Mier-Torrecilla et al 2010a - Revision history

Cinmemj at 09:56, 27 February 2019

2019-02-27T09:56:22Z

Scipediacontent at 09:15, 14 January 2019

2019-01-14T09:15:31Z

Scipediacontent at 09:13, 14 January 2019

2019-01-14T09:13:01Z

Scipediacontent at 09:11, 14 January 2019

2019-01-14T09:11:25Z

Cinmemj: Cinmemj moved page Draft Samper 466629376 to Mier-Torrecilla et al 2010a

2018-02-28T14:24:33Z

Cinmemj moved page Draft Samper 466629376 to Mier-Torrecilla et al 2010a

Cinmemj: Created page with "

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-<!-- metadata commented in wiki content
+Published in ''Int. J. Numer. Meth. Fluids'' Vol. 67 (11), pp. 1516-1539, 2010<br />
-==Advances in the simulation of multi-fluid flows   with the Particle Finite Element Method.  Application to bubble dynamics.==
+doi:10.1002/fld.2429
-−
-'''M.&nbsp;Mier-Torrecilla, S.R.&nbsp;Idelsohn<span id="fnc-1"></span>[[#fn-1|<sup>1</sup>]]and E.&nbsp;Oñate'''
--->
-−
 ==Abstract==

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 and a physical part (calculating how the flow variables change in time at each node) described by the following Stokes equations:
 {| class="formulaSCP" style="width: 100%; text-align: left;"
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-The latter can be further split into two parts: one related to viscosity effects, and the other related to the incompressibility. According to the incremental pressure segregation scheme <span id='citeF-25'></span>[[#cite-25|[25]]], and after implicit backward Euler time discretization, we propose the following splitting of equations ([[#5 PRESSURE SEGREGATION FOR THE MULTI-FLUID EQUATIONS|5]]) <span id='citeF-26'></span>[[#cite-26|[26]]]:
+The latter can be further split into two parts: one related to viscosity effects, and the other related to the incompressibility. According to the incremental pressure segregation scheme <span id='citeF-25'></span>[[#cite-25|[25]]], and after implicit backward Euler time discretization, we propose the following splitting of equations ([[#eq-11.g|11]]) <span id='citeF-26'></span>[[#cite-26|[26]]]:
 <ol>

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 Let <math display="inline">\Omega \subset \boldsymbol{R}^d</math>, <math display="inline">d\in \{ 2,3\} </math>, be a bounded domain containing two different fluids (see Figure [[#img-1|1]]). We denote time by <math display="inline">t</math>, the Cartesian spatial coordinates by <math display="inline">\boldsymbol{x}=\{ x_i\} _{i=1}^d</math>, and the vectorial operator of spatial derivatives by <math display="inline">\boldsymbol{\nabla }=\{ \partial _{x_i}\} _{i=1}^d</math>. The evolution of the velocity <math display="inline">u=u(\boldsymbol{x},t)</math> and the pressure <math display="inline">p=p(\boldsymbol{x},t)</math> is governed by the Navier-Stokes equations:
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
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 ===Boundary and interface conditions===
-In order for the Navier-Stokes problem ([[#eq-1.c|1.c]]) to be well-posed, suitable boundary conditions need to be specified. On the external boundary <math display="inline">\partial \Omega =\Gamma _D\cup \Gamma _N</math>, such that <math display="inline">\Gamma _D\cap \Gamma _N=\emptyset </math>, we consider the following:
+In order for the Navier-Stokes problem ([[#eq-1.c|1]]) to be well-posed, suitable boundary conditions need to be specified. On the external boundary <math display="inline">\partial \Omega =\Gamma _D\cup \Gamma _N</math>, such that <math display="inline">\Gamma _D\cap \Gamma _N=\emptyset </math>, we consider the following:
 <span id="eq-4"></span>

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 ===Boundary and interface conditions===
-In order for the Navier-Stokes problem ([[#2 GOVERNING EQUATIONS|2]]) to be well-posed, suitable boundary conditions need to be specified. On the external boundary <math display="inline">\partial \Omega =\Gamma _D\cup \Gamma _N</math>, such that <math display="inline">\Gamma _D\cap \Gamma _N=\emptyset </math>, we consider the following:
+In order for the Navier-Stokes problem ([[#eq-1.c|1.c]]) to be well-posed, suitable boundary conditions need to be specified. On the external boundary <math display="inline">\partial \Omega =\Gamma _D\cup \Gamma _N</math>, such that <math display="inline">\Gamma _D\cap \Gamma _N=\emptyset </math>, we consider the following:
 <span id="eq-4"></span>

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