were examined by Brownian dynamics simulations, revealing that confinement and hydrodynamic interaction play important roles on the hindered diffusion (Ando, Chow & Skolnick Reference Ando, Chow and Skolnick2013; Chow & Skolnick Reference Chow and Skolnick2015; Skolnick Reference Skolnick2016). A two-phase flow model was used to study localisation of P granules in a non-spherical cell, and the study revealed that the cytoplasmic streaming has significant effects on the localisation of P granules as well as the embryo division (Wang & Hu Reference Wang and Hu2017). More recently, Brownian diffusion and motion under external forces of spherical and non-spherical particles in cavities were investigated using a hybrid finite element and Green’s function approach (Li et al. Reference Li, Jiang, Singh, Heinonen, Hernández-Ortiz and de Pablo2020; Singh et al. Reference Singh, Li, Jiang, Hernández-Ortiz, Jaeger and de Pablo2020; Chen & Jiang Reference Chen and Jiang2022, Reference Chen and Jiang2023).
Most prior studies considered the confining geometry as a spherical cavity. This treatment corresponds to the scenario that cells and droplets exhibit spherical shape, and all particles are enclosed by a single cavity wall. In this study, we consider the confined space to be between two concentric spherical walls. For the application of particle movement within a cell, the outer (inner) wall represents the cell membrane (nucleus) (Cohen-Fix Reference Cohen-Fix2021). From a practical point of view, this model is more related to synthetic cells. Recently, synthetic cells with nucleus-like compartments were created, and particle dynamics between concentric spherical walls is key to intercompartment communication (Ioannou et al. Reference Ioannou, Monck, Ceroni, Brooks, Kuimova and Elani2024). While major numerical and theoretical models treat the cell as a cavity confined by a single wall filled with a Newtonian fluid at low Reynolds number (Aponte-Rivera & Zia Reference Aponte-Rivera and Zia2016; Aponte-Rivera et al. Reference Aponte-Rivera, Su and Zia2018; Daddi-Moussa-Ider et al. Reference Daddi-Moussa-Ider, Löwen and Gekle2018; Stein et al. Reference Stein, de Caino, Lauga, Shelley and Goldstein2021), this study is the first to consider the effect of an inner wall on particle dynamics. For most real biological cells, the structure (e.g. shapes of cell membrane and nucleus, location of nucleus, elastic/non-fluid-like elements), fluid properties, and boundary conditions can be more complex; however, fundamental physics obtained in this model system lays the initial step towards understanding intracellular particle dynamics, and could still hold in more complex situations. In this study, we consider that an external force is exerted on the particle; this force in a cell situation could be self-propelling forces on molecular motors or magnetic/electric forces on nanoparticles.
Apart from particle dynamics in cells, the studied system is also related to double emulsion system, and the two spherical walls correspond to interfaces between the middle phase and the inner/outer phase (Liu et al. Reference Liu, Wu, Ju, Xie, Wang, Niu and Chu2013). In one application, nanoparticles were dispersed in the middle phase, and the double emulsions provide confinement for directing self-assembly of magnetic nanoparticles, leading to formation of hierarchically structured magnetic single-hole hollow spheres (Wang et al. Reference Wang, Hou, Yao, Gao and Ge2016). During the self-assembly process, particle dynamics confined between two concentric spherical walls determines the structure and function of the hollow spheres. In this application, the external force on the particle could be gravitational force or magnetic forces.
Another application to which the studied system could be related is droplet-based microfluidics. In a recent study (Magrinya et al. Reference Magrinya, Palacios-Alonso, Llombart, Delgado-Buscalioni, Alexander-Katz, Arriaga and Aragones2025), a spherical magnetic particle is placed within a spherical droplet, and by using a rotating magnetic field, the particle can rotate at a designated angular velocity, which corresponds to rotation of the inner wall here. The rotating particle can drive the rotation of the droplet, which corresponds to rotation of the outer wall here. The angular velocity of the droplet depends on factors such as surrounding flow conditions and angular velocity of the particle. The macromolecule within the droplet correspond to the confined particle, and the external force on the particle is the gravitational force.
Here, we focus on a computational study on dynamics of a spherical particle under external forces in the low-Reynolds-number fluid confined between two concentric spherical walls. Two cases are considered. In the first case, the walls are stationary and the particle moves under externally applied forces. We will calculate hydrodynamic mobilities of the particle at various radial positions in the confined space; drift motion of the particle perpendicular to the external force will be studied; fluid vortices induced by particle motion will also be analysed. In the second case, either one wall or both walls rotate at a constant angular velocity, and the reasons for considering rotating walls are as follows. First, in microfluidic applications, the cell or droplet could rotate during its dynamics in the surrounding fluid, and the rotational motion could be modelled by rotating the outer wall considered in this work. Second, if we consider a rotary molecular motor or microroller (Davis & Crowdy Reference Davis and Crowdy2012; Delmotte Reference Delmotte2019; Grelier, Sivak & Ehrich Reference Grelier, Sivak and Ehrich2024; Sheng et al. Reference Sheng2024) rotating in the centre of the cell/droplet, then its rotational motion could be treated by rotation of the inner wall. We will then analyse trajectories of a non-neutrally buoyant particle in the flow caused by wall rotations. In this study, particle dynamics and fluid flow are solved numerically, and the numerical methods are described in the next section.
2. Methods
Figure 1 shows a schematic of the simulation system. A rigid spherical particle in a low-Reynolds-number fluid between two concentric spherical walls is considered. The radius of the inner (outer) wall is
$R_{\textit{i}}$
(
$R_o$
), and the particle radius is
$R_{\textit{p}}$
. In this work,
$R_o$
is set to 15, while
$R_{\textit{i}}$
and
$R_{\textit{p}}$
are varied. Values of
$R_{\textit{i}}$
(
$R_{\textit{p}}$
) are 2, 5 and 8 (0.75, 1.5, 2.25 and 3). For fluid dynamics, we assume that the Reynolds number (
$Re=\boldsymbol{\omega } R_o^2/\nu$
), Stokes number (
$Sk=\boldsymbol{\omega } R_{\textit{p}}^2/\nu$
) and frequency number (
$N_{\textit{Fr}}=R_o^2/\nu t^*$
) are much less than 1, where
$\boldsymbol{\omega }$
is the angular velocity of the rotating walls,
$\nu$
is the kinematic viscosity of the fluid, and
$t^*$
is the characteristic time imposed by external forcing (Pozrikidis Reference Pozrikidis1992). Based on the above assumptions, viscous forces are much more important than inertial acceleration and convective forces, so the full Navier–Stokes equations are simplified to the quasi-steady Stokes equation (Lee & Ladd Reference Lee and Ladd2007). We also neglect effects from the inertia of the fluid, such as convective inertia and the added mass effect (Lavrenteva et al. Reference Lavrenteva, Prakash and Nir2016).
For particle dynamics, ignoring Brownian motion, the balance equation for force and torque on the particles is
\begin{equation} m_{\textit{p}} \frac {{\textrm{d}} \boldsymbol{u}_{\textit{p}}}{{\textrm{d}} t} + 2m_{\textit{p}} \boldsymbol{\omega } \times \boldsymbol{u}_{\textit{p}} = \boldsymbol{F}^H + \boldsymbol{F}^{W} + \boldsymbol{F}^{C} + \boldsymbol{F}^{\textit{ext}}, \end{equation}
where
$m_{\textit{p}}$
is particle mass,
$\boldsymbol{u}_{\textit{p}}$
is particle velocity,
$t$
is time,
$\boldsymbol{F}^{H}$
is the hydrodynamic force/torque vector,
$\boldsymbol{F}^{W}$
is the force/torque vector from particle–wall interaction,
$\boldsymbol{F}^{C}$
is the centrifugal or centripetal force/torque that only exists for a non-neutrally buoyant particle when walls rotate, and
$\boldsymbol{F}^{\textit{ext}}$
are external forces/torques. Here,
$\boldsymbol{F}^{H}$
generally includes viscous forces, fluid acceleration
$m_{\textit{f}}\, {\textrm{d}} \boldsymbol{u}_{\textit{f}} / {\textrm{d}} t$
, and fluid Coriolis force
$2m_{\textit{f}}\kern1pt \boldsymbol{\omega } \times \boldsymbol{u}_{\textit{f}}$
, where
$m_{\textit{f}}=4\unicode{x03C0} \rho _{\textit{f}} R_{\textit{p}}^3 /3$
, and
$\rho _{\textit{f}}$
is fluid density (Maxey & Riley Reference Maxey and Riley1983; Michaelides Reference Michaelides1997; Rallabandi Reference Rallabandi2021; Tsai Reference Tsai2022). As quasi-steady flow and
$Re \ll 1$
are assumed, inertial terms in
$\boldsymbol{F}^{H}$
are neglected. On the left-hand side of (2.1), inertial terms are of order
$m_{\textit{p}} \boldsymbol{\omega } u_0$
, where
$u_0$
is particle velocity under
$F^{\textit{ext}}$
in unbounded fluid, and
$\rho _{\textit{p}}$
is particle density. This term can be further written as
$Sk (\rho _{\textit{p}}/\rho _{\textit{f}}) F^{\textit{ext}}$
. If
$Sk \ll 1$
and
$\rho _{\textit{p}}$
is not much larger than
$\rho _{\textit{f}}$
, then inertial terms on the left-hand side could be ignored. Under the above conditions, we consider
$F^{\textit{ext}}$
and
$F^C$
to be of the same order
$(\rho _{\textit{p}}-\rho _{\textit{f}} ) R_{\textit{p}}^3 \boldsymbol{\omega } ^2 R_c \sim F^{\textit{ext}}$
, and (2.1) is reduced to
\begin{equation} \boldsymbol{F}^H + \boldsymbol{F}^{W} + \boldsymbol{F}^{C} + \boldsymbol{F}^{\textit{ext}} = \boldsymbol{0}. \end{equation}
Schematic of the simulation system. The radius of the inner (outer) wall is
$R_{\textit{i}}$
(
$R_o$
), and the particle radius is
$R_{\textit{p}}$
. In this study,
$R_o=15$
; values of
$R_{\textit{i}}$
are 2, 5 and 8; values of
$R_{\textit{p}}$
are 0.75, 1.5, 2.25 and 3.0.
