Session Q26: General Fluid Dynamics: Theory and Mathematical Methods

The generic instability of relativistic fluids revisited: transverse mode fluctuations and diffussion fluxes in the Landau-Lifshitz frame

Stability of statistical fluctuations of relativistic fluids is a classical problem that has been addressed since the pioneering works of Hisckock and Lindblom back en 1985, who identified exponential growths of statistical fluctuations using a phenomenological approach.  In the present work we critically examine the role played by the heat flux in the onset of this type of instability  using the Landau-Lifschitz frame, in which the dissipative contributions to this variable are not contained in the spatial components of the energy-momentum tensor.  It is shown that the generic instabilities in the transverse mode are  eliminated if the corresponding constitutive equations are established within  a first order in the gradients approximation kinetic theory formalism, and that purely relativistic dissipative contributions to the diffusive fluxes arise the aforementioned  frame.   

Solution of Basic Flows Using a Second Order Theory of Fluids

The Navier-Stokes-Fourier (NSF) equations have proved very valuable in modeling fluid flows over the last two centuries. However, there are cases in which large variations in velocity and/or thermal fields occur where it has been shown that they do not provide accurate results. Using continuum mechanics principles, second order equations were derived and shown to reproduce experimental results of the shock structure of gases over a large range of Mach numbers (Paolucci & Paolucci JFM, 486, 686-710 (2018)). Computer experiments using the direct simulation Monte Carlo (DSMC) method have shown that at small Knudsen number the pressure and temperature profiles in force-driven compressible plane Poiseuille flow exhibits different qualitative behavior from the profiles obtained by NSF equations. We compare the DSMC measurements with numerical solutions of equations resulting from the second order theory. We find that the second order equations predict the bimodal temperature profile and recover many of the other anomalous features (e.g., non-constant pressure and non-zero parallel heat flux). Comparisons of the predictions coming from the second order theory are provided in order to critically assess its validity and usefulness.   

Late-time evolution of Rayleigh-Taylor instability in a domain of a finite size

We develop the theoretical analysis to systematically study the late-time evolution of Rayleigh-Taylor instability in a domain of a finite size. The nonlinear dynamics of fluids with similar and contrasting densities are considered for two-dimensional flows driven by sustained acceleration. The flows are periodic in the plane normal to the direction of acceleration and have no external mass sources. Group theory analysis is applied to accurately account for the mode coupling. Asymptotic nonlinear
solutions are found to describe the interface dynamics far from the boundaries and near the boundaries. The influence of the size of the domain on the diagnostic parameters of the flow is identified. In particular, it is shown that in a finite size the domain the flow is decelerating compared to spatially extended case. A close analysis of the shear present at the interface of the fluids is also studied both for a domain of a finite size and in the limit of an infinite domain. It is shown how the interfacial
shear acts as a natural parameter to the family of solutions. The theory outcomes for the numerical modelling and design of experiments on Rayleigh-Taylor instability are discussed.   

Analytical and Numerical Study of a Pulsatile Flow in a Porous Tube

Lateral leakage is encountered in many biological and chemical applications such as renal flows and filtration processes. In this paper we report a comprehensive analytical and numerical method to examine pulsatile flow in a porous-walled tube, with leakage flow rate or leakage pressure gradient prescribed. In the first scenario, analytical results have been obtained when the leakage flow rate is small as compared to the axial flow rate. Numerical simulations using ANSYS Fluent were also performed for cases where both the pulsatile Reynolds number based on the amplitude of the mean axial velocity and the leakage ratio, were varied. The results indicate that, the analytical solution for the axial velocity had an increasing deviation from the numerical results with the pulsatile Reynolds number, and the leakage ratio. Interestingly, the analytical radial velocity almost overlapped with its numerical counterpart, for all considered cases. In the second scenario where the pressure-gradient for the leakage is prescribed, an analytical solution was obtained. Importantly, it suggests the criteria, based on the wall permeability, for the application of the analytical method developed herein.   

Inviscid and viscous interaction of a line or a circular vortex filament with a solid sphere

Inviscid and viscous fluid flows induced by a line vortex or a circular vortex ring in the presence of a solid sphere of radius $a$ are studied. Euler equations are used for the inviscid fluid flow model while equations of low-Reynolds number flow (Stokes flow) are employed for the viscous fluid model. The velocity potential formulation yields exact solution for the line vortex-sphere non-viscous interaction problem. Our analytic solution reveals extreme values for the radial and axial velocities indicating a strong interaction. An expression for the force on the sphere is calculated in an integral form containing a logarithmic term. The force is higher when the line vortex is closer to the sphere and is weaker when it is placed farther. Analytic solution for the corresponding viscous problem is found using a general solution representation. It is observed that the interaction in the viscous case is weaker. Closed form solutions for the axisymmetric vortex ring-sphere problem are also determined using stream function methods. The radius and location parameters have significant impact on the interaction in all cases. Our results form a basis for the investigation of the motion of vortex lines and rings in the vicinity of a spherical boundary.