Local Well-posedness of the Contact Line Problem in 2-d Stokes Flow
نویسنده
چکیده
We consider the evolution of contact lines for viscous fluids in a two-dimensional open-top vessel. The domain is bounded above by a free moving boundary and otherwise by the solid wall of the vessel. The dynamics of the fluid are governed by the incompressible Stokes equations under the influence of gravity, and the interface between fluid and air is under the effect of capillary forces. Here we develop a local wellposedness theory of the problem in the framework of nonlinear energy methods. We utilize several techniques, including: energy estimates of a geometric formulation of the Stokes equations, a Galerkin method with a time-dependent basis for an –perturbed linear Stokes problem in moving domains, the contraction mapping principle for the –perturbed nonlinear full contact line problem, and a continuity argument for uniform energy estimates.
منابع مشابه
A Mixed Formulation of a Sharp Interface Model of Stokes Flow with Moving Contact Lines
Two-phase fluid flows on substrates (i.e. wetting phenomena) are important in many industrial processes, such as micro-fluidics and coating flows. These flows include additional physical effects that occur near moving (three-phase) contact lines. We present a new 2-D variational (saddlepoint) formulation of a Stokesian fluid with surface tension that interacts with a rigid substrate. The model ...
متن کاملGlobal Unique Solvability of Inhomogeneous Navier-stokes Equations with Bounded Density
In this paper, we prove the global existence and uniqueness of solution to d-dimensional (for d = 2, 3) incompressible inhomogeneous Navier-Stokes equations with initial density being bounded from above and below by some positive constants, and with initial velocity u0 ∈ H (R) for s > 0 in 2-D, or u0 ∈ H (R) satisfying ‖u0‖L2‖∇u0‖L2 being sufficiently small in 3-D. This in particular improves t...
متن کاملLocal Well-Posedness for Volume-Preserving Mean Curvature and Willmore Flows with Line Tension
We show the short-time existence and uniqueness of solutions for the motion of an evolving hypersurface in contact with a solid container driven by the volume-preserving mean curvature flow (MCF) taking line tension effects on the boundary into account. Difficulties arise due to dynamic boundary conditions and due to the contact angle and the non-local nature of the resulting second order, nonl...
متن کاملOn the Local Well-posedness of a 3D Model for Incompressible Navier-Stokes Equations with Partial Viscosity
In this short note, we study the local well-posedness of a 3D model for incompressible Navier-Stokes equations with partical viscosity. This model was originally proposed by Hou-Lei in [4]. In a recent paper, we prove that this 3D model with partial viscosity will develop a finite time singularity for a class of initial condition using a mixed Dirichlet Robin boundary condition. The local well-...
متن کاملControllability of 3d Swimmers in a Perfect Fluid
1. Introduction. Researches on animal locomotion in fluid have now a long history. Focusing on the area of Mathematical Physics, the modeling leads to a system of PDEs (governing the fluid flow) coupled with a system of ODEs (driving the rigid motion of the immersed body). The first difficulty mathematicians came up against was to prove the well-posedness of such systems. This task was carried ...
متن کامل