To model the rigid particle, we discretise the particle surface into a set of
$N$
nodes, and each node is connected to its neighbouring nodes by stiff springs. To avoid particle deformation, surface nodes are also connected to the particle centre by the springs. After discretisation, (2.2) is translated into the
$N$
surface nodes as
\begin{equation} \boldsymbol{f}_{\textit{i}}^H + \boldsymbol{f}_{\textit{i}}^S + \boldsymbol{f}_{\textit{i}}^W + \boldsymbol{f}_{\textit{i}}^{C} + \boldsymbol{f}_{\textit{i}}^{\textit{ext}} = \boldsymbol{0}, \end{equation}
where
$i=1, 2,\ldots,N$
,
$\boldsymbol{f}_{\textit{i}}^H$
is hydrodynamic force,
$\boldsymbol{f}_{\textit{i}}^{\textit{S}}$
is spring force,
$\boldsymbol{f}_{\textit{i}}^W$
is the force from particle–wall interaction,
$\boldsymbol{f}_{\textit{i}}^C$
is centrifugal or centripetal force, and
$\boldsymbol{f}_{\textit{i}}^{\textit{ext}}$
are external forces. The hydrodynamic force
$\boldsymbol{f}_{\textit{i}}^H$
is computed implicitly when we integrate the equation of motion for all nodes. The spring force on the
$i$
th node by the
$j$
th node is
\begin{equation} \boldsymbol{f}^S_{\textit{ij}} = k \big(|\boldsymbol{r}_{\textit{ij}}| - r_{0} \big) \frac {\boldsymbol{r}_{\textit{ij}}}{|\boldsymbol{r}_{\textit{ij}}|}, \end{equation}
where
$k$
is spring stiffness,
$r_0$
is equilibrium spring length,
$\boldsymbol{r}_{\textit{ij}}=\boldsymbol{r}_{\textit{i}} - \boldsymbol{r}_{\textit{j}}$
, and
$\boldsymbol{r}_{\textit{i}}$
and
$\boldsymbol{r}_{\textit{j}}$
are coordinates of the
$i$
th and
$j$
th nodes. Different
$k$
(250, 300 and 360) are tested, and results show that forces on the rigid particle are insensitive to these spring stiffnesses. In production simulations, we set
$k=250$
. In rotating wall simulations, centrifugal or centripetal force on the
$i$
th node of the particle is
\begin{equation} \boldsymbol{f}_{\textit{i}}^C = \frac {\rho _{b} V}{N}\boldsymbol{\omega } \times (\boldsymbol{\omega } \times \boldsymbol{r}_{\textit{i}}), \end{equation}
where
$\rho _b=\rho _{\textit{p}}-\rho _{\textit{f}}$
is buoyancy-corrected density, and
$V$
is particle volume. Here,
$\boldsymbol{f}_{\textit{i}}^C$
is centrifugal (centripetal) when
$\rho _{\textit{p}} \gt \rho _{\textit{f}}$
(
$\rho _{\textit{p}} \lt \rho _{\textit{f}}$
). The external force is
$\boldsymbol{f}_{\textit{i}}^{\textit{ext}}=\boldsymbol{F}^{\textit{ext}}/N$
. The particle–wall interaction is included only in rotating wall simulations, for enabling the entropic barrier near the wall that results in a region very close to the wall inaccessible to the particle (Sunol & Zia Reference Sunol and Zia2023), and preventing the particle from going out of the simulation domain. Also,
$\boldsymbol{f}_{\textit{i}}^W$
is the negative gradient of Lennard-Jones (LJ) potential
\begin{equation} U_{LJ}(d) = 4 \epsilon \left [ \left ( \frac {\sigma }{d}\right )^{12}-\left (\frac {\sigma }{d} \right )^6 \right ] \end{equation}
for
$d \leqslant 2^{1/6}\sigma$
, and
$\boldsymbol{f}_{\textit{i}}^W$
is zero for
$d \gt 2^{1/6}\sigma$
. Here,
$d$
is the radial distance between the surface node and the wall,
$\sigma = 0.89\bar {a}$
is set empirically such that each node has an excluded volume of radius
$\bar {a}$
, and
$\epsilon =k_BT$
, where
$k_B$
is the Boltzmann constant, and
$T$
is temperature (Li et al. Reference Li, Jiang, Singh, Heinonen, Hernández-Ortiz and de Pablo2020). Here, the hydrodynamic drainage force related to the thin film between the particle and the wall is not resolved with high accuracy. This force depends on factors such as thin film width, wettability and roughness of surfaces, and relative velocity between surfaces, to name a few; it usually occurs when the distance between two surfaces is in the range of a few nanometres to hundreds of nanometres, and the magnitude of the force is usually of the order of nanonewtons (Bonaccurso, Kappl & Butt Reference Bonaccurso, Kappl and Butt2002; Guriyanova et al. Reference Guriyanova, Semin, Rodrigues, Butt and Bonaccurso2009; Dagastine et al. Reference Dagastine, Webber, Manica, Stevens, Grieser and Chan2010). If the external force acting on the particle is in the nanonewton scale, then the hydrodynamic drainage force will play a major role in affecting particle trajectory near the wall: the duration for particle motion along the wall and the timings for the particle to attach to and detach from the wall will change. If the external force is on a scale that is much larger than nanonewtons, then the hydrodynamic drainage force will have a minor effect on particle trajectories. Nonetheless, in both scenarios, we expect that particle trajectories when hydrodynamic drainage force is accurately resolved will be qualitatively the same as those when only LJ potential is applied.
The equation of motion for all nodes on particle surface is
\begin{equation} {\textrm{d}}\boldsymbol{R} = \left [ \boldsymbol{U}_0 + \unicode{x1D648} \boldsymbol{\cdot }\boldsymbol{F} \right ] {\textrm{d}}t, \end{equation}
where
$\boldsymbol{R}= (\boldsymbol{r}_{1}, \boldsymbol{r}_{2},\ldots,\boldsymbol{r}_{\textit{N}})$
is a
$3N$
vector for nodal coordinates,
$\boldsymbol{U}_0$
is a
$3N$
vector with undisturbed (ambient) fluid velocity at nodal positions, and
$\unicode{x1D648}$
is the
$3N \times 3N$
mobility tensor. Here,
$\boldsymbol{U}= (\boldsymbol{u}_1,\boldsymbol{u}_2, ..,\boldsymbol{u}_{\textit{N}} )=\unicode{x1D648} \boldsymbol{\cdot }\boldsymbol{F}$
contains
$3N$
disturbed velocities, and
$\boldsymbol{F}= (\boldsymbol{f}_1,\boldsymbol{f}_2,\ldots,\boldsymbol{f}_{\textit{N}} )$
is a
$3N$
vector for non-hydrodynamic forces. The disturbed and ambient fields are solved separately, and the ambient field corresponds to the flow when the particle is absent. By integrating (2.7) for all surface nodes, we can model translational and rotational motion of the particle. The velocity field
$\boldsymbol{U}$
caused by nodal forces
$\boldsymbol{F}$
can be computed by solving the Stokes equation
\begin{equation} - \boldsymbol{\nabla }p+\mu\, {\nabla}^{2} \boldsymbol{u}=-\boldsymbol{f},\quad \boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{u}=0, \end{equation}
where
$p$
is fluid pressure,
$\mu$
is dynamic viscosity of the fluid,
$\boldsymbol{u}$
is fluid velocity, and
$\boldsymbol{f}(\boldsymbol{r})=\sum _{i=1}^{N} \boldsymbol{f}_{\textit{i}}(\boldsymbol{r}_{\textit{i}})\, \delta _s(\boldsymbol{r}-\boldsymbol{r}_{\textit{i}})$
is the force density acting on the fluid. To avoid singularity due to nodal point forces, a smoothing function
$\delta _{s}(\boldsymbol{r})$
is used to regularise point forces:
\begin{eqnarray} \delta _s(\boldsymbol{r}) = \frac {\xi _s^3}{\unicode{x03C0} ^{3/2}}\,{\textrm e}^{-\xi _s^2\, |\boldsymbol{r}|^2} \left [ \frac {5}{2} - \xi _s^2\, |\boldsymbol{r}|^2 \right ]\!. \end{eqnarray}
The regularisation parameter
$\xi _s$
is relevant to the characteristic nodal spacing on particle surface
$h$
, i.e.
