How inertial lift affects the dynamics of a microswimmer in Poiseuille flow

The transport of motile microorganisms is strongly influenced by fluid flows that are ubiquitous in biological environments. Here we demonstrate the impact of fluid inertia. We analyze the dynamics of a microswimmer in pressure-driven Poiseuille flow, where fluid inertia is small but non-negligible. Using perturbation theory and the reciprocal theorem, we show that in addition to the classical inertial lift of passive particles, the active nature generates a ‘swimming lift’, which we evaluate for neutral and pusher/puller-type swimmers. Accounting for fluid inertia engenders a rich spectrum of complex dynamics including bistable states, where tumbling coexists with stable centerline swimming or swinging. The dynamics is sensitive to the swimmer’s hydrodynamic signature and goes well beyond the findings at vanishing fluid inertia. Our work will have non-trivial implications on the transport and dispersion of active suspensions in microchannels.

The reciprocal theorem in fluid dynamics and transport phenomena

In the study of fluid dynamics and transport phenomena, key quantities of interest are often the force and torque on objects and total rate of heat/mass transfer from them. Conventionally, these integrated quantities are determined by first solving the governing equations for the detailed distribution of the field variables (i.e. velocity, pressure, temperature, concentration, etc.) and then integrating the variables or their derivatives on the surface of the objects. On the other hand, the divergence form of the conservation equations opens the door for establishing integral identities that can be used for directly calculating the integrated quantities without requiring the detailed knowledge of the distribution of the primary variables. This shortcut approach constitutes the idea of the reciprocal theorem, whose closest relative is Green’s second identity, which readers may recall from studies of partial differential equations. Despite its importance and practicality, the theorem may not be so familiar to many in the research community. Ironically, some believe that the extreme simplicity and generality of the theorem are responsible for suppressing its application! In this Perspectives piece, we provide a pedagogical introduction to the concept and application of the reciprocal theorem, with the hope of facilitating its use. Specifically, a brief history on the development of the theorem is given as a background, followed by the discussion of the main ideas in the context of elementary boundary-value problems. After that, we demonstrate how the reciprocal theorem can be utilized to solve fundamental problems in low-Reynolds-number hydrodynamics, aerodynamics, acoustics and heat/mass transfer, including convection. Throughout the article, we strive to make the materials accessible to early career researchers while keeping it interesting for more experienced scientists and engineers.

An active particle in a complex fluid

In this work, we study active particles with prescribed surface velocities in non-Newtonian fluids. We employ the reciprocal theorem to obtain the velocity of an active spherical particle with an arbitrary axisymmetric slip velocity in an otherwise quiescent second-order fluid. We then determine how the motion of a diffusiophoretic Janus particle is affected by complex fluid rheology, namely viscoelasticity and shear-thinning viscosity, compared to a Newtonian fluid, assuming a fixed slip velocity. We find that a Janus particle may go faster or slower in a viscoelastic fluid, but is always slower in a shear-thinning fluid as compared to a Newtonian fluid.

Dynamics of a treadmilling microswimmer near a no-slip wall in simple shear

Induction of flow is commonly used to control the migration of a microswimmer in a confined system such as a microchannel. The motion of a swimmer, in general, is governed by nonlinear equations due to non-trivial hydrodynamic interactions between the flow and the swimmer near a wall. This paper derives analytical expressions for the equations of motion governing a circular treadmilling swimmer in simple shear near a no-slip wall by combining the reciprocal theorem for Stokes flow with an exact solution for the dragging problem of a cylinder near a wall. We demonstrate that the reduced dynamical system possesses a Hamiltonian structure, which we use to show that the swimmer cannot migrate stably at a constant distance from a wall but only exhibit periodic oscillatory motion along the wall, or to escape from it. A treadmilling swimmer with the lowest two treadmilling modes is investigated in detail by means of a bifurcation analysis of the reduced dynamical system. It is found that the swimming direction of oscillatory motion is clarified by the sign of the Hamiltonian in the absence of flow, and that the induction of the flow suppresses upstream migration but aligns swimmer orientations in downstream migration. These results could inform strategies for the transport and control of micro-organisms and micromachines.

Electro-osmotic flow through a nanopore

Electro-osmotic pumping of fluid through a nanopore that traverses an insulating membrane is considered. The density of surface charge on the membrane is assumed to be uniform and sufficiently low for the Poisson–Boltzmann equation to be linearized. The reciprocal theorem gives the flow rate generated by an applied weak electric field, expressed as an integral over the fluid volume. For a circular hole in a membrane of zero thickness, an analytical result is possible up to quadrature. For a membrane of arbitrary thickness, the full Poisson–Nernst–Planck–Stokes system of equations is solved numerically using a finite volume method. The numerical solution agrees with the standard analytical result for electro-osmotic flux through a long cylindrical pore when the membrane thickness is large compared to the hole diameter. When the membrane thickness is small, the flow rate agrees with that calculated using the reciprocal theorem.

Boundary Conditions for Torsional Circular Shaft with Two-Dimensional Dodecagonal Quasicrystals

Through generalizing the method of a decay analysis technique determining the interior solution developed by Gregory and Wan, a set of necessary conditions on the end-data of torsional circular shaft in two-dimensional dodecagonal quasicrystals (2D dodecagonal QCs) for the existence of a rapidly decaying solution is established. By accurate solutions for auxiliary regular state, using the reciprocal theorem, these necessary conditions for the end-data to induce only a decaying elastostatic state (boundary layer solution) will be translated into appropriate boundary conditions for the torsional circular shaft in 2D dodecagonal QCs. The results of the present paper enable us to establish a set of boundary conditions.