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We study analytically the joint dispersion of Gaussian patches of salt and colloids in linear flows, and how salt gradients accelerate or delay colloid spreading by diffusiophoretic effects. Because these flows have constant gradients in space, the problem can be solved almost entirely for any set of parameters, leading to predictions of how the mixing time and the Batchelor scale are modified by diffusiophoresis. We observe that the evolution of global concentrations, defined as the inverse of the patches areas, are very similar to those obtained experimentally in chaotic advection. They are quantitatively explained by examining the area dilatation factor, in which diffusive and diffusiophoretic effects are shown to be additive and appear as the divergence of a diffusive contribution or of a drift velocity. An analysis based on compressibility is developed in the salt-attracting case, for which colloids are first compressed before dispersion, to predict the maximal colloid concentration as a function of the parameters. This maximum is found not to depend on the flow stretching rate nor on its topology (strain or shear flow), but only on the characteristics of salt and colloids (diffusion coefficients and diffusiophoretic constant) and the initial size of the patches.
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Motivated by problems arising in the pneumatic actuation of controllers for micro-electromechanical systems (MEMS), labs-on-a-chip or biomimetic soft robots, and the study of microrheology of both gases and soft solids, we analyze the transient fluid--structure interaction (FSIs) between a viscoelastic tube conveying compressible flow at low Reynolds number. We express the density of the fluid as a linear function of the pressure, and we use the lubrication approximation to further simplify the fluid dynamics problem. On the other hand, the structural mechanics is governed by a modified Donnell shell theory accounting for Kelvin--Voigt-type linearly viscoelastic mechanical response. The fluid and structural mechanics problems are coupled through the tubes radial deformation and the hydrodynamic pressure. For small compressibility numbers and weak coupling, the equations are solved analytically via a perturbation expansion. Three illustrative problems are analyzed. First, we obtain exact (but implicit) solutions for the pressure for steady flow conditions. Second, we solve the transient problem of impulsive pressurization of the tubes inlet. Third, we analyze the transient response to an oscillatory inlet pressure. We show that an oscillatory inlet pressure leads to acoustic streaming in the tube, attributed to the nonlinear pressure gradient induced by the interplay of FSI and compressibility. Furthermore, we demonstrate an enhancement in the volumetric flow rate due to FSI coupling. The hydrodynamic pressure oscillations are shown to exhibit a low-pass frequency response (when averaging over the period of oscillations), while the frequency response of the tube deformation is similar to that of a band-pass filter.
Active droplets swim as a result of the nonlinear advective coupling of the distribution of chemical species they consume or release with the Marangoni flows created by their non-uniform surface distribution. Most existing models focus on the self-propulsion of a single droplet in an unbounded fluid, which arises when diffusion is slow enough (i.e. beyond a critical Peclet number, $mbox{Pe}_c$). Despite its experimental relevance, the coupled dynamics of multiple droplets and/or collision with a wall remains mostly unexplored. Using a novel approach based on a moving fitted bispherical grid, the fully-coupled nonlinear dynamics of the chemical solute and flow fields are solved here to characterise in detail the axisymmetric collision of an active droplet with a rigid wall (or with a second droplet). The dynamics is strikingly different depending on the convective-to-diffusive transport ratio, $mbox{Pe}$: near the self-propulsion threshold (moderate $mbox{Pe}$), the rebound dynamics are set by chemical interactions and are well captured by asymptotic analysis; in contrast, for larger $mbox{Pe}$, a complex and nonlinear combination of hydrodynamic and chemical effects set the detailed dynamics, including a closer approach to the wall and a velocity plateau shortly after the rebound of the droplet. The rebound characteristics, i.e. minimum distance and duration, are finally fully characterised in terms of $mbox{Pe}$.
In a recent paper, Liu, Zhu and Wu (2015, {it J. Fluid Mech.} {bf 784}: 304) present a force theory for a body in a two-dimensional, viscous, compressible and steady flow. In this companion paper we do the same for three-dimensional flow. Using the fundamental solution of the linearized Navier-Stokes equations, we improve the force formula for incompressible flow originally derived by Goldstein in 1931 and summarized by Milne-Thomson in 1968, both being far from complete, to its perfect final form, which is further proved to be universally true from subsonic to supersonic flows. We call this result the textit{unified force theorem}, which states that the forces are always determined by the vector circulation $pGamma_phi$ of longitudinal velocity and the scalar inflow $Q_psi$ of transverse velocity. Since this theorem is not directly observable either experimentally or computationally, a testable version is also derived, which, however, holds only in the linear far field. We name this version the textit{testable unified force formula}. After that, a general principle to increase the lift-drag ratio is proposed.
Channel formation and branching is widely seen in physical systems where movement of fluid through a porous structure causes the spatiotemporal evolution of the medium in response to the flow, in turn causing flow pathways to evolve. We provide a simple theoretical framework that embodies this feedback mechanism in a multi-phase model for flow through a fragile porous medium with a dynamic permeability. Numerical simulations of the model show the emergence of branched networks whose topology is determined by the geometry of external flow forcing. This allows us to delineate the conditions under which splitting and/or coalescing branched network formation is favored, with potential implications for both understanding and controlling branching in soft frangible media.
Long, shallow microchannels embedded in thick soft materials are widely used in microfluidic devices for lab-on-a-chip applications. However, the bulging effect caused by fluid--structure interactions between the internal viscous flow and the soft walls has not been completely understood. Previous models either contain a fitting parameter or are specialized to channels with plate-like walls. This work is a theoretical study of the steady-state response of a compliant microchannel with a thick wall. Using lubrication theory for low-Reynolds-number flows and the theory for linearly elastic isotropic solids, we obtain perturbative solutions for the flow and deformation. Specifically, only the channels top wall deformation is considered, and the ratio between its thickness $t$ and width $w$ is assumed to be $(t/w)^2 gg 1$. We show that the deformation at each stream-wise cross-section can be considered independently, and that the top wall can be regarded as a simply supported rectangle subject to uniform pressure at its bottom. The stress and displacement fields are found using Fourier series, based on which the channel shape and the hydrodynamic resistance are calculated, yielding a new flow rate--pressure drop relation without fitting parameters. Our results agree favorably with, and thus rationalize, previous experiments.
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