$\xi _s\sim h^{-1}$
. This is to ensure that the regularised force density is spread over the length scale of the particle’s surface elements, preventing the fluid from penetrating the particle surface (Li et al. Reference Li, Jiang, Singh, Heinonen, Hernández-Ortiz and de Pablo2020). A no-slip boundary condition is applied on the wall: when the two walls are stationary,
$\boldsymbol{u}(\boldsymbol{r}_w)=\boldsymbol{0}$
, where
$\boldsymbol{r}_w$
is the position at the wall; when the two walls rotate,
$\boldsymbol{u}(\boldsymbol{r}_w)=\boldsymbol{\omega } \times \boldsymbol{r}_w$
. Reasons for imposing no-slip boundary conditions on inner and outer walls are as follows. When the inner wall represents a nucleus membrane in cells or a solid particle/microswimmer’s surface, it is considered as a solid–liquid interface where a no-slip condition usually applies (Swan & Brady Reference Swan and Brady2010; Aponte-Rivera & Zia Reference Aponte-Rivera and Zia2016); when the inner wall is considered as a liquid–liquid interface in double emulsion systems, a no-slip boundary condition is valid when the surfactant concentration is in intermediate values (Hu, Zhang & Wang Reference Hu, Zhang and Wang2010). For the outer wall, a no-slip condition is chosen such that the wall represents the cell membrane (Aponte-Rivera & Zia Reference Aponte-Rivera and Zia2016; Gibson et al. Reference Gibson, Zhang, Stilgoe, Nieminen and Rubinsztein-Dunlop2019) or the liquid–liquid interface in double emulsion systems when surfactant concentration is in intermediate values (Hu et al. Reference Hu, Zhang and Wang2010).
The above set of equations is solved using a general geometry Ewald-like method (Li et al. Reference Li, Jiang, Singh, Heinonen, Hernández-Ortiz and de Pablo2020; Singh et al. Reference Singh, Li, Jiang, Hernández-Ortiz, Jaeger and de Pablo2020), and its details can be found in the references cited. The method has been validated for sedimentation of a spherical particle between two parallel walls and particle dynamics in a spherical cavity (Li et al. Reference Li, Jiang, Singh, Heinonen, Hernández-Ortiz and de Pablo2020; Chen & Jiang Reference Chen and Jiang2023). In this work, we will use the method to study dynamics of a spherical particle between two concentric spherical walls. Dimensionless variables are used, which are based on a set of characteristic scales. The characteristic length scale is
$\bar {a}$
, the energy scale is
$k_BT$
, the force scale is
$k_BT/\bar {a}$
, and the time scale is
$\bar {a}^2\zeta /(k_BT)$
, where
$\zeta =6\unicode{x03C0} \mu \bar {a}$
is the friction coefficient in Stokes’ law.
For hydrodynamic mobilities of the particle, the finite element method is also used to compute spatial distributions of the mobilities, which are compared to those from the general geometry Ewald-like method. In the finite element computations, the two walls and particle surface are modelled as no-slip boundaries that enclose the fluid domain. A uniform boundary stress is applied on the particle surface to drive particle motion. We place the particle centre at a series of radial positions in the confined space to obtain radial distributions of the mobilities. At each radial position, the Stokes equation is solved by a steady-state solver using COMSOL Multiphysics 6.0. Non-uniform tetrahedral mesh is used to discretise the computational domain, and the mesh is finer near all boundaries. In test simulations, meshes are further refined to ensure that mesh sizes used in production simulations can balance numerical accuracy and computational cost. In production simulations, minimal mesh sizes are at the particle surface: they are 0.037, 0.074, 0.11 and 0.15 for
$R_{\textit{p}}/R_o$
values 0.05, 0.1, 0.15 and 0.2, respectively; the maximal mesh size in the domain is 3.9. During post-processing, particle velocities are extracted from numerical solutions, and are then used to calculate radial and transverse mobilities of the particle.
3. Results and discussions
3.1. Mobility
Figure 2 shows radial and transverse mobilities normalised by that in an unbounded fluid (
$M_r$
and
$M_t$
) as functions of scaled radial position of particle centre (
$\textit{r}/\textit{R}_o$
) at different confinement levels. In this study,
$R_o$
is fixed, and the confinement level is varied by changing either
$R_{\textit{i}}$
or
$R_{\textit{p}}$
. We consider
$R_{\textit{i}}/R_o$
(
$R_{\textit{p}}/R_o$
) to be 0.13, 0.33 and 0.53 (0.05, 0.1, 0.15 and 0.2). The results show that at a certain position accessible for the particle, the mobilities decrease as confinement level increases. This is because as the system becomes more confined, the space occupied by the fluid shrinks, and the no-slip wall affects particle motion by hydrodynamic interaction to a greater extent. At a particular position,
$M_r \lt M_t$
because the average particle–wall surface-to-surface distance along the radial direction is smaller than that along the transverse direction, and the effect of a no-slip wall on the particle’s radial motion is larger. The above observations are similar to those for particle motion in a spherical cavity (Aponte-Rivera & Zia Reference Aponte-Rivera and Zia2016; Chen & Jiang Reference Chen and Jiang2023). In most cases, radial and transverse mobilities computed by the general geometry Ewald-like method agree well with those computed by the finite element method. The difference between results from the two methods is more noticeable for the radial mobility at a larger
$R_{\textit{p}}/R_o$
, and the difference generally becomes larger as particle–wall distance decreases.
Normalised (a,c,e) radial and (b,d,f) transverse mobilities of the spherical particle as functions of scaled radial position of particle centre, for (a,b)
$R_{\textit{i}} / R_o=0.13$
, (c,d)
$R_{\textit{i}} / R_o=0.33$
, (e,f)
$R_{\textit{i}} / R_o=0.53$
. For each
$R_{\textit{i}}/R_o$
, the
$R_{\textit{p}} / R_o$
values are 0.05, 0.1, 0.15 and 0.2. Solid lines denote results by the general geometry Ewald-like method; dots denote results by the finite element method. Vertical dashed lines denote particle centres when the particle surface contacts the wall.
Figure 2 also shows that at a certain
$R_{\textit{p}}/R_o$
and
$R_{\textit{i}}/R_o$
, distributions of the mobilities are asymmetric about the radial centre of the confined space, and the position for the maximal mobility is closer to the inner wall. These are different from results for particle motion between two parallel walls, in which mobilities are symmetric about the centre plane, and the maximal mobility occurs in the centre (Swan & Brady Reference Swan and Brady2010); the results also differ from those in a spherical cavity where the maximal mobility is in the cavity centre (Aponte-Rivera & Zia Reference Aponte-Rivera and Zia2016; Chen & Jiang Reference Chen and Jiang2023). Observations here are due to the curvature effect of the spherical walls. For the particle between two walls, the inner wall is in a convex configuration, while the outer wall is in a concave configuration. When radial distances between the particle centre and walls are equal, the average surface-to-surface distance between the particle and the outer wall is smaller, and the hindrance on mobilities caused by the outer wall is stronger. In each case, between the inner (outer) wall and the position for the maximal mobility, the mobilities decrease as the particle moves closer to the inner (outer) wall.
To quantify the effect of curvature and justify the use of the planar wall limit when appropriate, we have compared normalised transverse mobility (
$M_t$
) in the middle between two spherical walls, with
$M_t$
in the middle between two parallel walls (Swan & Brady Reference Swan and Brady2010). Here,
$H$
is defined to be the radial distance between the inner and outer walls, which is also the distance between two parallel walls. When
$\textit{H}/\textit{R}_{\textit{p}}=5$
, the mobility is 99 % of that in the planar boundary limit; when
$\textit{H}/\textit{R}_{\textit{p}}=4$
, the mobility is 96 % of that in the planar boundary limit; when
$\textit{H}/\textit{R}_{\textit{p}}=3$
, the mobility is 91 % of that in the planar boundary limit. These results suggest that as the radial distance between two spherical walls increases or particle radius decreases, mobility between two spherical walls approaches that in the planar wall limit.
We further analyse mobilities at
$\textit{r}/\textit{R}_o=0.75$
, and the reason for choosing this position is that it is accessible for the particle under different
$R_{\textit{i}}/R_o$
. As
$R_{\textit{i}}/R_o$
increases from 0.13 to 0.53,
$M_r$
(
$M_t$
) at
$\textit{r}/\textit{R}_o=0.75$
decreases by 7 %, 15 %, 27 % and 34 % (5 %, 11 %, 18 % and 25 %) when
$R_{\textit{p}}/R_o$
is 0.05, 0.1, 0.15 and 0.2, respectively. As
$R_{\textit{p}}/R_o$
increases from 0.05 to 0.2,
$M_r$
(
$M_t$
) at
$\textit{r}/\textit{R}_o=0.75$
decreases by 70 %, 72 % and 79 % (50 %, 52 % and 61 %) when
$R_{\textit{i}}/R_o$
is 0.13, 0.33 and 0.53, respectively. These results indicate that when the size ratios are varied, changes in mobilities at the same position are more pronounced under a higher confinement level, and the change in
$M_r$
is more than that in
$M_t$
. It is also found that at the same position, varying
$R_{\textit{p}}/R_o$
leads to more changes in mobilities than varying
$R_{\textit{i}}/R_o$
does.
As mentioned in prior works (Li et al. Reference Li, Jiang, Singh, Heinonen, Hernández-Ortiz and de Pablo2020; Chen & Jiang Reference Chen and Jiang2023), the no-slip boundary condition in the general geometry Ewald-like method is satisfied for nodal points at least
$\sim \xi _{s}^{-1}=1.67$
away from the wall. When
$R_{\textit{i}}/R_o=0.13$
, expected ranges of radial positions for results to be accurate are [0.29, 0.84], [0.34, 0.79], [0.39, 0.74] and [0.44, 0.69] for
$R_{\textit{p}}/R_o$
values 0.05, 0.1, 0.15 and 0.2, respectively. When
$R_{\textit{i}}/R_o=0.33$
, the ranges are [0.49, 0.84], [0.54, 0.79], [0.59, 0.74] and [0.64, 0.69]. When
$R_{\textit{i}}/R_o=0.53$
, the ranges are [0.69, 0.84] and [0.74, 0.79] for
$R_{\textit{p}}/R_o$
values 0.05 and 0.1; however, when
$R_{\textit{p}}/R_o$
is increased to 0.15 and 0.2, it is difficult for the method to produce very accurate results in the confined space. The reasons for showing results outside the range for which they are accurate, and the value of the results, are as follows. Despite the fact that the results are accurate only in particular regions of the full flow domain, they provide the first step towards understanding particle mobility between two concentric spherical walls, and they can still capture essential features of the mobility qualitatively: mobility decreases as confinement level increases, and at the same particle–wall distance, the mobility near the inner wall is larger than that near the outer wall due to the curvature effect. Furthermore, although inaccurate, these data set the upper limit for mobilities very near the walls (numerical results are larger than analytical solutions very near the wall; Chen & Jiang Reference Chen and Jiang2023). To our knowledge, currently there are no known exact solutions for mobilities of a finite-size spherical particle between two concentric spherical walls. As a consequence, it is not known to what quantitative extent the errors would be compared to exact solutions. To help further assess the errors, we applied a readily available finite element method to the same system, with meshes heavily refined near the particle, and it was found that differences between results by our method and those by the finite element method are relatively small.
