Fluids with gravitationally stable density stratification caused by diffusing solutes are ubiquitous in nature and may spontaneously generate flows in the destabilizing presence of immersed bodies. In this paper we document the discovery of a counterintuitive phenomenon: objects in such fluids may self-induce suction forces producing near constant accelerations towards nearby walls, in the absence of any external forces. Intriguingly, the fundamental mechanism is akin to Aristotle’s ancient idea whereby motion is sustained through flows abhorring vacuum generation. Here we present an experimental, computational, and theoretical study to fully explore this new phenomenon. First, experiments exhibiting wall collapse are presented. Next, flow and density structures are measured and compared quantitatively to computational simulations with spheres and cylinders, both in free space and near symmetry-disrupting vertical walls. Further computations reveal a competition between the pressure and viscous stress forces that enable a “lubrication screening,” overcoming the resistance of a thin lubricating layer. In particular, a low pressure region in the gap develops and the particle spontaneously moves to fill the vacuum by being pushed along by ensuing flows. The resulting unexpected motion in a viscous dominated flow propels the particle almost all the way to the wall within a distance scale set by the stratified fluid properties, ultimately decelerating with a soft-landing. Lastly, extensions of these new phenomena to thin and porous geometries are discussed with theoretical and computational predictions showing how the wall-induced motion can be reversed by porosity, pushing the body away from the wall.

The vertical transport of solid material in a stratified medium is fundamental to a number of environmental applications, with implications for the carbon cycle and nutrient transport in marine ecosystems. In this work, we study the diffusion-limited settling of highly porous particles in a density-stratified fluid through a combination of experiment, analysis, and numerical simulation. By delineating and appealing to the diffusion-limited regime wherein buoyancy effects due to mass adaptation dominate hydrodynamic drag, we derive a simple expression for the steady settling velocity of a sphere as a function of the density, size, and diffusivity of the solid, as well as the density gradient of the background fluid. In this regime, smaller particles settle faster, in contrast with most conventional hydrodynamic drag mechanisms. Furthermore, we outline a general mathematical framework for computing the steady settling speed of a body of arbitrary shape in this regime and compute exact results for the case of general ellipsoids. Using hydrogels as a highly porous model system, we validate the predictions with laboratory experiments in linear stratification for a wide range of parameters. Last, we show how the predictions can be applied to arbitrary slowly varying background density profiles and demonstrate how a measured particle position over time can be used to reconstruct the background density profile.

We investigate diffusion-driven flows in a parallel-plate channel domain with linear density stratification, which arise from the combined influence of gravity and diffusion in density-stratified fluids. We compute the time-dependent diffusion-driven flows and perturbed density field using eigenfunction expansions under the Boussinesq approximation. In channel domain, the unsteady flow converges to a steady-state solution either monotonically or non-monotonically (highly oscillatory), depending on the relation between the Schmidt number and the non-dimensionalized stratified scalar diffusivity, while the flow in the half-space inclined plane problem exhibits oscillatory convergence for all parameters. To validate the Boussinesq approximation, we propose the quasi-Boussinesq approximation, which includes transverse density variation in the inertial term. Numerical solutions show that the relative difference between the Boussinesq and quasi-Boussinesq approximations is uniformly small. We also study the mixing of a passive tracer induced by the advection of the unsteady diffusion-driven flow and present the series representation of the time-dependent effective diffusion coefficient. For small Schmidt numbers, the effective diffusion coefficient induced by the unsteady flow solution can oscillate with an amplitude larger than the effective diffusion coefficient induced by the long-time-limiting steady-state flow. Interestingly, the unsteady flow solution can reduce the time-dependent effective diffusion coefficient temporally in some parameter regimes, below even that produced by pure molecular diffusion in the absence of a flow. However, at long times, the effective diffusion is significantly enhanced for large P\'eclet numbers.

Himalayan lakes represent critical water resources, culturally important waterbodies, and potential hazards. Some of these lakes experience dramatic water-level changes, responding to seasonal monsoon rains and post-monsoonal draining. To address the paucity of direct observations of hydrology in retreating mountain glacial systems, we describe a field program in a series of high altitude lakes in Sagarmatha National Park, adjacent to Ngozumba, the largest glacier in Nepal. In situ observations find extreme (>12 m) seasonal water-level changes in a 60-m deep lateral-moraine-dammed lake (lacking surface outflow), during a 16-month period, equivalent to a 5 × 10^6 m ^3 volume change annually. The water column thermal structure was also monitored over the same period. A hydraulic model is constructed, validated against observed water levels, and used to estimate hydraulic conductivities of the moraine soils damming the lake and improves our understanding of this complex hydrological system. Our findings indicate that lake level compared to the damming glacier surface height is the key criterion for large lake fluctuations, while lakes lying below the glacier surface, regulated by surface outflow, possess only minor seasonal water-level fluctuations. Thus, lakes adjacent to glaciers may exhibit very different filling/draining dynamics based on presence/absence of surface outflows and elevation relative to retreating glaciers, and consequently may have very different fates in the next few decades as the climate warms.

