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1. Finite-time blowup for a Navier–Stokes modelequation for the self-amplification of strain

   https://doi.org/10.2140/apde.2023.16.997  

   Miller, Evan (January 2023, Analysis & PDE)

   We consider a model equation for the Navier–Stokes strain equation which has the same identity for enstrophy growth and a number of the same regularity criteria as the full Navier–Stokes strain equation, and is also an evolution equation on the same constraint space. We prove finite-time blowup for this model equation, which shows that the identity for enstrophy growth and the strain constraint space are not sufficient on their own to guarantee global regularity for Navier–Stokes. The mechanism for the finite-time blowup of this model equation is the self-amplification of strain, which is consistent with recent research suggesting that strain self-amplification, not vortex stretching, is the main mechanism behind the turbulent energy cascade. Because the strain self-amplification model equation is obtained by dropping certain terms from the full Navier–Stokes strain equation, we will also prove a conditional blowup result for the full Navier–Stokes equation involving a perturbative condition on the terms neglected in the model equation. 

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2. Global well-posedness of the velocity–vorticity-Voigt model of the 3D Navier–Stokes equations

   https://doi.org/https://doi.org/10.1016/j.jde.2018.08.033  

   Larios, A; Pei, Y; Rebholz, L (February 2019, Journal of differential equations)

   The velocity–vorticity formulation of the 3D Navier–Stokes equations was recently found to give excellent numerical results for flows with strong rotation. In this work, we propose a new regularization of the 3D Navier–Stokes equations, which we call the 3D velocity–vorticity-Voigt (VVV) model, with a Voigt regularization term added to momentum equation in velocity–vorticity form, but with no regularizing term in the vorticity equation. We prove global well-posedness and regularity of this model under periodic boundary conditions. We prove convergence of the model's velocity and vorticity to their counterparts in the 3D Navier–Stokes equations as the Voigt modeling parameter tends to zero. We prove that the curl of the model's velocity converges to the model vorticity (which is solved for directly), as the Voigt modeling parameter tends to zero. Finally, we provide a criterion for finite-time blow-up of the 3D Navier–Stokes equations based on this inviscid regularization. 

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3. Efficient numerical scheme for a dendritic solidification phase field model with melt convection

   Chen, C; Yang, X. (March 2019, Journal of computational physics)

   In this paper, we consider numerical approximations for a dendritic solidification phase field model with melt convection in the liquid phase, which is a highly nonlinear system that couples the anisotropic Allen-Cahn type equation, the heat equation, and the weighted Navier-Stokes equations together. We first reformulate the model into a form which is suitable for numerical approximations and establish the energy dissipative law. Then, we develop a linear, decoupled, and unconditionally energy stable numerical scheme by combining the modified projection scheme for the Navier-Stokes equations, the Invariant Energy Quadratization approach for the nonlinear anisotropic potential, and some subtle explicit-implicit treatments for nonlinear coupling terms. Stability analysis and various numerical simulations are presented. 

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4. Simplified conservative discretization of the Cahn-Hilliard-Navier-Stokes equations

   https://doi.org/10.1016/j.jcp.2024.113382  

   Goulding, Jason; Ayazi, Mehrnaz; Shinar, Tamar; Schroeder, Craig (December 2024, Journal of Computational Physics)

   In this paper we construct a novel discretization of the Cahn-Hilliard equation coupled with the Navier-Stokes equations. The Cahn-Hilliard equation models the separation of a binary mixture. We construct a very simple time integration scheme for simulating the Cahn-Hilliard equation, which is based on splitting the fourth-order equation into two second-order Helmholtz equations. We combine the Cahn-Hilliard equation with the Navier-Stokes equations to simulate phase separation in a two-phase fluid flow in two dimensions. The scheme conserves mass and momentum and exhibits consistency between mass and momentum, allowing it to be used with large density ratios. We introduce a novel discretization of the surface tension force from the phase-field variable that has finite support around the transition region. The model has a parameter that allows it to transition from a smoothed continuum surface force to a fully sharp interface formulation. We show that our method achieves second-order accuracy, and we compare our method to previous work in a variety of experiments. 

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5. Validity of steady Prandtl layer expansions

   https://doi.org/10.1002/cpa.22109  

   Guo, Yan; Iyer, Sameer (November 2023, Communications on Pure and Applied Mathematics)

   Abstract Let the viscosity for the 2D steady Navier‐Stokes equations in the region and with no slip boundary conditions at . For , we justify the validity of the steady Prandtl layer expansion for scaled Prandtl layers, including the celebrated Blasius boundary layer. Our uniform estimates in ε are achieved through a fixed‐point scheme:for solving the Navier‐Stokes equations, where are the tangential and normal velocities at , DNS stands for of the vorticity equation for the normal velocityv, and the compatibility ODE for at . 

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