Tag - Fluid dynamics

Jincheng Yang: Layer separation and energy dissipation for 3D NSE at high Reynolds number

7th May, 2024

In this talk, we consider the 3D incompressible Navier-Stokes equation in a bounded domain, with a canonical example of Poiseuille flow in mind. We provide an unconditional upper bound for the boundary layer separation and energy dissipation of Leray-Hopf weak solutions, uniformly in high Reynolds numbers. We estimate layer separation by measuring the energy norm of the discrepancy between a (turbulent) low-viscosity Leray-Hopf solution and a fixed (laminar) regular Euler solution with similar initial conditions and body force. This is accomplished by a new nonlinear boundary vorticity estimate.

Maurizio Grasselli: Non-local Navier-Stokes-Cahn-Hilliard systems with unmatched densities

16th April, 2024

H. Abels, H. Garcke, and G. Grün (2012) proposed a diffuse interface model to describe liquid-liquid phase separation in incompressible binary fluids of different densities. This model consists of the Navier-Stokes system which is non-linearly coupled with an advective Cahn-Hilliard equation. In this talk, however, instead of taking the usual free energy functional, we consider a non-local version. Therefore, the resulting Cahn-Hilliard equation is a second-order (spatially) non-local equation. This system was already analysed by S. Frigeri (2016) who established the existence of a global weak solution. I intend to present some further results in dimension two which I have obtained jointly with C.G. Gal, A. Giorgini, and A. Poiatti (2023). These results are mainly concerned with strong solutions, uniqueness, and convergence to a single equilibrium. Some related open issues will also be discussed.

Matteo Fornoni: Optimal control for non-local Cahn-Hilliard tumour growth models

2nd April, 2024

We consider a non-local tumour growth model of phase-field type, describing the evolution of tumour cells through proliferation in the presence of a nutrient. The model consists of a coupled system, incorporating a non-local Cahn-Hilliard equation for the tumour phase variable and a reaction-diffusion equation for the nutrient. The optimal control problem aims to identify a suitable therapy, capable of guiding the evolution of the tumour towards a predefined target. We first establish novel regularity results for the PDE system by applying maximal regularity theory in weighted Lp spaces. Such a technique enables us to prove the local existence and uniqueness of a regular solution in a quite general framework, which also includes chemotaxis effects to some extent, but restricts us to regular double-well potentials. Then, by leveraging the time-regularisation properties of the weighted spaces and some global boundedness estimates, we further extend the solution to a global one. In a second version of the model, we add a viscous regularisation term which allows us to prove the existence and uniqueness of a global regular solution under more general hypotheses. Indeed, we can also include singular double-well potentials and cross-diffusive chemotactic effects, at the expense of some additional hypotheses on the controls. These results provide the foundation for addressing the optimal control problem in both cases. Specifically, we prove the existence of an optimal therapy and then, by studying the Fréchet-differentiability of the control-to-state operator and introducing the adjoint system, we derive first-order necessary optimality conditions. We finally discuss some open questions and future research directions.

Joaquim Serra: Stable phase transitions: open questions and new results

26th March, 2024

Surface tension and similar forces lead to area-minimizing interfaces in some physical phenomena, observable at macroscopic scales. However, this principle of surface area minimization does not uniformly apply across all scales, as the underlying physical energies often vary with scale. For example, describing a soap film as an area-minimizing surface becomes implausible at scales comparable to 5 nanometers, the size of a soap molecule. Similarly, the Allen-Cahn energy (i.e., scalar Ginzburg-Landau) exhibits scale-dependent behavior that mirrors area minimization only at larger scales. The regularity theory for absolute energy-minimizing minimal surfaces has been successfully extended to several scale-dependent models, including Allen-Cahn. Yet, extending these results to all stable configurations, which represent the states observable in nature, poses significant challenges. In the talk, I will discuss the pressing open questions and the latest findings regarding stable phase transitions in 3-dimensional environments.

Yoichiro Mori: Inextensible interface problem in 2D Stokes fluid

18th March, 2024

We consider the dynamics of a closed intextensible interface immersed in a 2D Stokes fluid, a model that has been used for 2D simulations of vesicle dynamics. In this model, a 1D closed interface exerts a bending force and the interface is subject to an inextensibility constraint. As part of the problem, one must solve for the unknown tension that ensures membrane inextensibility. Given a force exerted on the interface, we first show that the problem of determining the tension is soluble if and only if the interface is not a circle. Using this result, we prove local-in-time well-posedness for this problem. We will finally discuss open questions and future directions.

