Commit 895e75b8 by Jonathan Lambrechts

 \section{Fluid equations} \newcommand{\dpartial}[2]{\frac{\partial #1}{\partial #2}} \section{Averaging of variables and operators} \subsection{Averaging of variables and operators} Distribution function of the $\alpha$ phase: \xi_{\alpha}(\vc{r},t) = \begin{cases} 1 \quad \text{si} \quad \vc{r} \in dv_{\alpha} \\ ... ... @@ -78,7 +78,7 @@ c_{\alpha} \nabla \langle f \rangle^{\alpha} + \nabla c_{\alpha} = - \dfrac{1}{dv} \int_{s_{\alpha\beta}} n_{\alpha\beta} ds_{m} \section{Mass balance} \subsection{Mass balance} Micro~: \dfrac{\partial \rho}{\partial t} + \nabla \cdot \rho \vc{v} = 0 ... ... @@ -108,7 +108,7 @@ assuming a constant density in the fluid ($\langle \rho \section{Momentum Balance} \subsection{Momentum Balance} Momentum equation (Incompressible Navier Stokes,$ρ=\mbox{cst}$) integrated over the fluid domain: ρ\,\Big(\frac{∂v}{∂t}\Big)_{α} + ρ\,(vv)_{α} + (∇·τ)_α -(∇p)_α = 0 \label{eq:stokes_momentum} ... ...  ... ... @@ -52,12 +52,14 @@ \begin{document} \bibliographystyle{plainnat} \input{todo.tex} \newpage \input{equations.tex} \input{scontact.tex} \input{particle-fluid-interaction.tex} \input{stability.tex} \input{stability2.tex} \input{imex.tex} \input{scontact.tex} \newpage \bibliography{zotero} \end{document}  \noindent \section{Particle-fluid interaction forces} NB : all those papers deal with gaz-particle mixture in general and fluidized bed in particular. I'm not sure this can be applied to subsurface water. \section*{Particle-fluid interaction forces} While other forces can be seen in the literature \citep{zhu_discrete_2007}, the principal forces present in the gas-particle interactions are the drag force$G$and the pressure gradient force$F$. \subsection*{Pressure gradient force} \subsubsection*{Pressure gradient force} The pressure gradient force can be easily computed by F = -V_p∇p where$p$is the fluid pressure field and$V_p$the volume of the particle \citep{anderson_fluid_1967}. \subsection*{Drag force} \subsubsection*{Drag force} There exists many parameterization of the drag force, \citet{li_gas-particle_2003} did a systematic study that indicates that they all possess similar predictive capability. \citet{di_felice_fluid-particle_2012} uses the following simple formulation: \label{eq:fdp} ... ... @@ -28,7 +27,7 @@ dependency on the Reynolds number: for example, that proposed by \citet {di_feli β = 1.8 - 0.65\exp\big(-0.5(1.5 - \log \re_p)²\big). \section*{Discretization} \subsubsection*{Discretization} \begin{enumerate} \item$ε\$ is computed as the projection of the particles volume on the mesh nodes \item[] This is exactly like a quadrature rule where the particles position would be the quadrature points and the particle volume the corresponding weights. ... ... @@ -41,7 +40,7 @@ F_i &= ∑_pF_p(P_p, (∇P)_p)φ_i(x_p)\\ &=∑_p\left(\frac{∂F_p}{∂P_p}φ_j(x_p) + \frac{∂F_p}{∂(∇P)_p}{·}∇φ_j(x_p)\right)φ_i(x_p) \end{align*} \end{enumerate} \section{Comparison with Darcy} \subsubsection*{Comparison with Darcy} In our current implementation (Brinkman law), we have \[ F = -ρ\frac{με²}{K₁}(u_p - u_f) + \dots\text{, with }K₁ = \frac{k₁ε³}{(1-ε)²} ... ...