In the remainder of this study, the general geometry Ewald-like method is used because it enables efficient computations of particle dynamics in confined low-Reynolds-number fluids (Hernández-Ortiz et al. Reference Hernández-Ortiz, de Pablo and Graham2007). A notable feature of this method is that the computational time scales linearly with the number of nodes (
$N$
) on the particle surface, and the associated computational complexity is
$O(N)$
. We have compared computational times for obtaining mobility data in the middle between two spherical walls using two methods on a single CPU (AMD EPYC 7742) core: at the same level of accuracy, one time step takes approximately 30 s for the general geometry Ewald-like method, while one time step takes approximately 900 s for the finite element calculation with a stationary mesh. We note that when a particle is very close to the wall, meshes need to be heavily refined in the region between particle and wall in the finite element method to ensure accuracy, which increases computational cost. For dynamic simulations in finite element analysis, techniques such as moving mesh are needed to enable particle motion, which further increases computational cost, while in the general geometry Ewald-like method, enabling particle motion is straightforward by updating nodal positions every time step.
Normalised (a,c) radial and (b,d) transverse mobilities of the spherical particle under
$R_{\textit{p}}/R_o=0.05$
. In (a) and (b), mobilities are plotted as functions of the scaled radial position of the particle centre; in (c) and (d), mobilities are plotted as functions of the scaled radial distance between the particle centre and the inner wall. For the two walls,
$R_{\textit{i}}/R_o$
values are 0.13, 0.33 and 0.53. In (a) and (b), numerical and analytical results for a spherical particle in a spherical cavity (
$R_{\textit{i}}/R_o=0$
) are also plotted for comparison.
To further analyse the mobilities, we choose representative results from figure 2 and plot them in different ways in figure 3. While each plot in figure 2 corresponds to the same
$R_{\textit{i}}/R_o$
with different
$R_{\textit{p}}/R_o$
, each plot in figure 3 corresponds to the same
$R_{\textit{p}}/R_o$
(0.05) with different
$R_{\textit{i}}/R_o$
. Radial and transverse mobilities for a spherical particle in a spherical cavity (
$R_{\textit{i}}/R_o=0$
) from numerical and theoretical methods are also included for reference in figure 3(a,b). The results here are plotted in two ways: in figures 3(a) and 3(b), the mobilities are plotted as functions of the scaled radial position of the particle centre
$\textit{r}/\textit{R}_o$
; in figures 3(c) and 3(d), the mobilities are plotted as functions of the scaled radial distance between the particle centre and the inner wall
$(r-R_{\textit{i}})/R_o$
. A notable observation from figures 3(a) and 3(b) is that when
$\textit{r}/\textit{R}_o\gt 0.86$
(
$\textit{r}/\textit{R}_o\gt 0.90$
), radial (transverse) mobilities under different
$R_{\textit{i}}/R_o$
are nearly the same. This means that in the range of
$R_{\textit{i}}$
considered in this study (
$R_o$
remains the same), mobilities very close to the outer wall are insensitive to
$R_{\textit{i}}$
. In the limit
$R_{\textit{i}}/R_o=0$
, numerical results agree well with theoretical predictions in most regions of the cavity; more relevant comparisons and discussions were presented in a prior work (Chen & Jiang Reference Chen and Jiang2023). Different from mobilities near the outer wall, figures 3(c) and 3(d) show that at a certain particle–wall distance near the inner wall, mobilities are affected by
$R_{\textit{i}}$
, and increasing
$R_{\textit{i}}$
lowers mobilities due to enhanced confinement effect.
It would be desirable to compare the above numerical results with theoretical predictions to evaluate the validity of the point-particle approximation when
$R_{\textit{p}}$
is much smaller than both
$R_{\textit{i}}$
and
$R_o$
. However, to the best of our knowledge, theoretical results on mobilities of a point particle between two concentric spherical walls are not readily available in the literature. Nonetheless, a few studies are related to this issue. In one study, Stokes’ stream functions due to an axisymmetric singularity acting in a viscous fluid between two no-slip concentric spherical surfaces were derived analytically (Aderogba Reference Aderogba2009). In another study, a Green’s function solution of the Stokes equation due to delta-function forcing in a three-dimensional spherical-shell domain with no-slip boundary conditions was obtained, while the delta-function forcing is acting on an infinitely thin shell at a certain radial position (Kramer, Davies & Wilson Reference Kramer, Davies and Wilson2021). Based on prior results of particle mobilities in a spherical cavity (Chen & Jiang Reference Chen and Jiang2023), we expect that mobilities under the point-particle approximation would have better agreement with numerical results in the middle between two spherical walls than those very close to the walls; as
$R_{\textit{p}}$
decreases, differences between theoretical and numerical results would decrease; as the particle becomes closer to the wall, mobilities by the point-particle approximation would be smaller than numerical results.
Normalised mobility due to torque-to-translation coupling of the spherical particle under (a)
$R_{\textit{p}}/R_o=0.05$
and (b)
$R_{\textit{p}}/R_o=0.1$
. Mobilities are plotted as functions of the scaled radial position of the particle centre. Here,
$R_{\textit{i}}/R_o$
values are 0, 0.13, 0.33 and 0.53. Analytical results for a spherical particle in a spherical cavity (
$R_{\textit{i}}/R_o=0$
) are also plotted as red dots for comparison.
The numerical method can also capture components related to particle rotation in the grand mobility tensor. Here, we study the normalised mobility due to torque-to-translation coupling
$M_{\textit{UT}}$
. The direction of the torque vector (
$\boldsymbol{T}$
) is set to be in the positive
$y$
-direction, and the induced translation is in the
$z$
-direction. We first compute
$M_{\textit{UT}}$
for a spherical particle in a spherical cavity when
$R_{\textit{p}}/R_o$
values are 0.05 and 0.1, and compare numerical results with analytical data (Aponte-Rivera & Zia Reference Aponte-Rivera and Zia2016). Figures 4(a) and 4(b) show that general trends by numerical and analytical results are the same:
$M_{\textit{UT}}$
is zero at the cavity centre, and it increases as the particle moves towards the wall; after
$M_{\textit{UT}}$
reaches maximum near the wall, it decreases as the particle–wall distance decreases. It can be seen that numerical results agree well with theoretical predictions in most regions of the cavity; the difference between two results starts to increase as the particle gets very close to the wall. We then study
$M_{\textit{UT}}$
for the spherical particle between two concentric spherical walls, and results are plotted in figure 4. From the results, we can see a general trend for different
$R_{\textit{i}}/R_o$
and
$R_{\textit{p}}/R_o$
: near the inner (outer) wall, as the particle–wall distance increases,
$M_{\textit{UT}}$
decreases (increases) until it reaches a local minimum (maximum); as the particle–wall distance increases further,
$M_{\textit{UT}}$
turns to increase (decrease). It is also observed that the local minimum near the inner wall is very close zero, while the local maximum near the outer wall is non-zero.
3.2. Drift
As discussed in our prior work (Chen & Jiang Reference Chen and Jiang2023), for a spherical particle confined in a spherical cavity, the particle will drift perpendicular to the external force when the force is not along or perpendicular to the particle–cavity line of centres. Magnitudes of the particle’s main velocity (
$U_{m}$
) along the external force and the drift velocity (
$U_{d}$
) are
\begin{align} U_m(r,\theta )&=[M_r(r)\cos ^2\theta + M_t(r)\sin ^2\theta ]F^{\textit{ext}}/(6 \unicode{x03C0} \mu R_{\textit{p}}), \end{align}
\begin{align} U_{d}(r,\theta )&=\frac {1}{2}\left [M_t(r)-M_r(r)\right ]\sin (2\theta )\,F^{\textit{ext}}/(6 \unicode{x03C0} \mu R_{\textit{p}}), \\[6pt] \nonumber \end{align}
where
$r$
is the radial position of the particle centre,
$\theta$
is the angle between the external force and particle–cavity line of centres,
$M_r$
and
$M_t$
are normalised radial and transverse mobilities of the spherical particle, and
$F^{\textit{ext}}$
is the magnitude of the external force. A positive value of
$U_m$
means that the main velocity is in the direction of
$\boldsymbol{F}^{\textit{ext}}$
. The vector of the drift velocity is in the plane formed by the particle–cavity line of centres and
$\boldsymbol{F}^{\textit{ext}}$
, and a positive value of
$U_d$
means that the drift is in the same direction as an anticlockwise rotation of
$\boldsymbol{F}^{\textit{ext}}$
by 90
$^{\circ }$
in this plane. From (3.2), we can see that the drift is caused by the anisotropy of the mobility tensor (
$M_r \neq M_t$
). The anisotropy is induced by the confinements.