Pranava Jayanti: Mass transfer and global solutions in a micro-scale model of superfluidity

5th February, 2024

We investigate a micro-scale model of superfluidity derived by Pitaevskii in 1959 to describe the interacting dynamics between the superfluid and normal fluid phases of Helium-4. This system consists of the nonlinear Schrödinger equation and the incompressible, inhomogeneous Navier-Stokes equations, coupled to each other via a bidirectional non-linear relaxation mechanism. The coupling permits mass/momentum/energy transfer between the phases, and accounts for the conversion of superfluid into normal fluid. We prove the existence of solutions in 𝕋d (d=2,3) for a power-type non-linearity, beginning from small initial data. Depending upon the strength of the nonlinear self-interactions, we obtain solutions that are global or almost-global in time.

The main challenge is to control the inter-phase mass transfer in order to ensure the strict positivity of the normal fluid density, while obtaining time-independent a priori estimates. We present two different approaches (purely energy based, versus a combination of energy estimates and maximal regularity) based on the dimension.

Haithem Taha: A variational theory of aero-hydrodynamics

16th January, 2024

The Euler equation does not possess a unique solution for the flow over a 2-dimensional object. This problem has serious repercussions in aerodynamics; it implies that the inviscid aero-hydrodynamic lift force over a 2-dimensional object cannot be determined from first principles; a closure condition must be provided. The Kutta condition has been ubiquitously considered for such a closure in the literature, even in cases where it is not applicable (e.g. unsteady). In this talk, I will present a special variational principle that we revived from the history of analytical mechanics: Hertz’s principle of least curvature. Using this principle, we developed a novel variational formulation of Euler’s dynamics of ideal fluids that is fundamentally different from the previously developed variational formulations based on Hamilton’s principle of least action. Applying this new variational formulation to the century-old problem of the ideal flow over an airfoil, we developed a general (dynamical) closure condition that is, unlike the Kutta condition, derived from first principles. In contrast to the classical theory, the proposed variational theory is not confined to sharp edged airfoils; i.e., it allows, for the first time, theoretical computation of lift over arbitrarily smooth shapes, thereby generalizing the century-old lift theory of Kutta and Zhukovsky. Moreover, the new variational condition reduces to the Kutta condition in the special case of a sharp-edged airfoil, which challenges the widely accepted wisdom about the viscous nature of the Kutta condition. We also generalized this variational principle to Navier-Stokes’s via Gauss’s principle of least constraint, thereby discovering the fundamental quantity that Nature minimizes in every incompressible flow. We proved that the magnitude of the pressure gradient over the field is minimum at every instant! We call it the Principle of Minimum Pressure Gradient (PMPG). We proved that the Navier-Stokes equation is the necessary condition for minimizing the pressure gradient subject to the continuity constraint. Hence, the PMPG turns any fluid mechanics problem into a minimization one where fluid mechanicians need not to apply Navier-Stokes equations, but merely need to minimize the proposed action.

Michele Dolce: Stability threshold of the 2D Couette flow in a homogeneous magnetic field

7th December, 2023

A planar incompressible and electrically conducting fluid can be described by the 2D Navier-Stokes-MHD system. One simple yet physically relevant laminar state is the Couette flow with a constant homogeneous magnetic field, given by uE=(y,0), BE=(b,0) in the domain T×R. The goal is to estimate how large can be a perturbation of this state while still resulting in a solution close to the laminar regime, thereby preventing the onset of turbulence. We prove that Sobolev regular initial perturbations of size O(Re-2/3), with Re being the Reynolds number, remain close to uE, BE and exhibit dissipation enhancement. The latter quantifies the convergence towards an x-independent state on a time-scale O(Re-1/3), much faster than the standard diffusive one O(Re-1).

Emeric Roulley: Dynamics of vortex cap solutions on the rotating unit sphere

16th November, 2023

We provide an analytical proof for the existence of periodic vortex cap solutions for the homogeneous and incompressible Euler equations on the rotating unit 2-sphere. These solutions are piecewise constant absolute vorticity distributions, subject to the Gauss constraint and rotating uniformly around the vertical axis (rotation axis of the sphere). The emergence of such structures was numerically conjectured in the atmospheric community by Dritschel-Polvani in the 90s. Our proof is based on the bifurcation from zonal solutions given by spherical caps.