We analyze the pressure-driven flow of a viscoelastic fluid in arbitrarily shaped, narrow channels and present a theoretical framework for calculating the relationship between the flow rate $q$ and pressure drop $Delta p$. We utilize the Oldroyd-B mo
del and first identify the characteristic scales and dimensionless parameters governing the flow in the lubrication limit. Employing a perturbation expansion in powers of the Deborah number ($De$), we provide analytical expressions for the velocity, stress, and the $q-Delta p$ relation in the weakly viscoelastic limit up to $O(De^2)$. Furthermore, we exploit the reciprocal theorem derived by Boyko $&$ Stone (Phys. Rev. Fluids, vol. 6, 2021, pp. L081301) to obtain the $q-Delta p$ relation at the next order, $O(De^3)$, using only the velocity and stress fields at the previous orders. We validate our analytical results with two-dimensional numerical simulations of the Oldroyd-B fluid in a hyperbolic, symmetric contracting channel and find excellent agreement. For the flow-rate-controlled situation, both our theory and simulations reveal weak dependence of the velocity field on the Deborah number, so that the velocity can be approximated as Newtonian. In contrast to the velocity, the pressure drop strongly depends on the viscoelastic effects and decreases with $De$. Elucidating the relative importance of different terms in the momentum equation contributing to the pressure drop, we identify that a pressure drop reduction for narrow contracting geometries is primarily due to gradients in the viscoelastic shear stresses, while viscoelastic axial stresses have a minor effect on the pressure drop along the symmetry line.
In the study of ordinary differential equations (ODEs) of the form $hat{L}[y(x)]=f(x)$, where $hat{L}$ is a linear differential operator, two related phenomena can arise: resonance, where $f(x)propto u(x)$ and $hat{L}[u(x)]=0$, and repeated roots, wh
ere $f(x)=0$ and $hat{L}=hat{D}^n$ for $ngeq 2$. We illustrate a method to generate exact solutions to these problems by taking a known homogeneous solution $u(x)$, introducing a parameter $epsilon$ such that $u(x)rightarrow u(x;epsilon)$, and Taylor expanding $u(x;epsilon)$ about $epsilon = 0$. The coefficients of this expansion $frac{partial^k u}{partialepsilon^k}big{|}_{epsilon=0}$ yield the desired resonant or repeated root solutions to the ODE. This approach, whenever it can be applied, is more insightful and less tedious than standard methods such as reduction of order or variation of parameters. While the ideas can be introduced at the undergraduate level, we could not find any elementary or advanced text that illustrates these ideas with appropriate generality.
Under a steady DC electric field of sufficient strength, a weakly conducting dielectric sphere in a dielectric solvent with higher conductivity can undergo spontaneous spinning (Quincke rotation) through a pitchfork bifurcation. We design an object c
omposed of a dielectric sphere and an elastic filament. By solving an elasto-electro-hydrodynamic (EEH) problem numerically, we uncover an EEH instability exhibiting diverse dynamic responses. Varying the bending stiffness of the filament, the composite object displays three behaviours: a stationary state, undulatory swimming and steady spinning, where the swimming results from a self-oscillatory instability through a Hopf bifurcation. By conducting a linear stability analysis incorporating an elastohydrodynamic model, we theoretically predict the growth rates and critical conditions, which agree well with the numerical counterparts. We also propose a reduced model system consisting of a minimal elastic structure which reproduces the EEH instability. The elasto-viscous response of the composite structure is able to transform the pitchfork bifurcation into a Hopf bifurcation, leading to self-oscillation. Our results imply a new way of harnessing elastic media to engineer self-oscillations, and more generally, to manipulate and diversify the bifurcations and the corresponding instabilities. These ideas will be useful in designing soft, environmentally adaptive machines.
Oscillations of flagella and cilia play an important role in biology, which motivates the idea of functional mimicry as part of bio-inspired applications. Nevertheless, it still remains challenging to drive their artificial counterparts to oscillate
via a steady, homogeneous stimulus. Combining theory and simulations, we demonstrate a strategy to achieve this goal by using an elasto-electro-hydrodynamic instability. In particular, we show that applying a uniform DC electric field can produce self-oscillatory motion of a microrobot composed of a dielectric particle and an elastic filament. Upon tuning the electric field and filament elasticity, the microrobot exhibits three distinct behaviors: a stationary state, undulatory swimming and steady spinning, where the swimming behavior stems from an instability emerging through a Hopf bifurcation. Our results imply the feasibility of engineering self-oscillations by leveraging the elasto-viscous response to control the type of bifurcation and the form of instability. We anticipate that our strategy will be useful in a broad range of applications imitating self-oscillatory natural phenomena and biological processes.