The application of the motion of a vertically suspended mass-spring system released under tension is studied focusing upon the delay timescale for the bottom mass as a function of the spring constants and masses. This ``hang-time", reminiscent of the Coyote and Road Runner cartoons, is quantified using the far-field asymptotic expansion of the bottom mass' Laplace transform. These asymptotics are connected to the short time mass dynamics through Tauberian identities and explicit residue calculations. It is shown, perhaps paradoxically, that this delay timescale is maximized in the large mass limit of the top ``boulder". Experiments are presented and compared with the theoretical predictions. This system is an exciting example for the teaching of mass-spring dynamics in classes on Ordinary Differential Equations, and does not require any normal mode calculations for these predictions.

We consider a scalar field governed by an advection-diffusion equation (or a more general evolution equation) with rapidly fluctuating, Gaussian distributed random coefficients. In the white noise limit, we derive the closed evolution equation for the ensemble average of the random scalar field by three different strategies, i.e., Feynman-Kac formula, the limit of Ornstein-Uhlenbeck process, and evaluating the cluster expansion of the propagator on an n-simplex. With the evolution equation of ensemble average, we study the passive scalar transport problem with two different types of flows, a random periodic flow, and a random strain flow. For periodic flows, by utilizing the homogenization method, we show that the N -point correlation function of the random scalar field satisfies an effective diffusion equation at long times. For the strain flow, we explicit compute the mean of the random scalar field and show that the statistics of the random scalar field have a connection to the time integral of geometric Brownian motion. Interestingly, all normalized moment (e.g., skewness, kurtosis) of this random scalar field diverges at long times, meaning that the scalar becomes more and more intermittent during its decay.

Diffusion-driven flow is a boundary layer flow that results from the combined influence of gravity and diffusion, which exists in density-stratified fluids whenever a gravitational field is not parallel to the solid boundary. In this paper, we investigate the unsteady diffusion-driven flows that emerge in a parallel-plate channel domain with a linear density stratification. We first compute the time-dependent diffusion-driven flows and perturbed density field using eigenfunction expansions under the Boussinesq approximation. In channel domain, the unsteady flow converges to a steady-state solution either monotonically or nonmonotonically (highly oscillatory), depending on the relation between the Schmidt number and the nondimensionalized stratified scalar diffusivity, while the flow in the half-space inclined plane problem exhibits oscillatory convergence for all parameters. To validate the Boussinesq approximation, we propose the quasi-Boussinesq approximation, which includes transverse density variation in the inertial term. Numerical solutions show that the relative difference between the Boussinesq and quasi-Boussinesq approximations is uniformly small. We also study the mixing of a passive tracer induced by the advection of the unsteady diffusion-driven flow and present the series representation of the time-dependent effective diffusion coefficient. For small Schmidt numbers, the effective diffusion coefficient induced by the unsteady flow solution can oscillate with an amplitude larger than the effective diffusion coefficient induced by the long-time-limiting steady-state flow. Interestingly, the unsteady flow solution can reduce the time-dependent effective diffusion coefficient temporally in some parameter regimes, below even that produced by pure molecular diffusion in the absence of a flow. However, at long times, the effective diffusion is significantly enhanced for large Peclet numbers.

We study the long-time behavior of an advection-diffusion equation with a general time-varying (including random) shear flow imposing no-flux boundary conditions on channel walls. We derive the asymptotic approximation of the scalar field at long times by using the center manifold theory. We carefully compare it with existing time-varying homogenization theory as well as other existing center manifold based studies and present conditions on the flows under which our new approximations give a substantial improve-ment to these existing theories. A recent study [Ding et al., Physica D 432, 133118 (2022)] has shown that Gaussian random shear flows induce a deterministic effective diffusivity at long times and explicitly calculated the invariant measure. Here, with our established asymptotic expansions, we not only concisely demonstrate those prior conclusions for Gaussian random shear flows but also generalize the conclusions regarding determinism to a much broader class of random (non-Gaussian) shear flows. Such ergodicity-like results are important since they assure an experimentalist only need to perform a single realization of a random flow to observe the ensemble moment predictions at long times. Monte Carlo simulations are presented illustrating how the highly random behavior converges to the deterministic limit at long times. Counterintuitively, we present a case demonstrating that the random flow may not induce larger dispersion than its deterministic counterpart and in turn present rigorous conditions under which a random renewing flow induces a stronger effective diffusivity. Last, for white in time flows and renewing flows, we derive the invariant measure and explore its Peclet number dependencies.