In this work, we consider the drift of a spherical particle confined between two concentric spherical walls. Here,
$R_{\textit{p}}/R_o$
is chosen as 0.05, and
$\theta$
between the external force and the particle–cavity line of centres is varied from 15
$^\circ$
to 75
$^\circ$
, with interval 15
$^\circ$
. Under different
$R_{\textit{i}}/R_o$
, we obtain
$U_d$
at various radial positions between the two walls from numerical simulations, and results are given in figure 5(a–c). In addition, we use (3.2) along with the mobility data in figure 2 to calculate
$U_d$
, and find that results agree with those obtained directly from simulations.
From the results, we can see that
$U_d$
is relatively small in the middle region between the two walls. This is because
$|M_r-M_t|$
is relatively small in this region. As the particle moves from the position for minimal
$U_d$
in the middle region towards the wall,
$U_d$
increases because
$|M_r-M_t|$
becomes larger; very close to the wall, however,
$U_d$
decreases because the no-slip wall plays a dominant role here. At a specific particle position,
$U_d$
reaches maximum when
$\theta =45^\circ$
; as
$\theta$
increases from 0
$^\circ$
(45
$^\circ$
) to 45
$^\circ$
(90
$^\circ$
),
$U_d$
increases (decreases). This is because
$U_d$
has a sinusoidal relationship with
$\theta$
.
It is also observed that under certain particle–wall distance and
$\theta$
, the magnitude of
$U_d$
near the inner wall increases as
$R_{\textit{i}}/R_o$
increases. This is because near the inner wall, mobilities are affected by increased
$R_{\textit{i}}/R_o$
, as shown in figure 3, and
$|M_r-M_t|$
increases as
$R_{\textit{i}}/R_o$
increases. On the other hand, near the outer wall, mobilities remain almost unchanged at different
$R_{\textit{i}}/R_o$
in this study, so the magnitude of
$U_d$
under certain particle–wall distance and
$\theta$
remains nearly the same as
$R_{\textit{i}}/R_o$
varies.
We also calculate scaled drift velocity
$\bar {U}_d=U_d/U_m$
at various radial positions and
$R_{\textit{i}}/R_o$
, and results are shown in figure 5(d–f). Radial distributions of
$\bar {U}_d$
share features similar to those of
$U_d$
:
$\bar {U}_d$
is relatively small in the middle region, and becomes larger near the wall; near the inner wall, at certain particle–wall distance and
$\theta$
,
$\bar {U}_d$
increases as
$R_{\textit{i}}/R_o$
increases; near the outer wall, changes in
$\bar {U}_d$
are rather small as
$R_{\textit{i}}/R_o$
varies. Different from
$U_d$
, when
$\theta$
varies from 0
$^\circ$
to 90
$^\circ$
,
$\bar {U}_d$
at a certain radial position is asymmetric about
$\theta =45^\circ$
;
$\bar {U}_d$
values for
$\theta =30^\circ$
and
$\theta =45^\circ$
are nearly the same. Among all cases, the maximal
$\bar {U}_d$
is approximately 0.25 and occurs near the outer wall. The above results suggest that the particle’s drift motion in the confined space, especially near the wall, could significantly affect particle motion, and has to be considered when manipulating particulate transport in such an environment.
(a–c) Drift velocity
$U_d$
and (d–f) scaled drift velocity
$U_d/U_m$
, as functions of the scaled radial position of the particle centre when
$R_{\textit{i}}/R_o$
values are (a,d) 0.13, (b,e) 0.33, (c,f) 0.53. Here,
$\theta$
is the angle between the external force and the particle–cavity line of centres. The size ratio is
$R_{\textit{p}}/R_o=0.05$
.
3.3. Flow induced by particle motion
We then analyse the flow caused by particle motion under
$\boldsymbol{F}^{\textit{ext}}$
in an initially quiescent fluid, and the distributions of fluid velocities in the
$x$
–
$y$
plane and streamlines near the particle are presented in figure 6. The size ratios are chosen as
$R_{\textit{p}}/R_o=0.05$
and
$R_{\textit{i}}/R_o=0.53$
; the particle centre is initially at
$r_{\textit{p}}/R_o=0.76$
. In figure 6, the top row shows the case when
$\boldsymbol{F}^{\textit{ext}}$
is along the radial direction. We can see that the fluid velocity in the
$x$
-direction (
$u$
) is largest near the particle, and it decays away from the particle; the fluid velocity in the
$y$
-direction (
$v$
) is diagonally symmetric about the particle centre, with positive (negative) values in the upper right (lower right) and lower left (upper left) regions near the particle. From streamlines, we can see fluid vortices above and below the particle in the
$x$
–
$y$
plane; in the three-dimensional space, the vortex is around the particle, and vortex centres form a plane perpendicular to
$\boldsymbol{F}^{\textit{ext}}$
and the
$x$
-axis.
Distributions of fluid velocities in the
$x$
–
$y$
plane and streamlines near the particle when the particle moves under
$\boldsymbol{F}^{\textit{ext}}$
in an initially quiescent fluid. Here,
$R_{\textit{p}}/R_o=0.05$
,
$R_{\textit{i}}/R_o=0.53$
and
$r_{\textit{p}}/R_o=0.76$
, where
$r_{\textit{p}}$
is the radial position of the particle centre. The left-hand column shows fluid velocity in the
$x$
-direction (
$u$
), and the middle column shows fluid velocity in the
$y$
-direction (
$v$
); the right-hand column shows streamlines and absolute magnitude of the total velocity. In the top (bottom) row,
$\boldsymbol{F}^{\textit{ext}}$
is in the radial (transverse) direction; in the middle row, the angle between
$\boldsymbol{F}^{\textit{ext}}$
and particle–cavity line of centres is 45
$^\circ$
.
The middle row in figure 6 shows the case when the angle between
$\boldsymbol{F}^{\textit{ext}}$
and the particle–cavity line of centres is 45
$^{\circ }$
. Different from the prior case, it is observed that
$u$
and
$v$
are both largest near the particle, and they both decay away from the particle; neither
$u$
nor
$v$
exhibits diagonal symmetry about the particle centre. By visualising streamlines, we can observe the fluid vortex near the particle induced by particle motion under
$\boldsymbol{F}^{\textit{ext}}$
; in the three-dimensional space, the plane formed by vortex centres is found to be perpendicular to
$\boldsymbol{F}^{\textit{ext}}$
.
In the bottom row of figure 6,
$\boldsymbol{F}^{\textit{ext}}$
points in the transverse direction. It can be seen that
$u$
is diagonally symmetric about the particle centre, with positive (negative) values in the upper right (lower right) and lower left (upper left) regions near the particle;
$v$
is largest near the particle, and it decays away from the particle. The streamlines show the fluid vortex near the particle; vortex centres in the three-dimensional space form a plane perpendicular to
$\boldsymbol{F}^{\textit{ext}}$
and the
$y$
-axis. In this case, although the particle is in the middle of the confined space along the radial direction, and the radial distances between the particle centre and the two walls are equal, the vortex near the inner wall is larger than that near the outer wall. This is because of the curvature effect of the spherical walls: for the particle between the two walls, the outer wall is in a concave configuration that restricts the flow. In the
$x$
–
$y$
plane, fluid vortices near two sides of the particle are not symmetric about the particle centre. This is different from results in the top row of figure 6 showing that the fluid vortices in the
$x$
–
$y$
plane near two sides of the particle are symmetric about the particle centre.
The above results indicate that as the direction of
$\boldsymbol{F}^{\textit{ext}}$
varies, the plane formed by vortex centres in three-dimensional space remains perpendicular to
$\boldsymbol{F}^{\textit{ext}}$
. It is also observed that significant fluid circulation mainly occurs near the particle, which is different from results in the spherical or cubic cavity that notable fluid circulation is observed in the entire cavity (Aponte-Rivera & Zia Reference Aponte-Rivera and Zia2016; Chen & Jiang Reference Chen and Jiang2022).
3.4. Rotating walls
In this subsection, we study trajectories of the spherical particle in flow caused by rotation of the spherical walls. Three main scenarios are considered: in the first scenario, the two walls rotate with the same angular velocity in the same direction; in the second scenario, the two walls rotate in different directions, and the outer wall’s angular velocity
$\boldsymbol{\omega }_o$
is perpendicular to that of the inner wall
$\boldsymbol{\omega }_{\textit{i}}$
(
$\boldsymbol{\omega }_o \times \boldsymbol{\omega }_{\textit{i}}=0$
), with
$|\boldsymbol{\omega }_o|=|\boldsymbol{\omega }_{\textit{i}}|$
; in the third scenario, only the outer or inner wall rotates, while the other wall remains stationary. The particle is non-neutrally buoyant, and a rotation-induced centrifugal (centripetal) force exerts on the particle when
$\rho _{\textit{p}} \gt \rho _{\textit{f}}$
(
$\rho _{\textit{p}} \lt \rho _{\textit{f}}$
). In the centrifugal (centripetal) case, the buoyancy-corrected density
$\rho _b = \rho _{\textit{p}} - \rho _{\textit{f}}$
is 0.2 (
$-0.2$
). The particle also experiences an external force that is in the negative
$z$
-direction, and
$F^{\textit{ext}}=-10$
. The size ratio
$R_{\textit{p}}/R_o$
is set to 0.05, and
$R_{\textit{i}}/R_o$
varies as 0.13, 0.33 and 0.53 in each scenario.
Trajectories of the particle centre in the
$y$
–
$z$
plane when the two walls rotate at a constant angular velocity
$\boldsymbol{\omega } =1$
, for
$R_{\textit{i}}/R_o=0.13$
. (a) The particle is released at three positions
$r_0=(0, 8, 0)$
,
$(0, -7.5, 0)$
and
$(0, 9, 4.3)$
under the centrifugal force. (b) The particle is released at three positions
$r_0=(0, 8, 0)$
,
$(0, 2, 8)$
and
$(0, 5, 5)$
under the centripetal force. The external force acts in the negative
$z$
-direction, and
$F^{\textit{ext}}=-10$
. Black circles denote positions of the inner and outer walls. Dashed and dotted black lines denote loci for
$U_r=0$
and
$U_t=0$
, respectively. The particle radius is 0.75, and particles are marked by circles at initial particle positions.
