See Active matter for background.

• bacteria enhance diffusion as a result of the flow fields they produce

The path taken by a tracer will depend on the detailed spatial and temporal correlations of the velocity. Numerical simulations were conducted in Fluid transport by individual microswimmers. The striking feature of the tracer trajectories is their , a consequence of the angular dependence of the flow field. Mathematically, it is because all terms in the multipole expansion, except the Stokelet are exact derivatives. The way this works:

Consider a tracer whose velocity (in LAB frame) is much smaller than the swimmer's velocity. Then, in the rest frame of the swimmer, the tracer follows a path which is approximately straight, and parallel to the swimmer's motion (in Lab frame). It's velocity deviating from the straight-line motion is given by the dipolar field. And so it's total displacement in the LAB frame (_{total displacement in swimmer's rest frame, relative to straight line path}) is given by integrating the dipolar field approximately along the straight line from $-\infty$ to $\infty$. However, because $\frac{\partial G_{ij}}{\partial x_k} (\vec{r}) D_{jk}$ is a total derivative ($D_{jk}$ is constant), and $G_{ij}$ is $0$ at $-\infty$ and $\infty$, the total displacement is $0$.

The reason we need the tracer's velocity to be much smaller than the swimmer's velocity, for the above argument is that the total displacement is given by the integral of the velocity with respect to time, i.e. $\int \vec{v}(t) dt = \int_C \vec{v}(t) \frac{ds}{|\vec{V}+\vec{v}(t)|}$, where $\vec{v}(t)$ is the dipolar velocity field of the swimmer (_{i.e. velocity field in its rest frame, minus the overall constant, $\vec{V}$}). $\vec{V}$ is the swimmer's velocity in the LAB frame. $\vec{V}+\vec{v}(t)$ is thus the total velocity field in swimmer's rest frame. $ds$ is a distance element along the path $C$ that the particle traces. $|\vec{V}+\vec{v}(t)|$ is the instantaneous speed of the particle along this path, so that $dt = \frac{ds}{|\vec{V}+\vec{v}(t)|}$. Now, if $|\vec{V}| \gg |\vec{v}(t)|$, $dt \approx \frac{ds}{|\vec{V}|}$, so that the integral is approximately a line integral of $\vec{v}(t)$ along $C$. But, when we take $\vec{v}(t)$ into account, when $\vec{v}(t)$ is parallel to $\vec{V}$, $|\vec{V}+\vec{v}(t)|$ is larger, and the contribution in the integral is less; when $\vec{v}(t)$ is anti-parallel to $\vec{V}$, $|\vec{V}+\vec{v}(t)|$ is smaller, and the contribution in the integral is more. This means the particle has a displacement bias towards the direction of motion of the swimmer. This is called entrainment.

But, Why do the faraway tracers have a net negative displacement?

The entrainment effect is an example of Darwin drift. The Darwin drift volume has also been calculated for these active swimmers

Contribution to diffusion

We can estimate the contribution to diffusion from the entrainment effect. We know that the Diffusion coefficient can be expressed in 3D as:

$D_{\text{entr}} = \frac{\langle \Delta x^2 \rangle}{6t}$

The entrainment length (Darwin drift) is of order $a$ (the size of the swimmer), when close (within distance $a$) to the swimmer. Thus, $\langle \Delta x^2 \rangle \sim a^2$, whenever there is a swimmer within a volume $\sim a^3$. If there are $n$ swimmers per unit volume, the probability that a swimmer is in a given region of volume$a^3$ is approximately $n a^3$. Therefore, $\langle \Delta x^2 \rangle \sim a^5 n$. Now the characteristic time step $t \sim V/a$, is the time scale that the swimmer travelling at speed $V$ takes to traverse the distance $a$ throughout which the swimmer interacts with the tracer particle. Therefore,

$D_{\text{entr}} \approx \frac{1}{6}a^4 n V$

There is also a contribution to diffusion from the random reorientations that real bacteria perform at approximately regular intervals (in their run and tumble behaviour). Is the contribution to the diffusion constant from random reorientations, or finite run lengths? I think the former, due to the disappearance of $\lambda$, the run length from the expression

$D_{\text{rr}} = \frac{4\pi}{3}(\frac{\kappa}{V})^3 nV$

where $\kappa$ is a measure of the swimmer's dipole strength.

Because the addition of variances ($\langle \Delta x^2 \rangle$) for independent processes, we then have that the total diffusion coefficient is approximately the sum:

$D = D_{\text{rr}} + D_{\text{entr}} + D_{\text{thermal}}$

For different kinds of systems, some of these diffusions coefficients will dominate.

Swimmers in Poiseuille flow

Zottl and Stark paper. Swimmer equations of motion, for swimmer in background flow $\mathbf{v}_f$:

$\frac{d}{dt}\mathbf{r} = v_0 \hat{\mathbf{e}}+ \mathbf{v}_f$

$\frac{d}{dt} \hat{\mathbf{e}}= \frac{1}{2}\mathbf{\Omega}_f \times \hat{\mathbf{e}}$

where \hat{\mathbf{e}} is the swimming direction of the point swimmer. In the case of Poiseuille flow, the equation determining, the angle of the swimmer follows the nonlinear pendulum equation (with $\sin$).

When swimming upstream, any deviation for the centre line is subject to a restoring torque from the vorticity and hence the swimmer trajectory oscillates around the centre of the channel. Swimming downstream any perturbation about the centre line is amplified by the vorticity , and the swimmer tumbles in the flow. For sufficiently large velocities, it continues to tumble down-stream, otherwise it reaches the walls and and the simple theory must be supplemented by additional physics.

One can also describe the motion of the swimmer in simple shear flow, and when there is tendency to swim, on average, in a particular direction, "-taxis" For instance,

• towards gravity, gravitaxis
• towards light, phototaxis
• following a chemical gradient, chemotaxis.

One can use these ideas, with shear, and gravitaxis (together often termed gyrotaxis), to explain, for instance, the formation of thin layers of phytoplankton in the oceans.

Surfaces

Why micro-organisms often accumulate at surfaces

First note, that a simple self-propelled rod, or sphere, when it eventually hits a surface, will then tend to move parallel to it, and only scape it, when a rotational fluctuation changes its direction enough to swim away from it.

However, there is a less trivial effect, due to hydrodynamic interactions with the wall. These can be taken into account, because Stokes equations are linear, by considering an image swimmer at a position corresponding to the reflection of the swimmer on the wall, and pointing in the opposite direction (so as to satisfy the boundary condition of no normal flow at a free boundary (one that can slip; Like what? I mean, say a liquid-gas interface doesn't satisfy either no-slip or no normal flow, no? http://onlinelibrary.wiley.com/doi/10.1002/cpa.3160190405/abstract Its no normal stress and no tangential stress.)). The extra terms needed to satisfy the no-slip condition are more complicated, and form the Blake tensor. But doesn't the reversed mirror-image Stokelet cancel both the normal and tangential components of the velocity at the boundary?? No, because the Stokelet doesn't have the right symmetry, i think

Like what? I mean, say a liquid-gas interface doesn't satisfy either no-slip or no normal flow, no? http://onlinelibrary.wiley.com/doi/10.1002/cpa.3160190405/abstract Its no normal stress and no tangential stress.

However hydrodynamic interactions are not the only contribution. For rotating swimmers, like E. Coli, the effect the wall drag on torque is important; it makes the swimmer move in circles near the wall. See more at Physics of microswimmers—single particle motion and collective behavior: a review.