In the context of scalar transport and mixing in fluid dynamics, a scalar refers to a quantity that has only magnitude and no direction, which can vary from point to point within a fluid. Common examples of scalar quantities in fluid dynamics include temperature, concentration of a chemical substance, or density. These scalars are termed passive when their values do not influence the flow field directly—though, in practice, variations in temperature and concentration can impact fluid properties like viscosity and density, subsequently affecting the flow. This simplification, however, is useful for modeling purposes. The study of passive scalar transport and mixing is vital across numerous scientific and engineering disciplines, including the pharmaceutical and food industries, oceanography, atmospheric sciences, and civil engineering. This understanding aids in addressing challenges related to pollutant dispersion, chemical mixing, and thermal management. 

The passive scalar is typically governed by the advection-diffusion equation. A notable study by G.I. Taylor (1953) analyzed a solute advected by steady, pressure-driven flow in a circular pipe. He demonstrated that, at long times, the scalar distribution could be approximated by a one-dimensional diffusion equation with an enhanced constant effective diffusion coefficient, known as Taylor dispersion. This transformation from the original three-dimensional governing equation to the simplified one-dimensional equation reduces the number of independent variables from four to two. This exemplifies a fundamental mathematical technique: simplifying complex systems by eliminating "fast modes." My objective is to derive more precise reduced models of scalar dynamics, applicable in both deterministic and stochastic flow contexts.

The first case we considered is deterministic time-varying shear flows the result of random flow is discussed in the section of the scalar intermittency. We developed a theory on the effective diffusivity and skewness of the longitudinal distribution of a diffusing tracer advected a periodic time-varying shear flow by 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.  We identified optimal parameters for maximizing mixing, such as temperature, the frequency and phase of wall motion, tracer diffusivity, and fluid viscosity. These parameters not only enhance the mixing rate but also enable control over the concentration distribution by adjusting the phase of the wall motion, affecting the skewness sign. To experimentally validate our theory, we constructed a device featuring a computer-controlled robotic arm designed to generate the prescribed wall motion. This setup provides a practical and realizable experimental framework not only for studying time-varying random flows but also for exploring scalar intermittency, which concerns the statistics of scalars inherited from random flows, such as those found in turbulent environments.

Reference

1. *Lingyun Ding and Richard M McLaughlin. Correlation function of a random scalar field evolving with a rapidly fluctuating gaussian process. Journal of Statistical Physics, 190(12):201, 2023a. doi: https://doi.org/10.1007/s10955-023-03191-7

2. *Lingyun Ding and Richard M McLaughlin. Determinism and invariant measures for diffusing passive scalars advected by unsteady random shear flows. Physical Review Fluids, 7(7):074502, 2022a. doi: https://doi.org/10.1103/PhysRevFluids.7.074502

3. *Lingyun Ding and Richard M McLaughlin. 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. Physica D: Nonlinear Phenomena, 432:133118, 2022b. doi: https://doi.org/10.1016/j.physd.2021.133118

4. *Roberto Camassa, Lingyun Ding, Zeliha Kilic, and Richard M McLaughlin. Persisting asymmetry in the probability distribution function for a random advection–diffusion equation in impermeable channels. Physica D: Nonlinear Phenomena, 425:132930, 2021a. doi: https://doi.org/10.1016/j.physd.2021.132930

Particle-laden flows involve solid particles suspended in a fluid medium, a crucial phenomenon in both natural and industrial settings. These multiphase flows are central to various applications, from sediment transport in environmental science to process optimization in chemical manufacturing. The complex interactions between particles and fluid significantly influence flow properties and behavior, making their study essential for improving systems reliant on effective particle transport.

Diffusion-driven flow is an intriguing phenomenon resulting from molecular diffusion. In density-stratified fluids, such as oceans, whenever the impermeable solid boundary is non-parallel to the gravitational field, a flow emerges driven by the buoyancy force, ensuring that the density field satisfies the no-flux boundary condition.  Diffusion-driven flow has significant implications in various fields. In oceanography, the salinity-induced density stratification along the continental shelf promotes upwelling flow, crucial for the vertical exchange of oceanic properties. Similarly, fluids in narrow fissures can stratify due to non-uniform scalar distributions or vertical temperature gradients, such as Earth's geothermal gradient, causing significant solute dispersion over geological timescales. Additionally, when a wedge-shaped object is immersed in a stratified fluid, the resulting flow can propel the object forward, with such phenomena further explored in multiple studies. This flow mechanism also plays a role in particle attraction and self-assembly in stratified fluids and can measure molecular diffusivity. Lastly, it is involved in layer formation in double-diffusive systems due to insulating sloping boundaries.

We explore dispersion induced by diffusion-driven flow, initially examining the mixing of passive scalars advected by unsteady flows in parallel-plate channels with linear density stratification. We provide series representations for velocity and density fields, highlighting distinct oscillatory behaviors in bounded and semi-infinite domains. Additionally, we analyze passive tracer mixing and present an explicit expression for the time-dependent effective diffusion coefficient, which oscillates with an amplitude exceeding its steady-state value. Notably, the unsteady flow can temporarily reduce the effective diffusion coefficient below pure molecular diffusion, though it is significantly enhanced over longer periods. This research has been highlighted as an "Editors' Suggestion" in Physical Review Fluids.

Further, I investigate diffusion-driven flow in nonlinear density-stratified fluids, demonstrating that classical power series expansions fail to provide accurate approximations across a broad parameter range. I propose a novel asymptotic expansion method inspired by center manifold theory, which uses derivatives of the cross-sectional averaged density field for expanding velocity, density, and pressure fields. This method also leads to an evolution equation for the averaged density field, which resembles the traditional diffusion equation but with a diffusion coefficient replaced by a positive-definite function dependent on the derivative of the solution. This approach offers a robust framework for modeling slow manifold phenomena, such as convection in curved pipes and thin film fluids.

Many research works have demonstrated that, even at relatively modest volume fractions of particles, the presence of these particles significantly influences the flow dynamics, leading to distinct behaviors based on the volume fraction. Consequently, it is imperative to quantitatively model the interaction between the fluid and particles. Such models fall into two broad categories: discrete and continuum. Discrete models, which track individual particles, are relatively accurate but computationally intensive. On the other hand, continuum models treat fluid and particles as separate continuous phases, each governed by coupled equations. Our study consists of significant particle concentrations, making continuum models the preferred approach.

Two widely used continuum models for analyzing particle suspensions are the diffusive-flux model and the 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 a 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 system of coupled 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.

In the Research experience for undergraduate student program (REU) 2023 at UCLA, Sarah Burnett and I mentored four undergraduate students, Jonathan Woo, Wing Pok Lee, Luke Triplett, Yifan Gu on this project. After REU, they continued on the project and presented their results at the American Physical Society Annual Meeting Division of Fluid Dynamics. With the support from professor Andrea Bertozzi, we finally finished a paper on this topic, which is titled “Acomparative study of dynamic models for gravity-driven particle-laden flows” and is published on Applied Mathematics Letter.

1. Richard Mclaughlin 

2. Roberto Camassa

3. Andrea Bertozzi 

4. Marcus Roper

5. Qinghai Zhang

6. Zhi Lin

7. Jingfang Huang

8. Jeremy L. Marzuola

9. Sarah Cassie Burnett   

10. Francesca Bernardi