Boundary conditions for free surface inlet and outlet problems

AbstractWe investigate and compare the boundary conditions that are to be applied to free-surface problems involving inlet and outlets of Newtonian fluid, typically found in coating processes. The flux of fluid is a priori known at an inlet, but unknown at an outlet, where it is governed by the local behaviour near the film-forming meniscus. In the limit of vanishing capillary number $\mathit{Ca}$ it is well known that the flux scales with ${\mathit{Ca}}^{2/ 3} $, but this classical result is non-uniform as the contact angle approaches $\lrm{\pi} $. By examining this limit we find a solution that is uniformly valid for all contact angles. Furthermore, by considering the far-field behaviour of the free surface we show that there exists a critical capillary number above which the problem at an inlet becomes over-determined. The implications of this result for the modelling of coating flows are discussed.

Download Full-text

Displacement of a two-dimensional immiscible droplet adhering to a wall in shear and pressure-driven flows

Journal of Fluid Mechanics ◽

10.1017/s0022112098003462 ◽

1999 ◽

Vol 383 ◽

pp. 29-54 ◽

Cited By ~ 78

Author(s):

ANTHONY D. SCHLEIZER ◽

ROGER T. BONNECAZE

Keyword(s):

Capillary Number ◽

Contact Angles ◽

Fluid Viscosity ◽

Diffusive Transport ◽

Two Dimensional ◽

Parallel Plates ◽

Contact Lines ◽

Critical Capillary Number ◽

Pressure Driven Flow ◽

Pressure Driven

The dynamic behaviour and stability of a two-dimensional immiscible droplet subject to shear or pressure-driven flow between parallel plates is studied under conditions of negligible inertial and gravitational forces. The droplet is attached to the lower plate and forms two contact lines that are either fixed or mobile. The boundary-integral method is used to numerically determine the flow along and dynamics of the free surface. For surfactant-free interfaces with fixed contact lines, the deformation of the interface is determined for a range of capillary numbers, droplet to displacing fluid viscosity ratios, droplet sizes and flow type. It is shown that as the capillary number or viscosity ratio or size of the droplet increases, the deformation of the interface increases and above critical values of the capillary number no steady shape exists. For small droplets, and at low capillary numbers, shear and pressure-driven flows are shown to yield similar steady droplet shapes. The effect of surfactants is studied assuming a fixed amount of surfactant that is subject to convective–diffusive transport along the interface and no transport to or from the bulk fluids. Increasing the surface Péclet number, the ratio of convective to diffusive transport, leads to an accumulation of surfactant at the downstream end of the droplet and creates Marangoni stresses that immobilize the interface and reduce deformation. The no-slip boundary condition is then relaxed and an integral form of the Navier-slip model is used to examine the effects of allowing the droplet to slip along the solid surface in a pressure-driven flow. For contact angles less than or equal to 90°, a stable droplet spreads along the wall until a steady shape is reached, when the droplet translates across the wall at a constant velocity. The critical capillary number is larger for these droplets compared to those with pinned contact lines. For contact angles greater than 90°, the wetted area between a stable droplet and the wall decreases until a steady shape is reached. The critical capillary number for these droplets is less than that for pinned droplets. Above the critical capillary number the droplet completely detaches for a contact angle of 120°, or part of it is pinched off leaving behind a smaller attached droplet for contact angles less than or equal to 90°.

Download Full-text

Local Well-Posedness for Free Boundary Problem of Viscous Incompressible Magnetohydrodynamics

Mathematics ◽

10.3390/math9050461 ◽

2021 ◽

Vol 9 (5) ◽

pp. 461

Author(s):

Kenta Oishi ◽

Yoshihiro Shibata

Keyword(s):

Free Surface ◽

Free Boundary ◽

Boundary Conditions ◽

Stokes Equations ◽

Mhd Flow ◽

Transmission Conditions ◽

Well Posedness ◽

Anisotropic Space ◽

Free Boundary Conditions ◽

Incompressible Magnetohydrodynamics

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.