We present two accurate and efficient algorithms for solving the incompressible, irrotational Euler equations with a free surface in two dimensions with background flow over a periodic, multiply-connected fluid domain that includes stationary obstacles and variable bottom topography. One approach is formulated in terms of the surface velocity potential while the other evolves the vortex sheet strength. Both methods employ layer potentials in the form of periodized Cauchy integrals to compute the normal velocity of the free surface, are compatible with arbitrary parameterizations of the free surface and boundaries, and allow for circulation around each obstacle, which leads to multiple-valued velocity potentials but single-valued stream functions. We prove that the resulting second-kind Fredholm integral equations are invertible, possibly after a physically motivated finite-rank correction. In an angle-arclength setting, we show how to avoid curve reconstruction errors that are incompatible with spatial periodicity. We use the proposed methods to study gravity-capillary waves generated by flow around several elliptical obstacles above a flat or variable bottom boundary. In each case, the free surface eventually self-intersects in a splash singularity or collides with a boundary. We also show how to evaluate the velocity and pressure with spectral accuracy throughout the fluid, including near the free surface and solid boundaries. To assess the accuracy of the time evolution, we monitor energy conservation and the decay of Fourier modes and compare the numerical results of the two methods to each other. We implement several solvers for the discretized linear systems and compare their performance. The fastest approach employs a graphics processing unit (GPU) to construct the matrices and carry out iterations of the generalized minimal residual method (GMRES).

We study the long time behavior of an advection-diffusion equation with a random shear flow which depends on a stationary Ornstein-Uhlenbeck (OU) process in parallel-plate channels enforcing the no-flux boundary conditions. We derive a closed form formula for the long time asymptotics of the arbitrary N-point correlator using the ground state eigenvalue perturbation approach proposed in Bronski and McLaughlin (1997). In turn, appealing to the conclusion of the Hausdorff moment problem (Shohat and Tamarkin, 1943), we discover a diffusion equation with a random drift and deterministic enhanced diffusion possessing the exact same probability density function at long times. The strategy we presented is not only restricted to the parallel-plate channel domain. The same methods can derive effective equations for a straight channel with uniform arbitrary cross-section. Such equations enjoy many ergodic properties which immediately translate to ergodicity results for the original problem. In particular, we establish that the first two Aris moments using a single realization of the random field can be used to explicitly construct all ensemble averaged moments. Also, the first two ensemble averaged moments explicitly predict any long time centered Aris moment. Our formulae quantitatively depict the dependence of the deterministic effective diffusion on the interaction between spatial structure of flow and random temporal fluctuation. Further, this approximation provides many identities regarding the stationary OU process dependent time integral. We derive explicit formulae for the decaying passive scalar's long time limiting probability density function (PDF) for different types of initial conditions (e.g. deterministic and random). (C) 2021 Published by Elsevier B.V.

We study the motion of a rigid sphere falling in a two-layer stratified fluid under the action of gravity in the potential flow regime. Experiments at a moderate Reynolds number of approximately 20 to 450 indicate that a sphere with the precise critical density, higher than the bottom layer density, can display behaviors such as bounce or arrestment after crossing the interface. We experimentally demonstrate that such a critical sphere density increases linearly as the bottom fluid density increases with a fixed top fluid density. Additionally, the critical density approaches the bottom layer fluid density as the thickness of density transition layer increases. We propose an estimation of the critical density based on the potential energy. With assuming the zero layer thickness, the estimation constitutes an upper bound of the critical density with less than 0.043 relative difference within the experimental density regime 0.997 g/cm^3 ∼ 1.11 g/cm^3 under the zero layer thickness assumption. By matching the experimental layer thickness, we obtain a critical density estimation with less than 0.01 relative difference within the same parameter regime.

