arXiv subjects

Lingyun Ding

Publications and source records attributed to Lingyun Ding.

16 recordsLinked to original sources

arXiv2025

Thin gap approximations for microfluidic device design

Over 125 years ago, Henry Selby Hele-Shaw realized that the depth-averaged flow in thin gap geometries can be closely approximated by two-dimensional (2D) potential flow, in a surprising marriage between the theories of viscous-dominated and inviscid flows. Hele-Shaw approximation allows visualization of potential flows over 2D airfoils and also undergirds important discoveries in the dynamics of interfacial instabilities and convection, yet it has found little use in modeling flows in microfluidic devices, although these devices often have thin gap geometries. Here, we derive a Hele-Shaw approximation for the flow in the kinds of thin gap geometries created within microfluidic devices. Using the Method of Weighted Residuals (MWR), we reinterpret the Hele-Shaw approximation as the leading term of an orthogonal polynomial expansion that can be systematically extended to higher-order corrections. The resulting leading-order equation coincides with the previously derived 2D approximations, but our derivation is shorter and more direct. By extending the expansion beyond leading order, we obtain a new reduced model that captures non-parabolic gap-wise velocity profiles and out-of-plane flow effects. We provide substantial numerical evidence showing that approximate equations can successfully model real microfluidic and inertial-microfluidic device geometries. By reducing three-dimensional (3D) flows to 2D models, our validated model will allow for accelerated device modeling and design.

physics.flu-dyn↗

arXiv2025

Lagrangian Particle Classification and Lagrangian Flux Identities for a Moving Hypersurface

For a moving hypersurface in the flow of a nonautonomous ordinary differential equation in $n$-dimensional Euclidean spaces, the fluxing index of a passively-advected Lagrangian particle is the total number of times it crosses the moving hypersurface within a time interval. The problem of Lagrangian particle classification is to decompose the phase space into flux sets, equivalence classes of Lagrangian particles at the initial time. In the context of scalar conservation laws, the problem of Lagrangian flux calculation (LFC) is to find flux identities that relate the Eulerian flux of a scalar through the moving hypersurface, a spatiotemporal integral over the moving surface in a given time interval, to spatial integrals over donating regions at the initial time of the interval. In this work, we implicitly characterize flux sets via topological degrees, explicitly construct donating regions, prove the equivalence of flux sets and donating regions, and establish two flux identities; these analytical results constitute our solutions to the aforementioned problems. Based on a flux identity suitable for numerical calculation, we further proposed a new LFC algorithm, proved its convergence, and demonstrated its efficiency, good conditioning, and high-order accuracy by results of various numerical tests.

math.NA↗

arXiv2025

Long-Time Asymptotics of Passive Scalar Transport in Periodically Modulated Channels

This work investigates the long-time asymptotic behavior of a diffusing passive scalar advected by fluid flow in a straight channel with a periodically varying cross-section. The goal is to derive an asymptotic expansion for the scalar field and estimate the timescale over which this expansion remains valid, thereby generalizing Taylor dispersion theory to periodically modulated channels. By reformulating the eigenvalue problem for the advection-diffusion operator on a unit cell using a Floquet-Bloch-type eigenfunction expansion, we extend the classical Fourier integral of the flat-channel problem to a periodic setting, yielding an integral representation of the scalar field. This representation reveals a slow manifold that governs the algebraically decaying dynamics, while the difference between the scalar field and the slow manifold decays exponentially in time. Building on this, we derive a long-time asymptotic expansion of the scalar field. We show that the validity timescale of the expansion is determined by the real part of the eigenvalues of a modified advection-diffusion operator, which depends solely on the flow and geometry within a single unit cell. This framework offers a rigorous and systematic method for estimating mixing timescales in channels with complex geometries. We show that non-flat channel boundaries tend to increase the timescale, while transverse velocity components tend to decrease it. The approach developed here is broadly applicable and can be extended to derive long-time asymptotics for other systems with periodic coefficients or periodic microstructures.

