A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

On the Navier-Stokes equations

書評 Hermann Sohr : The Navier-Stokes Equations : An Elementary Functional Analytic Approach

The Cahn–Hilliard equation is a fundamental model that describes phase separation processes of binary mixtures. In recent years, several types of dynamic boundary conditions have been proposed in order to account for possible short-range interactions of the material with the solid wall. Our first aim in this paper is to propose a new class of dynamic boundary conditions for the Cahn–Hilliard equation in a rather general setting. The derivation is based on an energetic variational approach that combines the least action principle and Onsager’s principle of maximum energy dissipation. One feature of our model is that it naturally fulfills three important physical constraints: conservation of mass, dissipation of energy and force balance relations. Next, we provide a comprehensive analysis of the resulting system of partial differential equations. Under suitable assumptions, we prove the existence and uniqueness of global weak/strong solutions to the initial boundary value problem with or without surface diffusion. Furthermore, we establish the uniqueness of the asymptotic limit as $${t\to+\infty}$$
 and characterize the stability of local energy minimizers for the system.

An Energetic Variational Approach for the Cahn-Hilliard Equation with Dynamic Boundary Condition: Model Derivation and Mathematical Analysis

Governing equations of motion for a viscous incompressible material surface are derived from the balance laws of continuum mechanics. The surface is treated as a time-dependent smooth orientable manifold of codimension one in an ambient Euclidian space. We use elementary tangential calculus to derive the governing equations in terms of exterior differential operators in Cartesian coordinates. The resulting equations can be seen as the Navier-Stokes equations posed on an evolving manifold. We consider a splitting of the surface Navier-Stokes system into coupled equations for the tangential and normal motions of the material surface. We then restrict ourselves to the case of a geometrically stationary manifold of codimension one embedded in $\Bbb{R}^n$. For this case, we present new well-posedness results for the simplified surface fluid model consisting of the surface Stokes equations. Finally, we propose and analyze several alternative variational formulations for this surface Stokes problem, including constrained and penalized formulations, which are convenient for Galerkin discretization methods.

Incompressible fluid problems on embedded surfaces: Modeling and variational formulations

We consider a numerical approach for the incompressible surface Navier-Stokes equation on surfaces with arbitrary genus g(S). The approach is based on a reformulation of the equation in Cartesian coordinates of the embedding R3, penalization of the normal component, a Chorin projection method, and discretization in space by surface finite elements for each component. The approach thus requires only standard ingredients which most finite element implementations can offer. We compare computational results with discrete exterior calculus simulations on a torus and demonstrate the interplay of the flow field with the topology by showing realizations of the Poincare-Hopf theorem on n-tori.

Solving the incompressible surface Navier-Stokes equation by surface finite elements

Energetic variational approaches for incompressible fluid systems on an evolving surface

On L3,∞-stability of the Navier–Stokes system in exterior domains

We consider the governing equations for the motion of compressible fluid on an evolving surface from both energetic and thermodynamic points of view. We employ our energetic variational approaches to derive the momentum equation of our compressible fluid systems on the evolving surface. Applying the first law of thermodynamics and the Gibbs equation, we investigate the internal energy, enthalpy, entropy, and free energy of the fluid on the evolving surface. We also study conservative forms and conservation laws of our compressible fluid systems on the evolving surface. Moreover, we derive the generalized heat and diffusion systems on an evolving surface from an energetic point of view. This paper gives a mathematical validity of the surface stress tensor determined by the Boussinesq-Scriven law. Using a flow map on an evolving surface and applying the Riemannian metric induced by the flow map are key ideas to analyze fluid flow on the evolving surface.

On derivation of compressible fluid systems on an evolving surface

Errata to “Energetic variational approaches for incompressible fluid systems on an evolving surface”

We consider the dominant equations for the motion of the non-Newtonian fluid in a domain from an energetic point of view We apply our energetic variational approaches and the first law of thermodynamics to derive the generalized compressible non-Newtonian fluid system We also derive the generalized incompressible non-Newtonian fluid system by using our energetic variational approaches

Hajime Koba

Papers

Energetic variational approaches for incompressible fluid systems on an evolving surface

On L3,∞-stability of the Navier–Stokes system in exterior domains

On derivation of compressible fluid systems on an evolving surface

Errata to “Energetic variational approaches for incompressible fluid systems on an evolving surface”

Energetic variational approaches for non-Newtonian fluid systems