Nonlinear deformation and localized failure of bacterial streamers in creeping flows

We investigate the failure of bacterial floc mediated streamers in a microfluidic device in a creeping flow regime using both experimental observations and analytical modeling. The quantification of streamer deformation and failure behavior is possible due to the use of 200 nm fluorescent polystyrene beads which firmly embed in the extracellular polymeric substance (EPS) and act as tracers. The streamers, which form soon after the commencement of flow begin to deviate from an apparently quiescent fully formed state in spite of steady background flow and limited mass accretion indicating significant mechanical nonlinearity. This nonlinear behavior shows distinct phases of deformation with mutually different characteristic times and comes to an end with a distinct localized failure of the streamer far from the walls. We investigate this deformation and failure behavior for two separate bacterial strains and develop a simplified but nonlinear analytical model describing the experimentally observed instability phenomena assuming a necking route to instability. Our model leads to a power law relation between the critical strain at failure and the fluid velocity scale exhibiting excellent qualitative and quantitative agreeing with the experimental rupture behavior.

: Bacteria and particle count in one P. fluorescens streamer's failure zone (dotted box). The volume of zoomed picture is calculated by multiplying the area of the picture with depth of field and is approximately 2000 µm 3 . Bacteria cell is cylindrical and particles are spherical in geometry. Diameter of bacteria and particle are 0.2 µm and the height/ length of one bacteria is 8 µm. Volume of one bacteria is 0.25 µm 3 and one particle is 4.19 ×10 -3 µm 3 .

Estimating Experimental Uncertainty
Two separate sources of experimental uncertainty were identified. The first is the repeatability error accounting for the heterogeneity of the biomass itself. Due to the very nature of this error, it has to be evaluated by statistical means (i.e. from a number of repeated observations). For each of 2 cases, the experiments were repeated 2-4 times for all flow rates to yield relative uncertainty estimates. For example, for the P. fluorescens for the flow velocity (U=8.92 × 10 -4 m/s), the experiment was repeated 3 times thus yielding relative uncertainty for each U (Fig. SF2). Table  ST1 provides the complete list of repetitions for each case. Let this uncertainty be denoted by . The second source of uncertainty resulted from an error in the tracking process itself. This error could be estimated by visually determining the uncertainty in tracking the Lagrangian points and the maximum error was estimated to be approximately 4%. This determines the error envelope for a single tracking of couplets. Let this uncertainty be denoted by . The final error envelope ( ) for critical stretch ratio is given by:

Error bars in
Figs. 6b and 7 represent .

Kinematics
The streamer is assumed to be slender cylindrical body (long wavelength defect approximation) throughout till the onset of instability. In the reference and current configuration, the radius of the cylinder are , and length , respectively. We also assume the streamer to have uniform behavior throughout its material volume except at the far field boundaries before the onset of instability. We also assume that streamer deformation is purely inelastic after the elastic limit is reached and to remain in this state till the onset of instability. We assume that inelastic deformation is isochoric and describe the geometry of the current streamer (right cylinder) with radius and length . Thus, in the rate form we arrive at the following relationship: 2̇+̇= 0 (S1) Now since the deformation after elastic limit is purely inelastic, we define the axial creep strain as = / . Integrating it from the elastic limit where the dimensions of the cylinder are assumed to be × to current configuration gives us the following logarithmic inelastic strain: is the creep stretch. We will refer to the elastic limit configuration to be the reference configuration for the purpose of this analysis. Now denoting the aspect ratio in the current configuration to be = / and the reference configuration to be Ω = / , we can relate them using incompressibility: Thus this also shows that = 2/3 Ω −2/3 and = Ω 1/3 −1/3 . In the rate form Eq. (S3) becomes ̇= 3 2 ̇ (S4)

Mechanics
We first write the free energy rate for this system in the current configuration: ̇=̇( ,̇) 2 +̇− Δ̇−̇ (S5) where is the dissipation density function which can in general depend on the strain and strain rate, is the surface tension (assumed uniform and without any gradient Marangoni effects), ̇ is the rate of change of surface area, Δ̇ is the rate of work done by the pressure difference between the inside and outside of the streamer and is axial the fluidic traction force. We can write the rate of change of surface area as: ̇= (2 + 2 ) = 2 �̇+̇+� ≈̇=̇, ≫ 1 (S6) For the rate of pressure work, we have: ̇= Δ 2 ̇− 2̇= 2 � 2̇−̇� = −2 2 ̇ where = 0 − 0 is the pressure differential between outside 0 and inside pressure 0 . Now note that from the slender body approximation of resistive flow theory we have, = / ln and thus we get for the fluidic work rate, ̇= ln / ̇= 2 ln