Pressure Jump Conditions for Stokes Equations with Discontinuous Viscosity in 2D and 3D (original) (raw)
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The three-dimensional jump conditions for the pressure and velocity fields, up to the second normal derivative, across an incompressible/inextensible interface in the Stokes regime are derived herein. The fluid viscosity is only piecewise continuous in the domain while the embedded interface exerts singular forces on the surround fluids. This gives rise to discontinuous solutions in the pressure and velocity field. These jump conditions are required to develop accurate numerical methods, such as the Immersed Interface Method, for the solutions of the Stokes equations in such situations.
Derivation of Jump Conditions in Multiphase Incompressible Flows with Singular Forces
Plasma and Fusion Research, 2008
For conducting numerical simulations of plasma dynamics consisting of multiple phases, a new immersed interface method (IIM) scheme to solve multiphase flows with different viscosities and densities is being developed. The jump conditions for velocity, pressure, and their derivatives necessary for the finite difference approximations in the IIM are derived. The derivation results in sets of coupled equations that can be solved numerically by an iterative method.
Approximation of the multi-dimensional Stokes system with embedded pressure discontinuities
Advances in Computational Mathematics, 2014
Surface tension in multi-phase fluid flow engenders pressure discontinuities on phase interfaces. In this work we present a finite element method to solve viscous incompressible flows problems, especially designed to cope with such a situation. Taking as a model the Stokes system we study a finite element solution method based on a classical Galerkin-least-squares formulation with an added pressure jump term multiplied by the mesh step size. Both the velocity and the pressure are represented with continuous piecewise linear functions except for the latter field on the embedded interface. A suitable modification of the pressure space is employed in order to represent interface discontinuities. A priori error analyses point to optimal convergence rates for this approach.
Finite element methods for the Stokes system with interface pressure discontinuities
IMA Journal of Numerical Analysis, 2013
Surface tension in multi-phase fluid flow engenders pressure discontinuities on phase interfaces. In this work we present two finite element methods to solve viscous incompressible flows problems, especially designed to cope with such a situation. Taking as a model the two-dimensional Stokes system, we consider solution methods based on piecewise linear approximations of both the velocity and pressure, with either velocity bubble or penalty enrichment, in order to obtain stable discrete problems. Moreover a suitable modification of the pressure space is employed in order to represent interface discontinuities. A priori error analyses point to optimal convergence rates for both approaches, which justify observations from previous numerical experiments carried out in [3].
SIAM Journal on Scientific Computing, 2009
The method developed in this paper is motivated by Peskin's immersed boundary (IB) method, and allows one to model the motion of flexible membranes or other structures immersed in viscous incompressible fluid using a fluid solver on a fixed Cartesian grid. The IB method uses a set of discrete delta functions to spread the entire singular force exerted by the immersed boundary to the nearby fluid grid points. Our method instead incorporates part of this force into jump conditions for the pressure, avoiding discrete dipole terms that adversely affect the accuracy near the immersed boundary. This has been implemented for the two-dimensional incompressible Navier-Stokes equations using a high-resolution finite-volume method for the advective terms and a projection method to enforce incompressibility. In the projection step, the correct jump in pressure is imposed in the course of solving the Poisson problem. This gives sharp resolution of the pressure across the interface and also gives better volume conservation than the traditional IB method. Comparisons between this method and the IB method are presented for several test problems. Numerical studies of the convergence and order of accuracy are included.
A new enrichment space for the treatment of discontinuous pressures in multi-fluid flows
International Journal for Numerical Methods in Fluids, 2012
In this work, a new enrichment space to accomodate jumps in the pressure field at immersed interfaces in finite element formulations, is proposed. The new enrichment adds two degrees of freedom per element that can be eliminated by means of static condensation. The new space is tested and compared to other existing finite element space and with the classical P 1 space in several problems involving jumps in the viscosity and/or the presence of singular forces at interfaces not conforming with the element edges. The combination of this enrichment space with an additional enrichment to accomodate discontinuities in the pressure gradient has also been explored, exhibiting excelent results in problems involving jumps in the density or the volume forces. complicated mathematical structure of the multi-fluid problem, the multi-scale features of the flow, the existence of one or multiple internal interfaces and the unsteadiness of the flow, constitute major challenges for the analysis.