3.4.1. Two walls rotate in the same direction
In the scenario where the two walls rotate in the same direction, the inner and outer walls both rotate about the
$x$
-axis at a constant angular velocity
$\boldsymbol{\omega } =1$
. We consider the particle centre to be in the
$y$
–
$z$
plane at
$x=0$
(symmetry plane). The radial and transverse velocities of the particle in the rotating flow can be written as
\begin{align} U_r &=M_r(r)\, (F_{c} - F^{\textit{ext}}\sin \beta )/(6 \unicode{x03C0} \mu R_{\textit{p}}), \end{align}
\begin{align} U_t &=\boldsymbol{\omega } r-[M_t(r)\, F^{\textit{ext}}\cos \beta ]/(6 \unicode{x03C0} \mu R_{\textit{p}}), \end{align}
where
$\beta$
is the polar angle of the particle centre in the symmetry plane. We then solve for
$U_r=0$
and
$U_t=0$
to obtain two loci
$C_r$
and
$C_t$
, which are plotted as dashed (
$C_r$
) and dotted (
$C_t$
) black lines in figure 7. The reason for calculating the loci is to determine the stagnation point for the particle where
$C_r$
and
$C_t$
intersect. Compared to results in our prior study (Chen & Jiang Reference Chen and Jiang2023),
$C_r$
here is the same as that for the particle in a spherical cavity. This is because
$C_r$
is independent of the particle’s mobilities, and is not affected by the confining geometry or particle size. We can also see that in the centrifugal (centripetal) case,
$C_r$
is in the region
$z\gt 0$
(
$z\lt 0$
);
$C_r$
values in the two cases are symmetric about
$z=0$
, as
$C_r$
depends on the sign of the rotation-induced force
$F_c$
. For
$C_t$
, it is independent of
$F_c$
and thus remains the same in the centripetal and centrifugal cases. The stagnation points are at the intersection of
$C_r$
and
$C_t$
, and are identified as
$(0,9.7,3.7)$
in the centrifugal case, and
$(0,9.7,-3.7)$
in the centripetal case.
When
$R_{\textit{i}}/R_o=0.13$
, temporal evolutions of the particle centre in the centrifugal and centripetal cases are presented in figure 7. The particle is released at three different locations between the two walls. For the centrifugal case in figure 7(a), when the particle is released at
$r_0=(0, 9, 4.3)$
very close to the stagnation point, it rotates around the stagnation point, and gradually moves away from this point until it contacts the outer wall. When the particle is released at
$r_0=(0, 8, 0)$
, it mainly moves towards the positive
$y$
-direction until it contacts the outer wall. When the particle is released at
$r_0=(0, -7.5, 0)$
, it mainly moves in the negative
$z$
-direction until it contacts the outer wall. In all cases, after the particle contacts the outer wall, it moves along the wall towards the positive
$z$
-direction. At approximately
$z=8$
, the particle detaches from the outer wall and moves towards the inner wall. The particle then contacts the inner wall, moves along the wall for a while, detaches from the inner wall, and moves towards the outer wall again. From the simulation results, it is observed that no matter where the particle is released, it will eventually evolve into the same stable orbit. In the stable orbit, the particle exhibits periodic motion, attaching to and detaching from the two walls alternately.
For the centripetal case in figure 7(b), when the particle is released at
$r_0=(0, 8, 0)$
near the stagnation point, it rotates around this point, and gradually moves towards it without contacting the two walls. When the particle is released at
$r_0=(0, 2, 8)$
, it moves towards the inner wall, contacts the inner wall at approximately
$y=0$
, and moves along the wall at negative
$y$
-positions. After the particle detaches from the inner wall, it moves towards the outer wall, contacts the outer wall, and moves along the wall for a while. The particle then detaches from the outer wall, and gradually moves towards the stagnation point. When the particle is released at
$r_0=(0, 5, 5)$
, it first moves towards the inner wall, contacts the inner wall at a positive
$y$
-position, and moves along the wall for a while. After the particle detaches from the inner wall, it gradually moves towards the stagnation point without contacting the outer wall. It is observed that the duration between attaching to and detaching from the inner wall for the case
$r_0=(0, 2, 8)$
is longer than that for the case
$r_0=(0, 5, 5)$
. We also find that in the centripetal case, no matter where the particle is released, it will eventually arrive at the stagnation point.
Trajectories of the particle centre in the
$y$
–
$z$
plane when the two walls rotate at a constant angular velocity
$\boldsymbol{\omega } =1$
, for
$R_{\textit{i}}/R_o=0.53$
. (a) The particle is released at three positions
$r_0=(0, 10, 3)$
,
$(0, -9, 7)$
and
$(0, 11, 0)$
under the centrifugal force. (b) The particle is released at two positions
$r_0=(0, 9, 7)$
and
$(0, 11, 0)$
under the centripetal force. The external force acts in the negative
$z$
-direction, and
$F^{\textit{ext}}=-10$
. Black circles denote positions of the inner and outer walls. Dashed and dotted black lines denote loci for
$U_r=0$
and
$U_t=0$
, respectively. The particle radius is 0.75, and particles are marked by circles at initial particle positions.
When
$R_{\textit{i}}/R_o$
becomes 0.33, the trajectories are similar to those under
$R_{\textit{i}}/R_o=0.13$
; results and discussions for
$R_{\textit{i}}/R_o=0.33$
are in the supplementary material available at https://doi.org/10.1017/jfm.2025.10516. When
$R_{\textit{i}}/R_o$
increases to 0.53,
$C_t$
becomes quite different from the values in the prior two cases: it is closer to the inner wall, its shape changes, and its length decreases greatly. A major change in this case is that
$C_r$
and
$C_t$
no longer intersect, such that there is no stagnation point for the particle. Figure 8 plots temporal evolutions of the particle centre in centrifugal and centripetal cases when
$R_{\textit{i}}/R_o=0.53$
. The particle is released at three different locations between the two walls. For the centrifugal case in figure 8(a), when the particle is released at
$r_0=(0, 10, 3)$
, it first contacts and moves along the inner wall; after detaching from the inner wall, it moves towards and contacts the outer wall, and eventually evolves into the stable orbit. When the particle is released at
$r_0=(0, -9, 7)$
and
$(0, 11, 0)$
, it directly moves towards and contacts the outer wall, and then evolves into the stable orbit. The overall trajectories are similar to those in prior centrifugal cases, while the duration between attaching to and detaching from the inner wall becomes larger as
$R_{\textit{i}}$
increases further.
For the centripetal case in figure 8(b), after the particle is released from two locations at
$r_0=(0, 9, 7)$
and
$(0, 11, 0)$
, it directly moves towards and contacts the inner wall, and moves along the wall until detachment in the region
$y\lt 0$
and
$z\lt 0$
; then the particle moves towards and contacts the outer wall, and moves along the outer wall in the region
$y\gt 0$
; once the particle detaches from the outer wall, it moves towards and contacts the inner wall, and evolves into the stable orbit. As there exists no stagnation point, the particle no longer moves to a certain point, which is different from observations in prior centripetal cases when
$R_{\textit{i}}/R_o$
is 0.13 and 0.33 (figure 7(b) and supplementary figure S1
b). Instead, the particle evolves into a stable orbit similar to orbits in centrifugal cases.
3.4.2. Two walls rotate in different directions
Now we consider the scenario where the two walls rotate in different directions with the same angular velocity
$\boldsymbol{\omega } =1$
. Specifically, the outer wall remains rotating about the
$x$
-axis, while the inner wall rotates about the
$y$
-axis (
$\boldsymbol{\omega }_{\textit{i}}\times \boldsymbol{\omega }_o=0$
). Figure 9 shows projections of particle trajectories in the
$y$
–
$z$
and
$x$
–
$z$
planes when
$R_{\textit{i}}/R_o=0.33$
. In the centrifugal case, the particle is released at
$r_0=(0, 8, 6)$
; in the centripetal case, the particle is released at
$r_0=(0, 5, 5)$
. From projections of the trajectories in the
$y$
–
$z$
plane, we can see that the overall trends in the centrifugal and centripetal cases remain similar to those when
$\boldsymbol{\omega }_{\textit{i}}=\boldsymbol{\omega }_o$
, while the differences between them are discussed as follows. In the centrifugal case (figure 9
a), the particle first moves around the stagnation point without contacting the inner wall; when the particle moves along the inner wall in the region
$y\lt 0$
, its centre almost overlaps with the wall surface, as the particle shifts along the
$x$
-direction under the influence of the rotating inner wall (figure 9
b). In the centripetal case (figure 9
c), after the particle detaches from the inner wall, it rotates around the stagnation point in a larger circle compared to the trajectory when
$\boldsymbol{\omega }_{\textit{i}}=\boldsymbol{\omega }_o$
; the location of the stagnation point is slightly different from that when
$\boldsymbol{\omega }_{\textit{i}}=\boldsymbol{\omega }_o$
.
