Session D22: General Fluid Dynamics and Fluid Instability: Theory

Morphing Continuum Theory: A First Order Approximation to the Balance Laws

Morphing Continuum Theory is constructed under the framework of Rational Continuum Mechanics (RCM) for fluid flows with inner structure. This multiscale theory has been successfully emplyed to model turbulent flows. The framework of RCM ensures the mathematical rigor of MCT, but contains new material constants related to the inner structure. The physical meanings of these material constants have yet to be determined. Here, a linear deviation from the zeroth-order Boltzmann-Curtiss distribution function is derived. When applied to the Boltzmann-Curtiss equation, a first-order approximation of the MCT governing equations is obtained. The integral equations are then related to the appropriate material constants found in the heat flux, Cauchy stress, and moment stress terms in the governing equations. These new material properties associated with the inner structure of the fluid are compared with the corresponding integrals, and a clearer physical interpretation of these coefficients emerges. The physical meanings of these material properties is determined by analyzing previous results obtained from numerical simulations of MCT for compressible and incompressible flows. The implications for the physics underlying the MCT governing equations will also be discussed.   

The second-order description of rotational non-equilibrium effects in polyatomic gases

The conventional description of gases is based on the physical laws of conservation (mass, momentum, and energy) in conjunction with the first-order constitutive laws, the two-century old so-called Navier-Stokes-Fourier (NSF) equation based on a critical assumption made by Stokes in 1845 that the bulk viscosity vanishes. While the Stokes' assumption is certainly legitimate in the case of dilute monatomic gases, ever increasing evidences, however, now indicate that such is not the case, in particular, in the case of polyatomic gases---like nitrogen and carbon dioxide---far-from local thermal equilibrium. It should be noted that, from room temperature acoustic attenuation data, the bulk viscosity for carbon dioxide is three orders of magnitude larger than its shear viscosity. In this study, this fundamental issue in compressible gas dynamics is revisited and the second-order constitutive laws are derived by starting from the Boltzmann-Curtiss kinetic equation. Then the topology of the second-order nonlinear coupled constitutive relations in phase space is investigated. Finally, the shock-vortex interaction problem where the strong interaction of two important thermal (translational and rotational) non-equilibrium phenomena occurs is considered in order to highlight the rotational non-equilibrium effects in polyatomic gases.   

The relativistic binary mixture revisited.

Relativistic plasmas are relevant in both theoretical and experimental scenarios. The transport equations for dissipative relativistic mixtures for very large temperatures are not completely understood. In this work we use kinetic theory in order to analyze critically the establishment of the balance equations, and propose a novel formalism involving explicitly the concept of peculiar velocity with respect of the barycentric frame. The results are discussed emphasizing the advantages compared to other existing approaches to the problem.   

Modelling transport of magnetic particles across a liquid-liquid interface

Microfluidic technologies have facilitated the self-assembly of a variety of particle clusters with enhanced control. Magnetic particles have the unique advantage of allowing precise control of their motion through a non-invasive mechanism (by tuning the external magnetic force-field). Recent work has utilized the interface formed between immiscible liquid phases to enable such self-assembly. Here we consider a microfluidic set-up consisting of two fluids with different surface tensions and viscosities. Magnetic particles are introduced into one of the fluids and the external magnetic field pulls the particles through the interface into the second fluid. We analyses the features of the magnetic particle transport through the liquid--liquid interface for two different interface regimes: non-deformable, no-slip interfaces, and deformable, slip interfaces. The results of the model allow for tuning of the magnetic field and interfacial tension to facilitate a route for the formation of aggregates of a desired size.   

The complex fluid dynamics of simple diffusion.

Diffusion as the mass transport process responsible for mixing fluids at the atomic level is often underestimated in its complexity. An initial discontinuity between two species of different atomic masses exhibits a mass density discontinuity under isothermal pressure equilibrium implying equal species molar densities. The self-consistent kinetic transport processes across such an interface leads to a zero sum of mass flux relative to the center of mass and so diffusion alone cannot relax an initially stationary mass discontinuity nor broaden the density profile at the interface. The diffusive mixing leads to a molar imbalance which drives a center of mass velocity which moves the heavier species toward the lighter species leading to the interfacial density relaxation. Simultaneously, the species non-zero molar flux modifies the pressure profile in a transient wave and in a local perturbation. The resulting center of mass velocity has two components; one, associated with the divergence of the flow, persists in the diffusive mixing region throughout the diffusive mixing process, and two, travelling waves at the front of the pressure perturbations propagate away from the mixing region. The momentum in these waves is necessary to maintain momentum conservation in the center of mass frame. Thus, in a number of ways, the diffusive mixing provides feedback into the small scale advective motions. Numerical methods which diffuse all species assuming P-T equilibrium may not recover the subtle dynamics of mass transport at an interface.   

