Swimming at low Reynolds number in fluids with odd, or Hall, viscosity

Abstract

We apply the geometric theory of swimming at low Reynolds number to the study of nearly circular swimmers in two-dimensional fluids with non-vanishing Hall, or "odd", viscosity. The Hall viscosity gives an off-diagonal contribution to the fluid stress-tensor, which results in a number of striking effects. In particular, we find that a swimmer whose area is changing will experience a torque proportional to the rate of change of the area, with the constant of proportionality given by the coefficient η o of odd viscosity. After working out the general theory of swimming in fluids with Hall viscosity for a class of simple swimmers, we give a number of example swimming strokes which clearly demonstrate the differences between swimming in a fluid with conventional viscosity and a fluid which also has a Hall viscosity. A number of more technical results, including a proof of the torque-area relation for swimmers of more general shape, are explained in a set of appendices.The theory of swimming in classical fluids at low Reynolds number [1,2] is remarkable because of the connections it makes between seemingly disparate fields [3]. For example, the motion of swimmers with cyclic swimming strokes is determined purely from classical fluid dynamics, but it can be re-cast into an elegant geometric formulation reminiscent of Berry's phase physics and gauge fields [3][4][5]. In fact, the motion of tiny organisms in fluids with high viscosity can be captured by a "gauge-theory" of shapes. Since the initial work on the geometric formulation of swimming there have been generalizations to swimmers in quantum fluids [6] and even to swimmers in fluids on curved spaces [7,8]. The theory has also been successfully applied in practice to describe the swimming of robots [9] and microbots [10,11].In this article we focus on swimmers in 2D fluids with broken time-reversal symmetry, for example, fluids in magnetic fields or rotating fluids. We are not interested in the specific source of time-reversal breaking, but instead just consider a classical fluid with a microscopic source of local angular momentum (on a much smaller scale than the size of the swimmer) that gives rise to a non-vanishing Hall (or odd) viscosity coefficient [12,13] in addition to the usual isotropic viscosity coefficients. The Hall viscosity is an off-diagonal viscosity term that is dissipationless and produces forces perpendicular to the direction of the fluid flow. It can have a quantum mechanical origin in, for example, systems exhibiting the quantum Hall effect [12][13][14][15][16][17][18][19][20][21][22], or a classical origin in plasmas at finite-temperature [23].We will not focus on the microscopic origin of the Hall viscosity coefficient, but only assume it to be non-vanishing in conjunction with the usual viscosity coefficients. From this assumption we will determine the motion of swimmers at low Reynolds number in the presence of Hall viscosity. Specifically, we will consider the problem of swimmers with circular boundaries that move via deformations of thei...

Phenomenology of nonrelativistic parity-violating hydrodynamics in 2+1 dimensions

“…This does not mean, however, that parity-odd transport coefficients cannot be detected by studying these flows, as one can choose boundary conditions on momentum or energy flux, both of which include parity-violating corrections (see, e.g., [23]). So let us consider the simplest possible toy model of the consequences of parity violation on boundary conditions.…”

“…The large number of new terms in the conservation laws suggests that parity-odd effects may be phenomenologically important for certain classes of flows. For example, an external probe sitting in an incompressible fluid has been studied, where there are new stresses normal to the surface [23]. However, the effects of parity violation on the hydrodynamic flows themselves are poorly understood, as the Hall viscosity is effectively a "topological" surface term in the incompressible Navier-Stokes equation, and so more terms must be added to see new physics away from boundaries.…”

Boundary central charge from bulk odd viscosity: Chiral superfluids

“…leaving a single odd viscosity coefficient η xz (ω) = η (1) o (ω) [1][2][3][4][5]. A non-zero q, however, along with the tensors δ ij and ε ij , can be used to construct additional SO (2)-invariant odd viscosity tensors, beyond σ xz .…”

Free-Surface Variational Principle for an Incompressible Fluid with Odd Viscosity

“…There is an ambiguity in the definition of A, since A = A + dΛ gives us dA = dA. However, this gauge freedom does not affect the boundary action (34). [50] As an example let us consider A = −ydx for M given by y ≤ h(x, t).…”

“…These effects are subtle in the case when the classical twodimensional fluid is incompressible. Recent works have outlined some of observable consequences of the odd viscosity for incompressible flows [33][34][35][36][37][38]. In particular, in Ref.[38] the equations governing the Hamiltonian dynamics of surface waves were derived in the case where bulk vorticity is absent.Let us start by summarizing the main equations of an incompressible fluid dynamics with odd viscosity.…”