Governing equations of RII problem
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The full system of PDEs that govern the fluid mechanics
Governing equations
1.Conservation of mass in both phases
\[{\partial (1-\phi) \over \partial t} + \nabla \cdot ((1-\phi) \mathbf{v_s})=-\Gamma\] \[{\partial \phi \over \partial t} + \nabla \cdot (\phi \mathbf{v_l})=\Gamma\]where \(\phi\) is the porosity, \((1-\phi)\) is the fraction of solid phase, t is time, \(v_l\) is fluid velocity, and \(v_s\) is solid velocity, \(\Gamma\) is the melting rate. We assume that \(\Gamma\) is proportional to the undersaturation of soluble component in the melt, so \(\Gamma = R A(\phi,c_s)(c_l^{eq}-c_l)\), where R is a kinetic coefficient with units 1/times.
2.Darcy’s law (conservation of momentum for liquid)
\[\phi (\mathbf{v_l}-\mathbf{v_s}) = {k \over \mu} ((1-\phi) \Delta\rho g \mathbf{\hat z} -\nabla \mathcal{P})\]A Darcy flux \(\phi (\mathbf{v_l}-\mathbf{v_s})\) is driven by gravity \(g \mathbf{\hat z}\) associated with the density difference \(\Delta \rho\), and by compaction pressure gradient \(\nabla \mathcal{P}\). The permeability k is divided by liquid viscosity \(\mu\), depending on the porosity. \(k =k_0 (\phi/\phi_0)^n\), where \(k_0\) is a reference permeability at a reference porosity \(\phi_0\), and n is a constant.
3.Chemical component conservation equation, consisting of three contributions: diffusion, advection and chemical source term.
\[{\partial \over \partial t} (\phi c_l)+ \nabla \cdot (\phi \mathbf{v_l} c_l) = \nabla \cdot (\phi D \nabla c_l) + c_\Gamma \Gamma\]where D is diffusivity in the liquid phase; diffusion through the solid phase is negligible, \(C_\Gamma\), is the concentration of reactively produced melts.
The solid conservation,
\[{\partial \over \partial t} ((1-\phi) c_s) + \nabla \cdot ((1-\phi) \mathbf{v_s} c_s ) =- c_\Gamma \Gamma\]Non-dimensional Scales
Solubility is assumed to be a linear function of height, \({\partial c_{eq} \over \partial z}=\beta\), \(\beta\) is the solubility gradient(\(m^{-1}\))
Assuming zero solubility at the base area z=0, \(c_{eq}= \beta z\),
The concentration \(c_\Gamma\) is offset from the equilibrium concentration by \(\alpha\), \(c_\Gamma=\beta z +\alpha\),
Characteristic velocity,
\[w_0 = {k_0 \Delta \rho g \over \mu \phi_0}\]Other characteristic scales:
\((x,z) = H (x’,z')\), \(\nabla\)= \({1\over H } \nabla\),
\(\phi = \phi_0 \phi’\),
$k=k_0 (d’)^2 (\phi’) ^n$
\(\mathbf{v_l}=w_0 \mathbf{v_l}'=(k_0 \Delta \rho g/\mu \phi_0) \mathbf{v_l}'\),
\(\mathbf{v_s}=\phi_0 w_0 \mathbf{v_s}’\),
\(t=\alpha/(w_0\beta)t’\),
\(\mathcal{P} = \mathcal{P_0} \mathcal{P}’ =(\zeta_0 \phi_0 w_0 \beta/\alpha) \mathcal{P}'\),
\(c_l=\beta H c_l’\),
\(\Gamma = (\phi_0 w_0 \beta / \alpha) \Gamma’\),
$c_s = c_{s0} c_s’$,
$d = d_0 d’=bc_{s0} d’$,
\(\zeta=\zeta_0 \zeta’\),
Scaled reaction rate
Reaction rate \(\Gamma = R A(\phi,c_s)(c_l^{eq}-c_l)\),
\[{\phi_0 w_0 \beta \over \alpha}\Gamma = R A'(\beta H z-\beta H c_l)\\\]So the scaled Reaction rate \(\Gamma’\) is,
\[\Gamma={R \alpha \over \phi_0 w_0 \beta} A' (\beta H z-\beta H c_l)\\ \Gamma={\alpha R H \over \phi_0 w_0} A'(z-c_l) \\ \Gamma=Da A'(z-c_l), A’= {c_s (1-\phi_0 \phi) \over c_{s0}(1-\phi_0)}\\\]Non-dimensionalization step by step
1. The dimensionless mass conservation in the solid
Starting with the first equation (1), \({\partial (1-\phi) \over \partial t} + \nabla \cdot ((1-\phi) \mathbf{v_s)}=-\Gamma\),
\[-{\partial \phi \over \partial t} + (1- \phi ) \nabla \cdot \mathbf{v_s} - \mathbf{v_s}\nabla \phi =-\Gamma\]The compaction rate \(\nabla \cdot \mathbf{v_s}\) is related to the compaction pressure $\mathcal{P}$ according to the linear constitutive law.