We develop a theory of enhanced diffusivity and skewness of the longitudinal distribution of a diffusing tracer advected by a periodic time-varying shear flow in a straight channel. Although applicable to any type of solute and fluid flow, we restrict the examples of our theory to the tracer advected by flows which are induced by a periodically oscillating wall in a Newtonian fluid between two infinite parallel plates as well as flow in an infinitely long duct. These wall motions produce the well-known Stokes layer shear solutions which are exact solutions of the Navier–Stokes equations. With these, we first calculate the second Aris moment for all time and its long-time limiting effective diffusivity as a function of the geometrical parameters, frequency, viscosity, and diffusivity. Using a new formalism based upon the Helmholtz operator, we establish a new single series formula for the variance valid for all time. We show that the viscous dominated limit results in a linear shear layer for which the effective diffusivity is bounded with upper bound $$\kappa (1+A^2/(2L^2))$$ , where $$\kappa $$ is the tracer diffusivity, A is the amplitude of oscillation, and L is the gap thickness. Alternatively, for finite viscosities, we show that the enhanced diffusion is unbounded, diverging in the high-frequency limit. Non-dimensionalization and physical arguments are given to explain these striking differences. Asymptotics for the high-frequency behavior as well as the low viscosity limit are computed. We present a study of the effective diffusivity surface as a function of the non-dimensional parameters which shows how a maximum can exists for various parameter sweeps. Physical experiments are performed in water using particle tracking velocimetry to quantitatively measure the fluid flow. Using fluorescein dye as the passive tracer, we document that the theory is quantitatively accurate. Specifically, image analysis suggests that the distribution variance be measured using the full width at half maximum statistic which is robust to noise. Further, we show that the scalar skewness is zero for linear shear flows at all times, whereas for the nonlinear Stokes layer, exact analysis shows that the skewness sign can be controlled through the phase of the oscillating wall. Further, for single-frequency wall modes, we establish that the long-time skewness decays at the faster rate of $$t^{-3/2}$$ as compared with steady shear scalar skewness which decays at rate $$t^{-1/2}$$ . These results are confirmed using Monte-Carlo simulations.

In this paper, we study the effect of impermeable boundaries on the symmetry properties of a random passive scalar field advected by random flows. We focus on a broad class of nonlinear shear flows multiplied by a stationary, Ornstein-Uhlenbeck (OU) time varying process, including some of their limiting cases, such as Gaussian white noise or plug flows. For the former case with linear shear, recent studies (Camassa et al., 2019) numerically demonstrated that the decaying passive scalar's long time limiting probability distribution function (PDF) could be negatively skewed in the presence of impermeable channel boundaries, in contrast to rigorous results in free space which established the limiting PDF is positively skewed (McLaughlin and Majda, 1996). Here, the role of boundaries in setting the long time limiting skewness of the PDF is established rigorously for the above class using the long time asymptotic expansion of the N-point correlator of the random field obtained from the ground state eigenvalue perturbation approach proposed in Bronski and McLaughlin (1997). Our analytical result verifies the conclusion for the linear shear flow obtained from numerical simulations in Camassa et al. (2019). Moreover, we demonstrate that the limiting distribution is negatively skewed for any shear flow at sufficiently low Peclet number. We demonstrate the convergence of the Ornstein-Uhlenbeck case to the white noise case in the limit gamma -> infinity of the OU damping parameter, which generalizes the results for free space in Resnick (1996) to the channel domain problem. We show that the long time limit of the first three moments depends explicitly on the value of gamma, which is in contrast to the conclusion in Vanden Eijnden (2001) for the limiting PDF in free space. To find a benchmark for theoretical analysis, we derive the exact formula of the N-point correlator for a flow with no spatial dependence and Gaussian temporal fluctuation, generalizing the results of Bronski et al. (2007). The long time analysis of this formula is consistent with our theory for a general shear flow. All results are verified by Monte-Carlo simulations. (C) 2021 Elsevier B.V. All rights reserved.

