Casimir effect in swimmer suspensions

We show that the Casimir effect can emerge in microswimmer suspensions. In principle, two effects conspire against the development of Casimir effects in swimmer suspensions. First, at low Reynolds number, the force on any closed volume vanishes, but here the relevant effect is the drag by the flow produced by the swimmers, which can be finite. Second, the fluid velocity and the pressure are linear on the swimmer force dipoles, and averaging over the swimmer orientations would lead to a vanishing effect. However, being that the suspension is a discrete system, the noise terms of the coarse-grained equations depend on the density, which itself fluctuates, resulting in effective nonlinear dynamics. Applying the tools developed for other nonequilibrium systems to general coarse-grained equations for swimmer suspensions, the Casimir drag is computed on immersed objects, and it is found to depend on the correlation function between the rescaled density and dipolar density fields. By introducing a model correlation function with medium-range order, explicit expressions are obtained for the Casimir drag on a body. When the correlation length is much larger than the microscopic cutoff, the average drag is independent of the correlation length, with a range that depends only on the size of the immersed bodies.

Figure 1

Geometry used for the calculation of the Casimir drag. The bodies confine an active suspension in a region of size $L_x\times L_y\times L_z$. There are nonflux boundary conditions at $x=0$ and $x=L_x$, while the fields are periodic in $y$ and $z$. In Eq. (19) the cross sections of the bodies are modeled as circles of radius $R$. The total drag on the bodies results from the subtraction of the drag produced by the region inside (separation $L_x$) and outside (separation $L_x^{\prime }$) the bodies. To obtain the final expression, Eq. (23), the limit $L_x^{\prime }\to \infty$ is considered.