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We develop a numerical scheme to compute the dynamic electrophoretic velocity of a uniformly charged, dielectric, spherical colloidal particle under an unsteady electric field, for, in principle, arbitrary imposed electric field strength, β= a∗e∗E ∗∞/(k∗BT ∗). Here, a∗ is the characteristic size of the particle, E ∗∞ is the applied field strength, and the factor k∗BT ∗/e∗ corresponds to the thermal voltage. We focus our computations on Debye lengths comparable to the particle size; i.e., 1/(κ∗a∗)= O(1), where 1/κ∗ is the Debye length, and moderately-charged particles, i.e., σ∗= O(ϵ∗k∗BT ∗/(e∗a∗)), where σ∗ is the surface charge density on the colloid and ϵ∗ is the permittivity of the medium. For a suddenly applied field, the initial growth of the electrophoretic mobility (i.e., ratio of particle speed to field strength) occurs on the momentum diffusion time and is independent of β in the practically relevant case of large Schmidt number, Sc= ν∗/D∗ ≫ 1, where ν∗ is the kinematic viscosity and D∗ is the mean ion diffusion coefficient. Subsequently, the Debye cloud deforms on the ion diffusion timescale, leading to the mobility approaching its steady-state in a β-dependent manner. For the stopping problem, where the field is suddenly switched off, most of the change in the particle speed occurs on the momentum diffusion timescale, and the particle fully stops after the Debye cloud regains its equilibrium spherical shape. Under an oscillatory field, the field frequency has a large influence on both the amplitude and phase lag of the mobility, as it determines if the Debye cloud has sufficient time to deform within an oscillation cycle. The time-average flow about the particle resembles a stresslet. 

The squirmer is a popular model to analyse the fluid mechanics of a self-propelled object, such as a micro-organism. We demonstrate that some fore–aft symmetric squirmers can spontaneously self-propel above a critical Reynolds number. Specifically, we numerically study the effects of inertia on spherical squirmers characterised by an axially and fore–aft symmetric ‘quadrupolar’ distribution of surface-slip velocity; under creeping-flow conditions, such squirmers generate a pure stresslet flow, the stresslet sign classifying the squirmer as either a ‘pusher’ or ‘puller’. Assuming axial symmetry, and over the examined range of the Reynolds number$$Re$$(defined based upon the magnitude of the quadrupolar squirming), we find that spontaneous symmetry breaking occurs in the puller case above$$Re \approx 14.3$$, with steady swimming emerging from that threshold consistently with a supercritical pitchfork bifurcation and with the swimming speed growing monotonically with$$Re$$. 

Electrophoresis is the motion of a charged colloidal particle in an electrolyte under an applied electric field. The electrophoretic velocity of a spherical particle depends on the dimensionless electric field strength$$\beta =a^*e^*E_\infty ^*/k_B^*T^*$$, defined as the ratio of the product of the applied electric field magnitude$$E_\infty ^*$$and particle radius$$a^*$$, to the thermal voltage$$k_B^*T^*/e^*$$, where$$k_B^*$$is Boltzmann's constant,$$T^*$$is the absolute temperature, and$$e^*$$is the charge on a proton. In this paper, we develop a spectral element algorithm to compute the electrophoretic velocity of a spherical, rigid, dielectric particle, of fixed dimensionless surface charge density$$\sigma$$over a wide range of$$\beta$$. Here,$$\sigma =(e^*a^*/\epsilon ^*k_B^*T^*)\sigma ^*$$, where$$\sigma ^*$$is the dimensional surface charge density, and$$\epsilon ^*$$is the permittivity of the electrolyte. For moderately charged particles ($$\sigma ={O}(1)$$), the electrophoretic velocity is linear in$$\beta$$when$$\beta \ll 1$$, and its dependence on the ratio of the Debye length ($$1/\kappa ^*$$) to particle radius (denoted by$$\delta =1/(\kappa ^*a^*)$$) agrees with Henry's formula. As$$\beta$$increases, the nonlinear contribution to the electrophoretic velocity becomes prominent, and the onset of this behaviour is$$\delta$$-dependent. For$$\beta \gg 1$$, the electrophoretic velocity again becomes linear in field strength, approaching the Hückel limit of electrophoresis in a dielectric medium, for all$$\delta$$. For highly charged particles ($$\sigma \gg 1$$) in the thin-Debye-layer limit ($$\delta \ll 1$$), our computations are in good agreement with recent experimental and asymptotic results.