Download Full-text

On Fourier Series in Eigenfunctions of Elliptic Boundary Value Problems

Georgian Mathematical Journal ◽

10.1515/gmj.2003.401 ◽

2003 ◽

Vol 10 (3) ◽

pp. 401-410

Author(s):

M. S. Agranovich ◽

B. A. Amosov

Keyword(s):

Fourier Series ◽

Boundary Conditions ◽

Fourier Coefficients ◽

Elliptic Boundary ◽

A Priori ◽

Adjoint Problem ◽

Elliptic Boundary Value ◽

Right Hand ◽

The Difference ◽

The Right

Abstract We consider a general elliptic formally self-adjoint problem in a bounded domain with homogeneous boundary conditions under the assumption that the boundary and coefficients are infinitely smooth. The operator in 𝐿2(Ω) corresponding to this problem has an orthonormal basis {𝑢𝑙} of eigenfunctions, which are infinitely smooth in . However, the system {𝑢𝑙} is not a basis in Sobolev spaces 𝐻𝑡 (Ω) of high order. We note and discuss the following possibility: for an arbitrarily large 𝑡, for each function 𝑢 ∈ 𝐻𝑡 (Ω) one can explicitly construct a function 𝑢0 ∈ 𝐻𝑡 (Ω) such that the Fourier series of the difference 𝑢 – 𝑢0 in the functions 𝑢𝑙 converges to this difference in 𝐻𝑡 (Ω). Moreover, the function 𝑢(𝑥) is viewed as a solution of the corresponding nonhomogeneous elliptic problem and is not assumed to be known a priori; only the right-hand sides of the elliptic equation and the boundary conditions for 𝑢 are assumed to be given. These data are also sufficient for the computation of the Fourier coefficients of 𝑢 – 𝑢0. The function 𝑢0 is obtained by applying some linear operator to these right-hand sides.

Download Full-text

Influence of Boundary Conditions in Numerical Simulation of Free Surface Vortices

Energy Procedia ◽

10.1016/j.egypro.2015.11.836 ◽

2015 ◽

Vol 82 ◽

pp. 893-899 ◽

Cited By ~ 5

Author(s):

Luca Cristofano ◽

Matteo Nobili

Keyword(s):

Numerical Simulation ◽

Free Surface ◽

Boundary Conditions

Download Full-text

An improved a priori error analysis of Nitsche’s method for Robin boundary conditions

Numerische Mathematik ◽

10.1007/s00211-017-0927-1 ◽

2017 ◽

Vol 138 (4) ◽

pp. 1011-1026 ◽

Cited By ~ 1

Author(s):

Nora Lüthen ◽

Mika Juntunen ◽

Rolf Stenberg

Keyword(s):

Error Analysis ◽

Boundary Conditions ◽

A Priori ◽

Robin Boundary Conditions ◽

Nitsche’S Method ◽

A Priori Error Analysis ◽

Nitsche's Method ◽

Robin Boundary

Download Full-text

Boundaries Influence on the Flow Field Around an Oscillating Sphere

Volume 7: CFD and VIV ◽

10.1115/omae2013-11385 ◽

2013 ◽

Cited By ~ 3

Author(s):

Domenica Mirauda ◽

Antonio Volpe Plantamura ◽

Stefano Malavasi

Keyword(s):

Free Surface ◽

Flow Field ◽

Boundary Conditions ◽

Damping Ratio ◽

Water Channel ◽

Open Water ◽

Free Surface Flows ◽

The Body ◽

Sphere Diameter ◽

Oscillating Sphere

This work analyzes the effects of the interaction between an oscillating sphere and free surface flows through the reconstruction of the flow field around the body and the analysis of the displacements. The experiments were performed in an open water channel, where the sphere had three different boundary conditions in respect to the flow, defined as h* (the ratio between the distance of the sphere upper surface from the free surface and the sphere diameter). A quasi-symmetric condition at h* = 2, with the sphere equally distant from the free surface and the channel bottom, and two conditions of asymmetric bounded flow, one with the sphere located at a distance of 0.003m from the bottom at h* = 3.97 and the other with the sphere close to the free surface at h* = 0, were considered. The sphere was free to move in two directions, streamwise (x) and transverse to the flow (y), and was characterized by values of mass ratio, m* = 1.34 (ratio between the system mass and the displaced fluid mass), and damping ratio, ζ = 0.004. The comparison between the results of the analyzed boundary conditions has shown the strong influence of the free surface on the evolution of the vortex structures downstream the obstacle.

Download Full-text