Stability and structure of fields in a flow with a hydrodynamic discontinuity

We consider from a far field the evolution of a hydrodynamic discontinuity separating incompressible ideal fluids of different densities, with mass flow across this interface. By solving the boundary value problem and finding fundamental solutions of linearized dynamics, we directly link interface stability to structure of the flow fields. We find that the classic Landau system of equations for the Landau-Darrieus instability has a degenerate and singular character. Eliminating this degeneracy leads to appearance of a neutrally stable solution whose vortical field can seed the instability. We further find that the interface is stable if the flux of energy fluctuations produced by the perturbed interface is small compared to the flux of specific kinetic energy across the planar interface. The interface is unstable if the energy fluctuations flux is large compared to the kinetic energy flux. Landau’s solution is consistent with the latter case. Keywords: hydrodynamic instabilities, interfacial dynamics, mixing   

Surpassing the energy method for nonlinear fluid stability

A basic question in fluid stability is whether a laminar flow is nonlinearly stable to all perturbations. The typical way to verify stability, called the energy method, is to show that the energy of a perturbation must decay monotonically. The energy method is known to be overly conservative in many systems, particularly when turbulence arises subcritically, such as in parallel shear flows. The energy method is a special case of a Lyapunov function method in which the Lyapunov function is the perturbation energy. This talk will present a more general approach in which the Lyapunov functions (1) are not restricted to being quadratic but instead are higher-degree polynomials, and (2) can depend explicitly on the spectrum of the velocity field in the eigenbasis of the energy stability operator. The optimal construction of such Lyapunov functions is complicated but can be done with computer assistance by formulating a polynomial optimization problem, which in turn is formulated as a semidefinite program. This talk will describe the general framework of the method. A companion talk by Federico Fuentes will illustrate its application to planar Couette flow, where we have verified nonlinear stability at larger Reynolds numbers than is possible using the energy method.   

Global stability of plane Couette flow beyond the energy stability limit

This talk will present computations verifying that the laminar state of plane Couette flow is nonlinearly stable to all perturbations. The Reynolds numbers up to which this globally stability is verified are larger than those at which stability can be proven by the energy method, which is the typical method for demonstrating nonlinear stability of a fluid flow. This improvement is achieved by constructing Lyapunov functions that are more general than the energy. These functions are not restricted to being quadratic, and they are allowed to depend explicitly on the spectrum of the velocity field in the eigenbasis of the energy stability operator. The optimal choice of such a Lyapunov function is a convex optimization problem, and it can be constructed with computer assistance by solving a semidefinite program. This general method will be described in a companion talk by David Goluskin; the present talk focuses on its application to plane Couette flow.   

Impact of finite rate chemistry on the hydrodynamic stability of shear flows in turbulent lean premixed combustion

Recent experimental observations show that the dynamic response of a reactive flow is strongly impacted by the fuel chemistry. In order to gain insight into some of the underlying mechanisms we formulate a new linear stability model that incorporates the impact of finite rate chemistry on the hydrodynamic stability of shear flows. Contrary to previous studies which typically assume that the velocity field is independent of the kinetic rates, the velocity field in our study is coupled with the temperature field. Using this formulation, we reproduce previous results, e.g., most unstable global modes, obtained for non-reacting shear flow. Moreover, we show that these modes are significantly altered in frequency and gain by the presence of a reaction region within the shear layer. This qualitatively agrees with results of our recent experimental and numerical studies, which show that the flame surface location relative to the shear layer influences the stability characteristics in combustion tunnels. This study suggests a physical explanation for the observed impact of finite rate chemistry on shear flow stability.   

Exact and Approximate Solutions for Transient Squeezing Flow

In this paper, we report two novel theoretical approaches to examine a fast-developing flow in a thin fluid gap, which is widely observed in industrial applications and biological systems. The problem is featured by a very small Reynolds number and Strouhal number, making the fluid convective acceleration is negligible, while its local acceleration is not. We have developed an exact solution for this problem which shows that the flow starts with an inviscid limit when the viscous effect has no time to appear, and is followed by a subsequent developing flow, in which the viscous effect continues to penetrate into the entire fluid gap. An approximate solution is also developed using a boundary layer integral method. This solution precisely captures the general behavior of the transient fluid flow process, and agrees very well with the exact solution. We also performed numerical simulation using Ansys-CFX. Excellent agreement between the analytical and the numerical solutions is obtained, indicating the validity of the analytical approaches. The study presented herein fills the gap in the literature, and will have a broad impact in industrial and biomedical applications.