\[\nabla \cdot \mathbf{v_s} = \mathcal{P} / \zeta = \mathcal{C}\]Eq (1) becomes,
\[-{\partial \phi \over \partial t} + (1-\phi) P/\zeta - \mathbf{v_s} \nabla \cdot \phi = -\Gamma\]Put the dimensionless parameter for the dimensional one,
\[-{\partial (\phi_0 \phi) \over \partial (\alpha/(w_0\beta)t)} + (1-\phi_0 \phi){\zeta_0 \phi_0 w_0 \beta/\alpha P \over \zeta_0 \zeta} - {\phi_0 w_0 \mathbf{v_s} \over H} \cdot\nabla (\phi_0 \phi ) = -{\phi_0 w_0 \beta \over \alpha} \Gamma\] \[-{w_0\beta\phi_0 \over \alpha}{\partial \phi \over \partial t} + (1-\phi_0 \phi){\phi_0 w_0 \beta \over \alpha} {P \over \zeta}- {\phi_0^2 w_0 \mathbf{v_s}\over H}\cdot \nabla \phi = -{\phi_0 w_0 \beta \over \alpha} \Gamma\]Multiple by \({ \alpha\over\phi_0 w_0 \beta }\), and get,
\[-{\partial \phi \over \partial t} + (1-\phi_0 \phi) C - {\phi_0 \over M} \mathbf{v_s} \cdot \nabla \phi= -\Gamma\] \[{\partial \phi \over \partial t} +{\phi_0 \over M} \mathbf{v_s} \cdot \nabla \phi = (1- \phi_0 \phi) C +\Gamma\]The dimensionless reactive melting rate \(\Gamma\) is equal to the scaled undersaturation \(\chi\),
We simplify the equations by taking the limit of small porosity \(\phi_0 <<M << 1\), and \(M = {\beta H \over \alpha}\), so \({\phi_0 \over M} << 1\). We also assume $\zeta$ is a constant, the equation becomes,
\[{\partial \phi \over \partial t}= \mathcal{P} + \chi\]2. The dimensionless mass conservation in the liquid
The scaled eq(2) \({\partial \phi \over \partial t} + \nabla \cdot (\phi \mathbf{v_l})=\Gamma\) becomes,
\[{\partial (\phi_0\phi) \over \partial (\alpha/(w_0\beta)t)} + {1 \over H}\nabla \cdot (\phi_0 \phi w_0 \mathbf{v_l})= \phi_0 w_0 \beta / \alpha \Gamma\] \[{\phi_0 w_0\beta \over \alpha}{\partial \phi \over \partial t} + {\phi_0 w_0 \over H}\nabla \cdot ( \phi \mathbf{v_l})= {\phi_0 w_0\beta \over \alpha} \Gamma\]Multiple by \({ \alpha\over\phi_0 w_0 \beta }\) and get,
\[{\partial \phi \over \partial t} + {\alpha \over \beta H}\nabla \cdot ( \phi \mathbf{v_l})= \Gamma\]\(M= {\beta H \over \alpha}\), so \({\alpha\over \beta H} = 1/M\),
\[M {\partial \phi \over \partial t} + \nabla \cdot ( \phi \mathbf {v_l})= M\chi\]3. The dimensionless Darcy’s law
eq (3) Darcy’s law, \(\phi (\mathbf{v_l}-\mathbf{v_s}) = {k \over \mu} ((1-\phi) \Delta\rho g\hat z -\nabla \mathcal{P})\),
\[\phi_0\phi (w_0v_l-\phi_0 w_0 v_s) = K_0 K ((1-\phi_0\phi) \Delta\rho g\hat z - {1 \over H} \nabla ((\zeta \phi_0 w_0 \beta/\alpha) \mathcal{P}))\]Taking the limit of small porosity \(\phi_0 <<M << 1\), we can neglect the \(\phi_0 w_0 v_s\) and \(\phi_0\phi \Delta\rho g\hat z\),
\[\phi_0 w_0 \phi v_l = K_0 K (\Delta\rho g\hat z - {1 \over H} \nabla ({\zeta \phi_0 w_0 \beta \over\alpha} P)\]Divided by \(\phi_0 w_0\),
