Actuated rheology of magnetic micro-swimmers suspensions: Emergence of motor and brake states

We study the effect of magnetic field on the rheology of magnetic micro-swimmers suspensions. We use a model of a dilute suspension under simple shear and subjected to a constant magnetic field. Particle shear stress is obtained for both pusher and puller types of micro-swimmers. In the limit of low shear rate, the rheology exhibits a constant shear stress, called actuated stress, which only depends on the swimming activity of the particles. This stress is induced by the magnetic field and can be positive (brake state) or negative (motor state). In the limit of low magnetic fields, a scaling relation of the motor-brake effect is derived as a function of the dimensionless parameters of the model. In this case, the shear stress is an affine function of the shear rate. The possibilities offered by such an active system to control the rheological response of a fluid are finally discussed.

Figure 1

Left: 3D parametrization of a bacterium in spherical coordinates $\left(\theta ,\varphi \right)$. The magnetic field $\mathbf{B}=B\mathbf{b}$ is contained in the $\left(x,y\right)$ plane and its orientation in this plane is given by the angle $\alpha$. Right: 3D representation of the orientation distribution function for ${\text{Pe}}_m=1,\alpha ={45}^{\circ }$ and ${\text{Pe}}_H={10}^{-4}$ (a), ${\text{Pe}}_H=4$ (b), and ${\text{Pe}}_H=10$ (c).

Figure 2

Rescaled particle shear stress ${\tilde{\mathrm{\Sigma }}}_p$, derived numerically from Eq. (8), as a function of the rotational Péclet number ${\text{Pe}}_H[\alpha ={45}^{\circ },{135}^{\circ }$; ${\text{Pe}}_m=1$; $\mathcal{A}=10$; $\epsilon =1$ (puller) and $-1$ (pusher)]. The active contribution ${\tilde{\mathrm{\Sigma }}}_{\mathrm{act}}$ is color labeled and referred to Eq. (8). When ${\text{Pe}}_H\ll 1,{\tilde{\mathrm{\Sigma }}}_p$ is equal to the active stress and tends to a constant value, the actuated stress. Depending on the sign of the actuated stress, the suspension can be turned to brake and motor states. For each of the four graphs, the brake and motor states are interpreted by a sketch in which we show the orientation of the elongated particles relative to the flow direction. The red (brake) and green (motor) lines and arrows correspond to the stream lines created by the hydrodynamic dipole of each particle (represented by a black ellipsoid).

Figure 3

Phase diagram in the $({\text{Pe}}_m,\alpha )$ space of the motor-brake effect for puller and pusher swimmers. Positive values and negative values of the actuated stress ${\tilde{\mathrm{\Sigma }}}_0$ are respectively red and green labeled and correspond to brake and motor states. The relative magnitude of the actuated stress is indicated by a logarithmic color gradient.

Figure 4

(a) Example of relation ${\tilde{\mathrm{\Sigma }}}_p\left({\text{Pe}}_H\right)$ obtained from a full numerical solution of the problem for $\mathcal{A}=10,\alpha ={45}^{\circ }$, and ${\text{Pe}}_m=0.6$ (symbols). The solid line is a linear fit taking into account the data up to the limit ${\text{Pe}}_H^{\max }$ such that the goodness of the fit $R^2$ remains larger than 0.999. The slope of the curve yields ${\text{Pe}}_H^{*}$ and the intercept ${\tilde{\mathrm{\Sigma }}}_0$. In regime (i), the actuated stress dominates. In regime (ii), the stress is essentially linear in shear rate, the actuated stress is negligible. In regime (iii), the affine expansion is no longer valid because other stress terms contribute to the shear stress. When the calculated ${\text{Pe}}_H^{*}>{\text{Pe}}_H^{\max }$, then we conclude that the regime ${\tilde{\mathrm{\Sigma }}}_p={\tilde{\mathrm{\Sigma }}}_0+{\eta }_p{\text{Pe}}_H$ is never reached: the stress is constant at low ${\text{Pe}}_H$, up to a transition to a regime not anymore dominated by the active and drag stresses. (b) Validity of the affine expansion of the stress (10) in the parameter space $({\text{Pe}}_m,\alpha )$ for $\mathcal{A}=10$. The color code corresponds to the value of $\mathrm{\Delta }={log}_{10}\left({\text{Pe}}_H^{\max }\right)-{log}_{10}\left({\text{Pe}}_H^{*}\right)$, which is the range of validity, in decades of ${\text{Pe}}_H$, for which Eq. (10) holds. When parameters are chosen in the regions in blue, Eq. (10) is no longer valid and other terms have to be taken into account in the expansion of the shear stress.

Figure 5

Rescaled particle shear stress ${\tilde{\mathrm{\Sigma }}}_p/\left|{\tilde{\mathrm{\Sigma }}}_0\right|$ as a function of the rescaled Péclet number ${\widetilde{\text{Pe}}}_H$ for different values of $\alpha$, for ${\text{Pe}}_m=0.1$, 0.4, and 1 and for both pusher and puller swimmers. The $\alpha$ values are indicated by the color code. Values $\alpha =0^{\circ },{90}^{\circ },{180}^{\circ }$ are excluded because they correspond to the situation ${\tilde{\mathrm{\Sigma }}}_0=0$, for which the master equation (11) is not defined. The numerical solution displays, for a wide range of ${\text{Pe}}_H$ and ${\text{Pe}}_m$, a collapse onto expression (8), represented in black solid (pushers) and dotted (pullers) lines. Negative values of shear rate correspond to a gradient of flow velocity in the $-y$ direction and vorticity in the $z$ direction. By reversing the shear flow, the actuated stress keeps its sign and the motor-brake effect is reversed (from motor to brake state and vice versa). Inset: rescaled actuated stress $\left|{\tilde{\mathrm{\Sigma }}}_0/\left[\frac{1}{2}sin\left(2\alpha \right)\right]\right|$ as a function of the magnetic Péclet number ${\text{Pe}}_m$. It exhibits a scaling in ${\text{Pe}}_m^2$ for ${\text{Pe}}_m\ll 1$ as predicted analytically. For ${\text{Pe}}_m\gg 1$, it saturates.

Figure 6

Computation of the particle shear stress and active stress for spherical micro-swimmers. The motor-brake effect is recovered also for these kind of particles and the value of the actuated stress is identical to the one of rod-shaped particles.