physics.flu-dyn↗

arXiv2025

Semi-implicit-explicit Runge-Kutta method for nonlinear differential equations

A semi-implicit-explicit (semi-IMEX) Runge-Kutta (RK) method is proposed for the numerical integration of ordinary differential equations (ODEs) of the form $\mathbf{u}' = \mathbf{f}(t,\mathbf{u}) + G(t,\mathbf{u}) \mathbf{u}$, where $\mathbf{f}$ is a non-stiff term and $G\mathbf{u}$ represents the stiff terms. Such systems frequently arise from spatial discretizations of time-dependent nonlinear partial differential equations (PDEs). For instance, $G$ could involve higher-order derivative terms with nonlinear coefficients. Traditional IMEX-RK methods, which treat $\mathbf{f}$ explicitly and $G\mathbf{u}$ implicitly, require solving nonlinear systems at each time step when $G$ depends on $\mathbf{u}$, leading to increased computational cost and complexity. In contrast, the proposed semi-IMEX scheme treats $G$ explicitly while keeping $\mathbf{u}$ implicit, reducing the problem to solving only linear systems. This approach eliminates the need to compute Jacobians while preserving the stability advantages of implicit methods. A family of semi-IMEX RK schemes with varying orders of accuracy is introduced. Numerical simulations for various nonlinear equations, including nonlinear diffusion models, the Navier-Stokes equations, and the Cahn-Hilliard equation, confirm the expected convergence rates and demonstrate that the proposed method allows for larger time step sizes without triggering stability issues.

math.NA↗

arXiv2024

A comparative study of dynamic models for gravity-driven particle-laden flows

The dynamics of viscous thin-film particle-laden flows down inclined surfaces are commonly modeled with one of two approaches: a diffusive flux model or a suspension balance model. The diffusive flux model assumes that the particles migrate via a diffusive flux induced by gradients in both the particle concentration and the effective suspension viscosity. The suspension balance model introduces non-Newtonian bulk stress with shear-induced normal stresses, the gradients of which cause particle migration. Both models have appeared in the literature of particle-laden flow with virtually no comparison between the two models. For particle-laden viscous flow on an incline, in a thin-film geometry, one can use lubrication theory to derive a compact dynamic model in the form of a $2\times 2$ system of conservation laws. We can then directly compare the two theories side by side by looking at similarities and differences in the flux functions for the conservation laws, and in exact and numerical simulations of the equations. We compare the flux profiles over a range of parameters, showing fairly good agreement between the models, with the biggest difference involving the behavior at the free surface. We also consider less dense suspensions at lower inclination angles where the dynamics involve two shock waves that can be clearly measured in experiments. In this context the solutions differ by no more than about 10%, suggesting that either model could be used for this configuration.

physics.flu-dyn↗

arXiv2024

Equilibrium theory of bidensity particle-laden suspensions in thin-film flow down a spiral separator

Spiral gravity separators are designed to separate multi-species slurry components based on differences in density and size. Previous studies have investigated steady-state solutions for mixtures of liquids and single particle species in thin-film flows. However, these models are constrained to single-species systems and cannot describe the dynamics of multi-species separation. In contrast, our analysis extends to mixtures containing two particle species of differing densities, revealing that they undergo radial separation, which is an essential mechanism for practical applications in separating particles of varying densities. This work models gravity-driven bidensity slurries in a spiral trough by incorporating particle interactions, using empirically derived formulas for particle fluxes from previous bidensity studies on inclined planes. Specifically, we study a thin-film bidensity slurry flowing down a rectangular channel helically wound around a vertical axis. Through a thin-film approximation, we derive equilibrium profiles for the concentration of each particle species and the fluid depth. Additionally, we analyze the influence of key design parameters, such as spiral radius and channel width, on particle concentration profiles. Our findings provide valuable insights into optimizing spiral separator designs for enhanced applicability and adaptability.

