Session Q26: General Fluid Dynamics: Theory

Geometric control of asymmetries for passive tracer transport in domains with triangular and rounded triangular cross-sections

We investigate the dispersion of passive tracers in laminar pressure-driven shear flows through triangular and ``trefoil'' capillary pipes, characterized as near-triangular cross-sections with smoothed corners and concave sides. We show through asymptotic analyses and Monte-Carlo simulations how the longitudinal asymmetry of the tracer distribution changes in time. For triangular cross-sections, varying from equilateral to isosceles, we observe a loading shift in the concentration profile from back-loaded to front-loaded, corresponding to a sign-change (from positive to negative) in the cross-sectionally averaged skewness. This change becomes more striking as the isosceles domains flatten. For the trefoil domains we explore, behavior aligns with previous studies: bulky cross-sections with aspect ratio ∼1 maintain the same back-loaded concentration profile (and positive skewness) throughout their time-evolution, while flatter cross-sections with aspect ratio «1 present a sign-change in the skewness corresponding to a back- to front-loading shift before symmetrizing on the longest timescales. Open questions and future directions will be discussed.   

New Families of Conservation Laws based on Extended Formulations of the Reynolds Transport Theorem

The Reynolds transport theorem describes the transport of a conserved quantity by a fluid volume, providing all integral conservation laws of fluid mechanics. Recently, extended versions were proved for transformations of a conserved quantity in an n-dimensional manifold or coordinate space, based on general coordinates X, parameters C and tensor field V. These can be used to derive new integral conservation laws in different spaces, including for volumetric, velocimetric and velocivolumetric spaces based on the ordered triples (u,x,t), where u is the Eulerian velocity and x is position. These require different fluid densities, here labelled ρ(x,t) in volumetric space [SI units: kg m^-3], д(u,t) in velocimetric space [kg (m s^-1)^-3] and ζ(u,x,t) in velocivolumetric space [kg (m s^-1)^-3 m^-3]. Such fluid densities can be defined from their underlying probability density functions p(x|t), p(u|t) and p(u,x|t) by convolution. The extended formulation is used to derive 11 tables of conservation laws for different choices of X and C, for the eight common conserved quantities of fluid mechanics (fluid mass, species mass, linear and angular momentum, energy, charge, entropy and probability). The findings considerably expand the set of known conservation laws of fluid mechanics.

    

The search for finite-time singularity solutions of the Euler equations for incompressible and inviscid fluids.

The search for finite-time singularity solutions of Euler equations is considered for the case of an incompressible and inviscid fluid. Under the assumption that a finite-time blow-up solution may be spatially anisotropic as time goes by such that the flow contracts more rapidly into one direction than into the other, it can be shown that the dynamics of an axially symmetric flow with swirl may be approximated to a simpler hyperbolic system. By using the method of characteristics, it can be shown that generically the velocity flow exhibits multivalued solutions appearing on a rim at a finite distance from the axis of rotation, which displays a singular behavior in the radial derivatives of velocities. Moreover, the general solution shows a genuine blow-up, as a consequence of smooth initial data. This singularity is closely related to the singular solution found by T Elgindi in 2022 for a non-smooth initial data. These singularities are generic for a vast number of smooth finite-energy initial conditions and are characterized by a local singular behavior of velocity gradients and accelerations.

    

What Does Nature Minimize In Every Incompressible Flow?

Here, we revive Gauss' principle of least constraint, which is not commonly found in textbooks of classical physics. It asserts that the motion of a constrained mechanical system is such that the the magnitude of constraint forces must be minimum at every instant. It is an actual minimal principle in contrast to the stationary principle of least action. For incompressible fluids, the pressure gradient is a constraint force; its main role is ensure the continuity constraint. Hence, according to Gauss' principle, the integral of the magnitude of the pressure gradient is minimum at every instant. We prove that Navier-Stokes equations represent the necessary condition for minimization of the pressure gradient. We call it The Principle of Minimum Pressure Gradient (PMPG). It turns any fluid mechanics problem into a minimization problem. We demonstrate this intriguing property by solving classical fluid mechanics problems (e.g., channel flow, Stokes' 2nd problem) without resorting to Navier-Stokes' equations, rather by minimizing the pressure gradient with respect to the free parameters. The PMPG is applicable to non-Newtonian fluids with arbitrary forcing (e.g., electromagnetic). In this case, Navier-Stokes' equations must be modified, while the PMPG is applicable verbatim. Moreover, its inviscid version, which provides a minimization formulation of Euler's euqations, may provide closure for two-dimensional problems where Euler's equations do not have a unique solution. For example, the century-old problem of the ideal flow over an airfoil with an arbitrary shape (not necessary with a sharp trailing edge) is not solvable using classical techniques. In contrast, the PMPG provides a unique solution (i.e., a unique circulation that minimizes the pressure gradient). Interestingly, this unique circulation reduces to Kutta's in the special case of a sharp-edged airfoil, which challenges the accepted wisdom about the viscous nature of the Kutta condition.    