Projections of the particle centre’s trajectories in the (a,c)
$y$
–
$z$
and (b,d)
$x$
–
$z$
planes when
$R_{\textit{i}}/R_o=0.33$
. The outer (inner) wall rotates about the
$x$
-axis (
$y$
-axis) at
$\boldsymbol{\omega } =1$
. (a,b) In the centrifugal case, the particle is released at
$r_0=(0, 8, 6)$
. (c,d) In the centripetal case, the particle is released at
$r_0=(0, 5, 5)$
. Trajectories when
$\boldsymbol{\omega }_{\textit{i}}=\boldsymbol{\omega }_o$
are also included for comparison. The external force acts in the negative
$z$
-direction, and
$F^{\textit{ext}}=-10$
. Black circles denote positions of the inner and outer walls. The particle radius is 0.75, and particles are marked by circles at initial particle positions.
From figures 9(b) and 9(d), we can directly observe particle motion along the
$x$
-direction. Due to rotation of the inner wall about the
$y$
-axis, the fluid velocity in the
$x$
-direction (
$u$
) near the inner wall is positive (negative) when
$z\gt 0$
(
$z\lt 0$
). As a consequence, in both centrifugal and centripetal cases, the particle moves in the positive (negative)
$x$
-direction when its centre is in the region
$z\gt 0$
(
$z\lt 0$
). In the centrifugal case (figure 9
b), the particle can also exhibit motion along the
$x$
-direction under the condition that it moves along the outer wall and it is not in the symmetry plane. This is caused by the repulsive force exerting on the particle from particle–wall interaction between the particle and the outer wall. Trajectories under
$R_{\textit{i}}/R_o=0.13$
are similar to the above results, and are discussed in § 2 of the supplementary material.
Projections of the particle centre’s trajectories in the (a,c)
$y$
–
$z$
and (b,d)
$x$
–
$z$
planes when
$R_{\textit{i}}/R_o=0.53$
. The outer (inner) wall rotates about the
$x$
-axis (
$y$
-axis) at
$\boldsymbol{\omega } =1$
. The particle is released at
$r_0=(0, 11, 0)$
in the centrifugal and centripetal cases. Trajectories when
$\boldsymbol{\omega }_{\textit{i}}=\boldsymbol{\omega }_o$
are also included for comparison. The external force acts in the negative
$z$
-direction, and
$F^{\textit{ext}}=-10$
. Black circles denote positions of the inner and outer walls. The particle radius is 0.75, and particles are marked by circles at initial particle positions.
When
$R_{\textit{i}}/R_o=0.53$
, particle motions are plotted in figure 10. For projections of the trajectories in the
$y$
–
$z$
plane, the overall trends become different from those when
$\boldsymbol{\omega }_{\textit{i}}=\boldsymbol{\omega }_o$
. In both centrifugal and centripetal cases, the main difference is that after the particle contacts the inner wall, it moves along the wall in the clockwise direction at
$y\gt 0$
. In the centrifugal case (figure 10
a), during contact with the inner wall, the particle centre nearly overlaps with the surface of the inner wall due to particle motion along the
$x$
-direction; after the particle detaches from the inner wall, it moves towards and contacts the outer wall, and gradually evolves into a stable orbit. In the centripetal case (figure 10
c), after the particle detaches from the inner wall, it rotates in a circle, and moves towards the stagnation point, which does not exist when
$\boldsymbol{\omega }_{\textit{i}}=\boldsymbol{\omega }_o$
. The recurrence of this stagnation point is mainly caused by the flow induced by rotation of the inner wall.
Figure 10(b,d) show projections of particle trajectories in the
$x$
–
$z$
plane. In the centrifugal case (figure 10
b), after the particle contacts the outer wall at approximately
$z=0$
, its
$z$
-position increases due to rotation of the outer wall. On the one hand, the increasing
$z$
-position makes the particle move in the positive
$x$
-direction, and leaves the
$y$
–
$z$
plane under the flow induced by rotation of the inner wall. On the other hand, as the particle shifts away from the
$y$
–
$z$
plane, it also experiences a repulsive force from particle–wall interaction, which tends to move the particle in the negative
$x$
-direction and back to the
$y$
–
$z$
plane. The combination of these two effects leads to a relatively slow motion towards the positive
$x$
-direction, even when the particle’s
$z$
-position reaches larger values. When the particle detaches from the outer wall, the particle–wall interaction becomes ineffective, and the particle starts to move rapidly in the positive
$x$
-direction until its
$z$
-position becomes zero. When the particle centre is at
$z\lt 0$
, the particle moves towards the negative
$x$
-direction. In the stable orbit, the particle attaches to and detaches from the two walls periodically, moving back and forth in the
$x$
-direction. In the centripetal case (figure 10
d), the particle centre remains in the region
$z\lt 0$
, so the particle moves in the negative
$z$
-direction due to rotation of the inner wall, until it reaches the stagnation point.
3.4.3. One wall rotates
In the last scenario, we consider only the inner or outer wall rotating about the
$x$
-axis at a constant angular velocity
$\boldsymbol{\omega } =1$
, while the other wall remains stationary. Figure 11 shows particle trajectories in the
$y$
–
$z$
plane when
$R_{\textit{i}}/R_o=0.33$
; trajectories under
$\boldsymbol{\omega }_{\textit{i}}=\boldsymbol{\omega }_o$
are also shown for comparison. In the centrifugal case (figure 11
a), the particle is released at
$r_0=(0,8,6)$
. When
$\boldsymbol{\omega} _{\textit{i}}=0$
and
$\boldsymbol{\omega} _o=1$
, the particle trajectory is similar to that when
$\boldsymbol{\omega }_{\textit{i}}=\boldsymbol{\omega }_o$
, while there are minor differences. In this case, the particle first moves around the stagnation point without contacting the inner wall; the
$z$
-position for contacting the outer wall is slightly below that when
$\boldsymbol{\omega }_{\textit{i}}=\boldsymbol{\omega }_o$
; the
$y$
-position for contacting the inner wall in the stable orbit is slightly larger than that when
$\boldsymbol{\omega }_{\textit{i}}=\boldsymbol{\omega }_o$
. We note that, for
$R_i/R_o=0.13$
(supplementary figure S3
a), trajectories when only outer wall rotates and both walls rotate in the same direction are nearly the same. These results are due to the fact that as
$R_{\textit{i}}/R_o$
increases from 0.13 to 0.33, the rotation of the inner wall affects the flow and particle motion to a greater extent. When
$\boldsymbol{\omega} _{\textit{i}}=1$
and
$\boldsymbol{\omega} _o=0$
, the particle trajectory is very different from the other two cases in figure 11(a), but is similar to that when only the inner wall rotates in supplementary figure S3
a: the particle mainly moves in the negative
$z$
-direction under
$\boldsymbol{F}^{\textit{ext}}$
, while it also shifts in the positive
$y$
-direction due to rotation of the inner wall; after contacting the outer wall, it moves in the clockwise direction, and arrives at the stagnation point near
$y=0$
and
$z=-13$
.
Trajectories of the particle centre in the
$y$
–
$z$
plane when only the inner or outer wall rotates at a constant angular velocity
$\boldsymbol{\omega } =1$
under
$R_{\textit{i}}/R_o=0.33$
. Trajectories when
$\boldsymbol{\omega }_{\textit{i}}=\boldsymbol{\omega }_o$
are also included for reference. (a) The particle is released at
$r_0=(0, 8, 6)$
under the centrifugal force. (b) The particle is released at
$r_0=(0, 5, 5)$
under the centripetal force. The external force acts in the negative
$z$
-direction, and
$F^{\textit{ext}}=-10$
. Black circles denote positions of the inner and outer walls. The particle radius is 0.75, and particles are marked by circles at initial particle positions.
In the centripetal case (figure 11
b), the particle is released at
$r_0=(0, 5, 5)$
. When
$\boldsymbol{\omega } _{\textit{i}}=0$
and
$\boldsymbol{\omega } _o=1$
, the particle trajectory is similar to that when
$\boldsymbol{\omega }_{\textit{i}}=\boldsymbol{\omega }_o$
: it first contacts and moves along the inner wall; after detaching from the inner wall, it moves around the stagnation point and eventually arrives at this point. The main differences are: when
$\boldsymbol{\omega } _{\textit{i}}=0$
and
$\boldsymbol{\omega } _o=1$
, the particle moves in a larger circle around the stagnation point, and the position of the stagnation point becomes slightly different. When
$\boldsymbol{\omega } _{\textit{i}}=1$
and
$\boldsymbol{\omega } _o=0$
, the particle trajectory is initially the same as in the other two cases: it first contacts and moves along the inner wall in the region
$y\gt 0$
. The trajectory differs after the particle detaches from the inner wall: when only the inner wall rotates, the particle moves in the negative
$z$
-direction under the action of
$\boldsymbol{F}^{\textit{ext}}$
, and also moves in the negative
$y$
-direction due to rotation of the inner wall; after contacting the outer wall, it gradually moves to the stagnation point near the bottom at
$y=0$
and
$z=-13$
. Being similar to the above results, results and discussions for trajectories under
$R_{\textit{i}}/R_o=0.13$
are presented in § 3 of the supplementary material.
Trajectories of the particle centre in the
$y$
–
$z$
plane when only the inner or outer wall rotates at a constant angular velocity
$\boldsymbol{\omega } =1$
under
$R_{\textit{i}}/R_o=0.53$
. Trajectories when the two walls rotate are also included for reference. (a) The particle is released at
$r_0=(0, 11, 0)$
under the centrifugal force. (b) The particle is released at
$r_0=(0, 11,0)$
under the centripetal force. The external force acts in the negative
$z$
-direction, and
$F^{\textit{ext}}=-10$
. Black circles denote positions of the inner and outer walls. The particle radius is 0.75, and particles are marked by circles at initial particle positions.
