Well-posedness for the linearized motion of an incompressible liquid with free surface boundary

In this paper, we consider the motion of incompressible magnetohydrodynamics (MHD) with resistivity in a domain bounded by a free surface. An electromagnetic field generated by some currents in an external domain keeps an MHD flow in a bounded domain. On the free surface, free boundary conditions for MHD flow and transmission conditions for electromagnetic fields are imposed. We proved the local well-posedness in the general setting of domains from a mathematical point of view. The solutions are obtained in an anisotropic space Hp1((0,T),Hq1)∩Lp((0,T),Hq3) for the velocity field and in an anisotropic space Hp1((0,T),Lq)∩Lp((0,T),Hq2) for the magnetic fields with 2<p<∞, N<q<∞ and 2/p+N/q<1. To prove our main result, we used the Lp-Lq maximal regularity theorem for the Stokes equations with free boundary conditions and for the magnetic field equations with transmission conditions, which have been obtained by Frolova and the second author.

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Marangoni instability of a thin liquid film heated from below by a local heat source

Journal of Fluid Mechanics ◽

10.1017/s0022112002003014 ◽

2003 ◽

Vol 475 ◽

pp. 377-408 ◽

Cited By ~ 99

Author(s):

SERAFIM KALLIADASIS ◽

ALLA KIYASHKO ◽

E. A. DEMEKHIN

Keyword(s):

Free Surface ◽

Liquid Film ◽

Discrete Spectrum ◽

Essential Spectrum ◽

Marangoni Number ◽

Thin Liquid Film ◽

Nonlinear Partial Differential Equations ◽

Spanwise Direction ◽

Surface Boundary ◽

Integral Boundary

We consider the motion of a liquid film falling down a heated planar substrate. Using the integral-boundary-layer approximation of the Navier–Stokes/energy equations and free-surface boundary conditions, it is shown that the problem is governed by two coupled nonlinear partial differential equations for the evolution of the local film height and temperature distribution in time and space. Two-dimensional steady-state solutions of these equations are reported for different values of the governing dimensionless groups. Our computations demonstrate that the free surface develops a bump in the region where the wall temperature gradient is positive. We analyse the linear stability of this bump with respect to disturbances in the spanwise direction. We show that the operator of the linearized system has both a discrete and an essential spectrum. The discrete spectrum bifurcates from resonance poles at certain values of the wavenumber for the disturbances in the transverse direction. The essential spectrum is always stable while part of the discrete spectrum becomes unstable for values of the Marangoni number larger than a critical value. Above this critical Marangoni number the growth rate curve as a function of wavenumber has a finite band of unstable modes which increases as the Marangoni number increases.

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Linear and non-linear potential-flow calculations of free-surface waves with lift and induced drag

Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science ◽

10.1243/0954406001523795 ◽

2000 ◽

Vol 214 (6) ◽

pp. 801-812

Author(s):

C-E Janson

Keyword(s):

Free Surface ◽

Potential Flow ◽

Lift Force ◽

Radiation Condition ◽

Linear Potential ◽

Surface Boundary ◽

Model Approximation ◽

Non Linear ◽

Lifting Surfaces ◽

The Waves

A potential-flow panel method is used to compute the waves and the lift force from surface-piercing and submerged bodies. In particular the interaction between the waves and the lift produced close to the free surface is studied. Both linear and non-linear free-surface boundary conditions are considered. The potential-flow method is of Rankine-source type using raised source panels on the free surface and a four-point upwind operator to compute the velocity derivatives and to enforce the radiation condition. The lift force is introduced as a dipole distribution on the lifting surfaces and on the trailing wake, together with a flow tangency condition at the trailing edge of the lifting surface. Different approximations for the spanwise circulation distribution at the free surface were tested for a surface-piercing wing and it was concluded that a double-model approximation should be used for low speeds while a single-model, which allows for a vortex at the free surface, was preferred at higher speeds. The lift force and waves from three surface-piercing wings, a hydrofoil and a sailing yacht were computed and compared with measurements and good agreement was obtained.

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Well-posedness of a nonstationary axisymmetric hydrodynamic problem with free surface

Siberian Mathematical Journal ◽

10.1134/s0037446617040024 ◽

2017 ◽

Vol 58 (4) ◽

pp. 564-577 ◽

Cited By ~ 1

Author(s):

V. N. Belykh

Keyword(s):

Free Surface ◽

Well Posedness ◽

Hydrodynamic Problem

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Fourth‐order finite‐difference P-SV seismograms

Geophysics ◽

10.1190/1.1442422 ◽

1988 ◽

Vol 53 (11) ◽

pp. 1425-1436 ◽

Cited By ~ 920

Author(s):

Alan R. Levander

Keyword(s):

Free Surface ◽

Finite Difference ◽

Fourth Order ◽

Equations Of Motion ◽

Dispersion Analysis ◽

Numerical Dispersion ◽

Staggered Grid ◽

Order System ◽

Surface Boundary ◽

Lamb’S Problem

I describe the properties of a fourth‐order accurate space, second‐order accurate time, two‐dimensional P-SV finite‐difference scheme based on the Madariaga‐Virieux staggered‐grid formulation. The numerical scheme is developed from the first‐order system of hyperbolic elastic equations of motion and constitutive laws expressed in particle velocities and stresses. The Madariaga‐Virieux staggered‐grid scheme has the desirable quality that it can correctly model any variation in material properties, including both large and small Poisson’s ratio materials, with minimal numerical dispersion and numerical anisotropy. Dispersion analysis indicates that the shortest wavelengths in the model need to be sampled at 5 gridpoints/wavelength. The scheme can be used to accurately simulate wave propagation in mixed acoustic‐elastic media, making it ideal for modeling marine problems. Explicitly calculating both velocities and stresses makes it relatively simple to initiate a source at the free‐surface or within a layer and to satisfy free‐surface boundary conditions. Benchmark comparisons of finite‐difference and analytical solutions to Lamb’s problem are almost identical, as are comparisons of finite‐difference and reflectivity solutions for elastic‐elastic and acoustic‐elastic layered models.

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