1. Ferry, J.D., Sawyer, W., Ashworth, J.: Behavior of concentrated polymer solutions under periodic stresses. Journal of Polymer Science 2(6), 593{611 (1947) 2. Drazin, P.G., Riley, N.: The Navier-Stokes equations: a classification of flows and exact solutions. 334. Cambridge University Press (2006) 3. Ding, L., Hunt, R., Woodie, H., McLaughlin, R.M., 2020. Enhanced diffusivity and skewness of a diffusing tracer in the presence of an oscillating wall. arXiv:2008.10717 The flow induced by the wall motion was originally presented by Ferry[1,2]. The figure below shows the comparison of Particle Tracking Velocimetry (PTV) data with the Ferry wave analytical solution. Each slice corresponds with a time series of the shear velocity over a duration of one period taken at different distances between the fixed wall and the moving wall with gap thickness L = 0.16 cm. The velocity of the wall motion takes the form v=A ꞷ cos(ꞷt). Left panel has wall oscillation amplitude A = 2 cm, right panel has A = 1 cm, other parameters: frequency ꞷ = 2π ∗ 0:01/s, dynamic viscosity ν = 0.0113 poise, and L = 0.16 cm. We present a study of shear dispersion. Here we consider the diffusing tracer advected by flows which are induced by a periodically oscillating wall in a linear stratified sodium chloride solution fluid between two infinite parallel walls. The figure below shows a schematic of the experimental setup from three different views.

We study the long time behavior of an advection-diffusion equation with a random shear flow which depends on a stationary Ornstein-Uhlenbeck (OU) process in parallel-plate channels enforcing the no-flux boundary conditions. We derive a closed form formula for the long time asymptotics of the arbitrary $N$-point correlator using the ground state eigenvalue perturbation approach proposed in \cite{bronski1997scalar}. In turn, appealing to the conclusion of the Hausdorff moment problem \cite{shohat1943problem}, we discover a diffusion equation with a random drift and deterministic enhanced diffusion possessing the exact same probability distribution function at long times. Such equations enjoy many ergodic properties which immediately translate to ergodicity results for the original problem. In particular, we establish that the first two Aris moments using a single realization of the random field can be used to explicitly construct all ensemble averaged moments. Also, the first two ensemble averaged moments explicitly predict any long time centered Aris moment. Our formulae quantitatively depict the dependence of the deterministic effective diffusion on the interaction between spatial structure of flow and random temporal fluctuation. This noteably contrasts the white noise case from the OU case. Further, this approximation provides many identities regarding the stationary OU process dependent time integral. We derive explicit formulae for the decaying passive scalar's long time limiting probability distribution function (PDF) for different types of initial conditions (e.g. deterministic and random). All results are verified by Monte-Carlo simulations.

An extremely broad and important class of phenomena in nature involves the settling and aggregation of matter under gravitation in fluid systems. Here, we observe and model mathematically an unexpected fundamental mechanism by which particles suspended within stratification may self-assemble and form large aggregates without adhesion. This phenomenon arises through a complex interplay involving solute diffusion, impermeable boundaries, and aggregate geometry, which produces toroidal flows. We show that these flows yield attractive horizontal forces between particles at the same heights. We observe that many particles demonstrate a collective motion revealing a system which appears to solve jigsaw-like puzzles on its way to organizing into a large-scale disc-like shape, with the effective force increasing as the collective disc radius grows. Control experiments isolate the individual dynamics, which are quantitatively predicted by simulations. Numerical force calculations with two spheres are used to build many-body simulations which capture observed features of self-assembly.

We consider the evolution of a decaying passive scalar in the presence of a Gaussian white noise fluctuating linear shear flow known as the Majda Model. We focus on deterministic initial data and establish the short, intermediate, and long time symmetry properties of the evolving point wise probability measure (PDF) for the random passive scalar. We identify, for the cases of both point source and line source initial data, regions in the x–yplane outside of which the PDF skewness is sign definite for all time, while inside these regions we observe multiple sign changes corresponding to exchanges in symmetry between hot and cold leaning states using exact representation formula for the PDF at the origin, and away from the origin, using numerical evaluation of the exact available Mehler kernel formulae for the scalar’s statistical moments. A new, rapidly convergent Monte-Carlo method is developed, dubbed Direct Monte-Carlo (DMC), using the available random Green’s functions which allows for the fast construction of the PDF for single point statistics, as well as multi-point statistics including spatially integrated quantities natural for full Monte-Carlo simulations of the underlying stochastic differential equations (FMC). This new method demonstrates the full evolution of the PDF from short times, to its long time, limiting and collapsing universal distribution at arbitrary points in the plane. Further, this method provides a strong benchmark for FMC and we document numbers of field realization criteria for the FMC to faithfully compute this complete dynamics. Armed with this benchmark, we apply the FMC to a channel with a no-flux boundary condition enforced on parallel planes and observe a dramatically different long time state resulting from the existence of the wall. In particular, the channel case collapsing invariant measure has negative skewness, with random states leaning heavily towards the hot state, in stark contrast to free space, where the limiting skewness is positive, with its states leaning heavily towards the cold state.