In figure 12, we plot trajectories of the particle when
$R_{\textit{i}}/R_o=0.53$
. In the centrifugal case (figure 12
a), the particle is released at
$r_0=(0, 11, 0)$
. When
$\boldsymbol{\omega } _{\textit{i}}=0$
and
$\boldsymbol{\omega } _o=1$
, the particle moves in the clockwise direction along the inner wall in the region
$y\gt 0$
. This is different from the trajectory when
$\boldsymbol{\omega }_{\textit{i}}=\boldsymbol{\omega }_o$
, in which the particle moves in the anticlockwise direction along the inner wall. One common behaviour that the two cases share is that the particle eventually evolves into a stable orbit, attaching to and detaching from the two walls periodically. When
$\boldsymbol{\omega } _{\textit{i}}=1$
and
$\boldsymbol{\omega } _o=0$
, the particle trajectory is similar to those in supplementary figure S3
a and figure 11(a) when only the inner wall rotates: initially, the particle mainly moves in the negative
$z$
-direction under
$\boldsymbol{F}^{\textit{ext}}$
, while it also shifts in the positive
$y$
-direction due to rotation of the inner wall; after contacting the outer wall, it moves in the clockwise direction and arrives at the stagnation point near the bottom. One difference is that the
$y$
-position of the stagnation point here becomes a little larger, which is caused by the enhanced fluid velocity as
$R_{\textit{i}}$
increases.
In the centripetal case (figure 12
b), the particle is released at
$r_0=(0, 11, 0)$
. When
$\boldsymbol{\omega } _{\textit{i}}=0$
and
$\boldsymbol{\omega } _o=1$
, the particle contacts and moves along the inner wall for a relatively short period of time; after detaching from the inner wall, it moves around the stagnation point and gradually reaches this point. On the contrary, there are no stagnation points when both walls rotate and
$\boldsymbol{\omega }_{\textit{i}}=\boldsymbol{\omega }_o$
. When
$\boldsymbol{\omega } _{\textit{i}}=1$
and
$\boldsymbol{\omega } _o=0$
, the particle moves in the negative
$z$
-direction under the influence of
$\boldsymbol{F}^{\textit{ext}}$
; it also moves in the negative
$y$
-direction due to the flow induced by rotation of the inner wall. After the particle contacts the outer wall, it eventually arrives at the stagnation point near the bottom.
In many results on particle trajectories, especially those in the centrifugal case, the particle is very close to the confining walls where mobilities are of less accuracy. If mobilities near the walls with very good accuracy are included, then we expect that the particle trajectory will be quantitatively different, but qualitative behaviour will remain unchanged. Quantitatively, particle motion along the wall at
$y\gt 0$
(
$y\lt 0$
) will be faster (slower). This is because the mobility very close to the wall becomes lower, and particle velocity due to the external force, which opposes (promotes) particle motion due to the rotating flow at
$y\gt 0$
(
$y\lt 0$
), becomes smaller. If a correct hydrodynamic drainage force is further included, then we expect that there are minor changes in the timing of collisions with the wall, the duration for particles being attached to the wall, and locations for the particle to attach to or detach from the wall, but the overall behaviour will be qualitatively the same.
4. Conclusions
In this work, we studied dynamics of a spherical particle in a low-Reynolds-number fluid confined between two concentric spherical walls using numerical simulations. Particle motion under an external force in an initially quiescent fluid was studied first. For hydrodynamic mobilities, it was found that the radial and transverse mobilities
$M_r$
and
$M_t$
reach maxima near the radial centre of the confined space, and the position for the maximal mobility is closer to the inner wall. When the external force is not parallel or perpendicular to particle–cavity line of centres, the particle exhibits a drift motion perpendicular to the external force. We observed that the maximal drift velocity occurs near the wall, and the influence of confinement level on drift velocity near the inner wall is greater than that near the outer wall. Flow field and streamlines during particle motion were visualised, and it was found that fluid circulation mainly occurs near the particle, which differs from results for particle motion in a spherical cavity where fluid circulation occurs in the entire cavity.
We also studied trajectories of a non-neutrally buoyant particle in the confined space when walls rotate. Three main scenarios are considered: (i) both walls rotate in the same direction with the same angular velocity
$\boldsymbol{\omega }_{\textit{i}}=\boldsymbol{\omega }_o$
; (ii) the two walls rotate in different directions (
$\boldsymbol{\omega }_{\textit{i}} \times \boldsymbol{\omega }_o=0$
), with
$|\boldsymbol{\omega }_{\textit{i}}| = |\boldsymbol{\omega }_o|$
; (iii) only one wall rotates, while the other wall is kept stationary. Under the centrifugal force, the overall trends in most trajectories are similar: the particle eventually evolves into a stable orbit, attaching to and detaching from the inner and outer walls periodically. Under the centripetal force, in most cases, particles released at different positions will finally arrive at the stagnation point between the two walls. These observations are similar to those for particle motion in a spherical cavity, except that the particle can contact and move along the inner wall in this study.
There are a few cases where the trajectories do not follow the overall trends. When
$\boldsymbol{\omega }_{\textit{i}}=\boldsymbol{\omega }_o$
and
$R_{\textit{i}}/R_o=0.53$
,
$C_r$
and
$C_t$
no longer intersect, and there is no stagnation point in the symmetry plane. As a result, the particle no longer moves to a stagnation point in the centripetal case; instead, it evolves into a stable orbit similar to that in the centrifugal case. However, under
$R_{\textit{i}}/R_o=0.53$
, the stagnation point can reappear when the two walls rotate in different directions (figure 10
c,d) and when only the outer wall rotates (figure 12
b). When
$\boldsymbol{\omega } _{\textit{i}}=1$
and
$\boldsymbol{\omega } _o=0$
(only the inner wall rotates), in both centrifugal and centripetal cases, the external force
$\boldsymbol{F}^{\textit{ext}}$
plays a dominant role in particle motion, and the particle finally arrives at a stagnation point near the bottom in the confined space.
In the current literature, most numerical and theoretical studies on particle dynamics under total confinement consider only a single outer wall. To our knowledge, this paper is the first to study mobility, drift and trajectories of particles between two spherical walls. This work goes beyond prior simpler assumptions of a single outer wall, and lays a step towards more realistic simulations of particle dynamics in intracellular environments and microfluidic encapsulation systems. Compared to our prior work, this paper extends the numerical model and analysis by studying rotation-to-translation coupling of the particle.
The main insights of this study are summarised below.
-
(i) Due to the curvature effect of spherical walls, radial distributions of radial and transverse mobilities are asymmetric about the radial centre between two spherical walls; at the same particle–wall distance, mobility near the inner wall is larger.
-
(ii) Increasing inner wall radius makes mobilities near the inner wall decrease, while mobilities near the outer wall are almost unchanged. This applies to both translation-to-translation and rotation-to-translation components of the grand mobility tensor.
-
(iii) Inclusion of the inner wall does not affect loci for
$U_r=0$
(
$C_r$
), and
$C_r$
is independent of the inner wall radius. On the contrary, the inner wall can alter loci for
$U_t=0$
(
$C_t$
), and the length of
$C_t$
and the region enclosed by
$C_t$
shrinks as the inner wall radius increases. -
(iv) Inclusion of the inner wall can affect particle trajectories: instead of moving near the cavity centre in a portion of the trajectory within a spherical cavity, the particle here can move along the surface of the inner wall in both centripetal and centrifugal cases.
-
(v) Rotation of the inner wall can alter particle trajectories: in the centrifugal case, the region that the stable orbit spans could shrink when the inner wall rotates; in the centripetal case, the vanished stagnation point can reappear when inner and outer walls rotate in different directions.
This study could improve understanding and manipulation of particulate transport in low-Reynolds-number fluids under total confinement, which has applications in intracellular transport and encapsulation technologies. Based on results and insights in this study, future research could explore effects of particle shape and concentration, magnitudes and directions of the two walls’ rotation, and non-concentric walls on particle dynamics in the confined space. We expect that researchers in fields of synthetic cells, double emulsions and droplet-based microfluidics will use results and insights from this study to better control particle movement in synthetic cells, improve effectiveness and efficiency of particle self-assembly in double emulsions, and create new functions of droplet-based microfluidics.
Supplementary material
Supplementary material is available at https://doi.org/10.1017/jfm.2025.10516.
Acknowledgements
We acknowledge computational resources provided by the ORISE Supercomputer.
Funding
This work was supported by the National Key R & D Programme of China (2022YFF0503504), the Strategic Priority Research Programme of the Chinese Academy of Sciences (XDB0620102, XDA0470203), and the International Cooperation Grant from the Chinese Academy of Sciences (025GJHZ2022023MI).
Declaration of interests
The authors report no conflict of interest.
Appendix A
We perform an error analysis in relation to the error introduced by discretising the particle surface. Here, the numerical result is compared with analytical solution for the radial mobility
$M_r$
at the cavity centre
$\textit{r}/\textit{R}_o=0$
under
$R_{\textit{i}}/R_o=0$
. We calculate the relative error as
$|(M_r^{\textit{num}}-M_r^{\textit{ana}})/M_r^{\textit{ana}}|$
, where
$M_r^{\textit{num}}$
and
$M_r^{\textit{ana}}$
are radial mobilities from numerical and analytical methods, respectively. The relative error is plotted as a function of the total number of vertices on the particle surface (
$N_{v\textit{ertices}}$
), and the results are shown in figure 13. It is observed that the relative error increases as
$N_{v\textit{ertices}}$
decreases. The minimal (maximal)
$N_{v\textit{ertices}}$
tested is 20 (363), and the resulting relative error is 7.4 % (0.14 %). It is also found that the relative error nearly scales as
$N_{v\textit{ertices}}^{-1}$
.
Relative error of the numerical result for
$M_r$
at
$\textit{r}/\textit{R}_o=0$
under
$R_{\textit{i}}/R_o=0$
as a function of total number of vertices on the particle surface.