physics.flu-dyn↗

arXiv2024

Diffusion-driven flows in a non-linear stratified fluid layer

Diffusion-driven flow is a boundary layer flow arising from the interplay of gravity and diffusion in density-stratified fluids when a gravitational field is non-parallel to an impermeable solid boundary. This study investigates diffusion-driven flow within a nonlinearly density-stratified fluid confined between two tilted parallel walls. We introduce an asymptotic expansion inspired by the center manifold theory, where quantities are expanded in terms of derivatives of the cross-sectional averaged stratified scalar (such as salinity or temperature). This technique provides accurate approximations for velocity, density, and pressure fields. Furthermore, we derive an evolution equation describing the cross-sectional averaged stratified scalar. This equation takes the form of the traditional diffusion equation but replaces the constant diffusion coefficient with a positive-definite function dependent on the solution's derivative. Numerical simulations validate the accuracy of our approximations. Our investigation of the effective equation reveals that the density profile depends on a non-dimensional parameter denoted as $γ$ representing the flow strength. In the large $γ$ limit, the system is approximated by a diffusion process with an augmented diffusion coefficient of $1+\cot^{2}θ$, where $θ$ signifies the inclination angle of the channel domain. This parameter regime is where diffusion-driven flow exhibits its strongest mixing ability. Conversely, in the small $γ$ regime, the density field behaves like pure diffusion with distorted isopycnals. Lastly, we show that the classical thin film equation aligns with the results obtained using the proposed expansion in the small $γ$ regime but fails to accurately describe the dynamics of the density field for large $γ$.

physics.flu-dyn↗

arXiv2024

Optimal Stirring Strategies for Passive Scalars in a Domain with a General Shape and No-Flux Boundary Condition

Multiscale metrics such as negative Sobolev norms are effective for quantifying the degree of mixedness of a passive scalar field advected by an incompressible flow in the absence of diffusion. In this paper we introduce a mix norm that is motivated by Sobolev norm $H^{-1}$ for a general domain with a no-flux boundary. We then derive an explicit expression for the optimal flow that maximizes the instantaneous decay rate of the mix norm under fixed energy and enstrophy constraints. Numerical simulations indicate that the mix norm decays exponentially or faster for various initial conditions and geometries and the rate is closely related to the smallest non-zero eigenvalue of the Laplace operator. These results generalize previous findings restricted for a periodic domain for its analytical and numerical simplicity. Additionally, we observe that periodic boundaries tend to induce a faster decay in mix norm compared to no-flux conditions under the fixed energy constraint, while the comparison is reversed for the fixed enstrophy constraint. In the special case of even initial distributions, two types of boundary conditions yield the same optimal flow and mix norm decay.

math.OC↗

arXiv2023

Shear dispersion of multispecies electrolyte solutions in the channel domain

In multispecies electrolyte solutions, even in the absence of an external electric field, differences in ion diffusivities induce an electric potential and generate additional fluxes for each species. This electro-diffusion process is well-described by the advection-Nernst-Planck equation. This study aims to analyze the long-time behavior of the governing equation under electroneutrality and zero current conditions and investigate how the diffusion-induced electric potential and shear flow enhance the effective diffusion coefficients of each species in channel domains. The some exact solutions of the effective equation and the asymptotic analyses for ions with large diffusivity discrepancies are presented. Furthermore, there are several interesting properties of the effective equation. First, it is a generalization of the Taylor dispersion, with a nonlinear diffusion tensor replacing the scalar diffusion coefficient. Second, the effective equation exhibits a scaling relation, revealing that the system with a weak flow is equivalent to the system with a strong flow under scaled physical parameters. Third, in the case of injecting a lower-concentration electrolyte solution into a pre-existing solution with the same ion species, the effective equation simplifies to a multidimensional diffusion equation. However, when introducing the electrolyte solution into a channel filled with deionized water, the ion-electric interaction results in several phenomena do not present in the advection-diffusion equation, including upstream migration of some species, spontaneous separation of ions, and non-monotonic dependence of the effective diffusivity on Péclet numbers. Last, the dependence of effective diffusivity on concentration and ion diffusivity suggests a method to infer the concentration ratio of each component and ion diffusivity by measuring the effective diffusivity.