Boundary Effects on Ideal Fluid Forces and Kelvin's Minimum Energy Theorem

The electrostatic force on a charge above a neutral conductor is generally attractive. Surprisingly, that force becomes repulsive in certain geometries (Levin & Johnson 2011), a result that follows from an energy theorem in electrostatics. Based on the analogous minimum energy theorem of Kelvin (1849), valid in the theory of ideal fluids, we show corresponding effects on steady and unsteady fluid forces in the presence of boundaries. We present a model of a body approaching a boundary, where the unsteady force is typically repulsive (Lamb 1975, §137). We also present a model of a Bernoulli suction gripper, for which the steady force is typically attractive. Both the unsteady and steady forces are shown to reverse sign when boundaries approximate flow streamlines, at energy minima predicted by Kelvin's theorem.   

A Fully Second Order Constitutive Theory of Fluids

A fully second order continuum theory of fluids is developed. The conventional balance equations of mass, linear momentum, energy and entropy are used. Constitutive equations are assumed to depend on density, temperature and velocity, and their derivatives up to second order. The principle of equipresence is used along with the Coleman-Noll procedure to derive restrictions on the constitutive equations by utilizing the second law. The entropy flux is not assumed to be equal to the heat flux over the temperature. We obtain explicit results for all constitutive quantities up to quadratic nonlinearity so as to satisfy the Clausius-Duhem inequality. Our results are shown to be consistent but much more general than other published results.   

A Theoretical and Experimental Study of an Oscillatory Flow Through a Compliant Tube

In this paper, a theoretical and experimental study of an oscillatory flow through a compliant tube is reported. This study attempts to mimic the flow through arteries, in order to achieve a better understanding of the interactions between flow pulsatility and arterial compliance by simultaneously measuring the flow rate and the tube’s shape evolution. The experimental setup consists of a horizontally submerged elastic PDMS tube mounted between two fixed ends, each instrumented with a pressure sensor to monitor the instantaneous pressure. The device providing the flow was custom designed and manufactured by our team to deliver a purely oscillatory flow. A high-speed camera is used to observe the tube’s undulating shape. A model is developed to describe the tube’s wall motion, informed by its tension, stiffness, and density. Based on the pressure data, the governing equation is then solved numerically to obtain the predicted evolution of the tube’s profile, and the solution is compared to data acquired from the footage of the tube. This study showcases a setup where a compliant tube’s flow conditions are measured in tandem with its deformation, offering a unique avenue through which fluid-structure interaction models in compliant ducts can be tested and refined.

    

A Fourier-Chebyshev pseudospectral methods for the investigation of potential singularities in the axisymmetric 3D Euler equation

We present a Fourier-Chebyshev pseudospectral scheme we to investigate the potential singularity in the axisymmetric 3D Euler equation (cf.  G. Luo and T. Hou, Proc. Natl. Acad. Sci. USA 111, 12968 (2014). We demonstrate that: (a) the singularity time is preceded in pseudospectral simulations bythe formation of localised oscillatory structures called tygers; and  (b)  a generalisation of the analyticity-strip method  can be used to track the movement of the complex singularities of the velocity field, from which we can get an estimate for the singularity time. We explore generalizations of this method to other ideal hydrodynamical equations.   

Drag is Due to Flux of Spanwise Vorticity from Solid Surfaces

Wall drag is exactly related to vorticity flux from solid surfaces [1]. Potential flow, with the same in-flow and out-flow and with no-flow through the wall, is used for comparison. Total drag from skin friction and pressure is given instantaneously by flux of spanwise vorticity across potential-flow streamlines, integrated over flow volume. This follows from incompressible Navier-Stokes for any Reynolds number and generally for interior flows through pipes, channels, etc. and for exterior flows past solid bodies. Drag is thus due entirely to flux away from the wall of spanwise vorticity, generated at the body surface by streamwise pressure gradients and body acceleration and then transported through the flow by nonlinear advection, stretching and viscous diffusion. Streamwise vorticity does not contribute directly. Our result is inspired by the Josephson-Anderson relation for drag in inviscid superfluids due to motion of quantized vortices. Drag for classical fluids in the infinite Reynolds limit is likewise explained by Euler solutions with vorticity flux, distinct from d'Alembert's. Drag reduction at any Reynolds number can be achieved only by reducing spanwise vorticity flux