Recent experiments have demonstrated that small-scale rotary devices installed in a microfluidic channel can be driven passively by the underlying flow alone without resorting to conventionally applied magnetic or electric fields. In this work, we co
nduct a theoretical and numerical study on such a flow-driven watermill at low Reynolds number, focusing on its hydrodynamic features. We model the watermill by a collection of equally-spaced rigid rods. Based on the classical resistive force (RF) theory and direct numerical simulations, we compute the watermills instantaneous rotational velocity as a function of its rod number $N$, position and orientation. When $N geq 4$, the RF theory predicts that the watermills rotational velocity is independent of $N$ and its orientation, implying the full rotational symmetry (of infinity order), even though the geometrical configuration exhibits a lower-fold rotational symmetry; the numerical solutions including hydrodynamic interactions show a weak dependence on $N$ and the orientation. In addition, we adopt a dynamical system approach to identify the equilibrium positions of the watermill and analyse their stability. We further compare the theoretically and numerically derived rotational velocities, which agree with each other in general, while considerable discrepancy arises in certain configurations owing to the hydrodynamic interactions neglected by the RF theory. We confirm this conclusion by employing the RF-based asymptotic framework incorporating hydrodynamic interactions for a simpler watermill consisting of two or three rods and we show that accounting for hydrodynamic interactions can significantly enhance the accuracy of the theoretical predictions.
We develop a general hydrodynamic theory describing a system of interacting actively propelling particles of arbitrary shape suspended in a viscous fluid. We model the active part of the particle motion using a slip velocity prescribed on the otherwi
se rigid particle surfaces. We introduce the general framework for particle rotations and translations by applying the Lorentz reciprocal theorem for a collection of mobile particles with arbitrary surface slip. We then develop an approximate theory applicable to widely separated spheres, including hydrodynamic interactions up to the level of force quadrupoles. We apply our theory to a general example involving a prescribed slip velocity, and a specific case concerning the autonomous motion of chemically active particles moving by diffusiophoresis due to self-generated chemical gradients.
From biofilm and colony formation in bacteria to wound healing and embryonic development in multicellular organisms, groups of living cells must often move collectively. While considerable study has probed the biophysical mechanisms of how eukaryotic
cells generate forces during migration, little such study has been devoted to bacteria, in particular with regard to the question of how bacteria generate and coordinate forces during collective motion. This question is addressed here for the first time using traction force microscopy. We study two distinct motility mechanisms of Myxococcus xanthus, namely twitching and gliding. For twitching, powered by type-IV pilus retraction, we find that individual cells exert local traction in small hotspots with forces on the order of 50 pN. Twitching of bacterial groups also produces traction hotspots, however with amplified forces around 100 pN. Although twitching groups migrate slowly as a whole, traction fluctuates rapidly on timescales <1.5 min. Gliding, the second motility mechanism, is driven by lateral transport of substrate adhesions. When cells are isolated, gliding produces low average traction on the order of 1 Pa. However, traction is amplified in groups by a factor of ~5. Since advancing protrusions of gliding cells push on average in the direction of motion, we infer a long-range compressive load sharing among sub-leading cells. Together, these results show that the forces generated during twitching and gliding have complementary characters and both forces are collectively amplified in groups.
Biologically important membrane channels are gated by force at attached tethers. Here, we generically characterize the non-trivial interplay of force, membrane tension, and channel deformations that can affect gating. A central finding is that minute
conical channel deformation under force leads to significant energy release during opening. We also calculate channel-channel interactions and show that they can amplify force sensitivity of tethered channels.
The elementary topological T1 process in a two-dimensional foam corresponds to the flip of one soap film with respect to the geometrical constraints. From a mechanical point of view, this T1 process is an elementary relaxation process through which
the entire structure of an out-of-equilibrium foam evolves. The dynamics of this elementary relaxation process has been poorly investigated and is generally neglected during simulations of foams. We study both experimentally and theoretically the T1 dynamics in a dry two-dimensional foam. We show that the dynamics is controlled by the surface viscoelastic properties of the soap films (surface shear plus dilatational viscosity, ms+k, and Gibbs elasticity e), and is independent of the shear viscosity of the bulk liquid. Moreover, our approach illustrates that the dynamics of T1 relaxation process provides a convenient tool for measuring the surface rheological properties: we obtained e = 32+/-8 mN/m and ms+k = 1.3+/-0.7 mPa.m.s for SDS, and e = 65+/-12 mN/m and ms+k = 31+/-12 mPa.m.s for BSA, in good agreement with values reported in the literature.