physics.flu-dyn↗

arXiv2023

Enhanced diffusivity and skewness of a diffusing tracer in the presence of an oscillating wall

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 general fluid flow, we restrict the examples of our theory to the tracer advected by flows that 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. We first derived the formula of the flow produced by the wall motions. Second, we 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. We show that the viscous-dominated limit results in a linear shear layer for which the effective diffusivity is bounded. For finite viscosities, the enhanced diffusion diverges in the high-frequency limit. We present a study of the effective diffusivity surface as a function of the non-dimensional parameters which shows how a maximum can exist 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 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, the skewness sign can be controlled through the oscillating phase. Last, for single-frequency wall motion, the long-time skewness decays at the faster rate as compared with the case with steady flow.

physics.flu-dyn↗

arXiv2023

Dispersion induced by unsteady diffusion-driven flow in parallel-plate channel

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 Péclet numbers.

physics.flu-dyn↗

arXiv2023

Critical density triplets for the arrestment of a sphere falling in a sharply stratified fluid

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 $g/cm^{3}$ $\sim $ 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.

physics.flu-dyn↗

arXiv2022

Quadrature by Two Expansions for Evaluating Helmholtz Layer Potentials

In this paper, a Quadrature by Two Expansions (QB2X) numerical integration technique is developed for the single and double layer potentials of the Helmholtz equation in two dimensions. The QB2X method uses both local complex Taylor expansions and plane wave type expansions to achieve a resulting representation which is numerically accurate for all target points (interior, exterior, or exactly on the boundary) inside a leaf box in the fast multipole method (FMM) hierarchical tree structure. Compared to the original Quadrature by Expansion (QBX) method, the QB2X method explicitly includes the nonlinearity from the boundary geometry in the plane wave expansions, thereby providing for higher order representations of both the boundary geometry and density functions in the integrand, with its convergence following standard FMM error analysis. Numerical results are presented to demonstrate the performance of the QB2X method for Helmholtz layer potentials and its comparison with the original QBX method for both flat and curved boundaries with various densities. The QB2X method overcomes the challenges of the original QBX method, and is better suited for efficient solutions of the Helmholtz equation with complex geometries.

math.NA↗

arXiv2022

Correlation function of a random scalar field evolving with a rapidly fluctuating Gaussian process

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.

math.AP↗

arXiv2020

Ergodicity and invariant measures for a diffusing passive scalar advected by a random channel shear flow and the connection between the Kraichnan-Majda model and Taylor-Aris Dispersion

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.

math.AP↗

arXiv2019

Quadrature by Two Expansions: Evaluating Laplace Layer Potentials using Complex Polynomial and Plane Wave Expansions

The recently developed quadrature by expansion (QBX) technique accurately evaluates the layer potentials with singular, weakly or nearly singular, or even hyper singular kernels in the integral equation reformulations of partial differential equations. The idea is to form a local complex polynomial or partial wave expansion centered at a point away from the boundary to avoid the singularity in the integrand, and then extrapolate the expansion at points near or even exactly on the boundary. In this paper, in addition to the local complex Taylor polynomial expansion, we derive new representations of the Laplace layer potentials using both the local complex polynomial and plane wave expansions. Unlike in the QBX, the local complex polynomial expansion in the new quadrature by two expansions (QB2X) method only collects the far-field contributions and its number of expansion terms can be analyzed using tools from the classical fast multipole method. The plane wave type expansion in the QB2X method better captures the layer potential features near the boundary. It is derived by applying the Fourier extension technique to the density and boundary geometry functions and then analytically utilizing the Residue Theorem for complex contour integrals. The internal connections of the layer potential with its density function and curvature on the boundary are explicitly revealed in the plane wave expansion and its error is bounded by the Fourier extension errors. We present preliminary numerical results to demonstrate the accuracy of the QB2X representations and to validate our analysis.

math.NA↗