\[\phi(\mathbf{v_l} -\phi_0 \mathbf{v_s}) ={k_0 k \over \mu} ({1\over \phi_0 w_0 }(1- \phi_0\phi) \Delta \rho g \mathbf{\hat{z}} - {1 \over H} \nabla ({\zeta_0 \beta \over\alpha} \mathcal{P}))\]Darcy’s flux \(\phi_0 w_0= {k_0 \Delta \rho g \over \mu}\), so \(k_0 = {\phi_0 w_0\mu \over \Delta \rho g}\),
\[\phi(\mathbf{v_l} -\phi_0 \mathbf{v_s}) = k((1-\phi_0 \phi) \mathbf{\hat z}- {k_0 \zeta_0 \beta \over\alpha H} \nabla \mathcal{P})\]Stiffness \(S = M {\delta^2 \over H^2} = {\beta H \over \alpha } {K_0 \zeta \over H^2} = {\beta k_0 \zeta \over \alpha H}\),
\[\phi(\mathbf{v_l} -\phi_0 \mathbf{v_s}) = k((1-\phi_0 \phi)\mathbf{\hat z}- S\nabla \mathcal{P})\]Taking the limit of small porosity \(\phi_0 <<M << 1\), we can neglect the \(\phi \phi_0\) term and get,
\[\phi \mathbf {v_l} = k (\hat z - S \nabla \mathcal{P})\]4. The dimensionless Chemical component conservation
Eq (5) \({\partial \over \partial t} (\phi c_l)+ \nabla \cdot (\phi \mathbf{v_l} c_l) = \nabla \cdot (\phi D \nabla c_l) + c_\Gamma \Gamma\)
\[\phi{\partial c_l\over \partial t} +c_l{\partial \phi\over \partial t}+ \phi \mathbf{v_l} \nabla \cdot c_l + c_l \nabla \cdot (\phi \mathbf{v_l}) = \nabla \cdot (\phi D \nabla c_l) + c_\Gamma \Gamma\]Use eq(2) \({\partial \phi \over \partial t} + \nabla \cdot (\phi \mathbf{v_l})=\Gamma\) to simply eq(28)
\[\phi{\partial c_l\over \partial t} +\phi \mathbf{v_l} \cdot \nabla c_l = \nabla \cdot (\phi D \nabla c_l) + (c_\Gamma - c_l)\Gamma\]Scaled eq(4),
\[\phi_0\phi{\partial (\beta H c_l)\over \partial (\alpha/(w_0\beta) t)} +\phi_0w_0 \phi \mathbf{v_l} \cdot {1 \over H}\nabla (\beta H c_l) = {1 \over H} \nabla \cdot (\phi_0 \phi D {1 \over H} \nabla (\beta H c_l)) + \alpha \phi_0 w_0 \beta / \alpha \Gamma\] \[{\phi_0 w_0\beta\beta H \over \alpha}\phi{\partial c_l\over \partial t} +\phi_0w_0 \beta \phi \mathbf{v_l} \cdot \nabla c_l = {\phi_0 \beta D \over H} \nabla \cdot ( \phi \nabla c_l) + \phi_0 w_0 \beta \Gamma\]Divide the whole equation by \(\phi_0 w_0 \beta\), and \(M= {\beta H \over \alpha}\),
\[M\phi{\partial c_l\over \partial t} +\phi \mathbf{v_l} \cdot \nabla c_l = {D \over w_0 H} \nabla \cdot ( \phi \nabla c_l) +\Gamma\]Since \(Pe = {w_o H \over D}\), eq(28) becomes,
\[\phi M{\partial c_l\over \partial t} +\phi \mathbf{v_l} \cdot \nabla c_l = {1 \over Pe} \nabla \cdot ( \phi \nabla c_l) +\Gamma\]Given \(M= {\beta H \over \alpha} <<1\), we can neglect the first term. Given undersaturation \(\chi = Da(z-c_l)\), \(c_l = z - {\chi \over Da}\),
\[\nabla c_l = \nabla(z- {\chi \over Da})=\hat {z}-{\nabla\chi \over Da}\]\(\nabla \cdot (\phi \nabla c_l)={\partial \over \partial x} (\phi {\partial c_l\over \partial x}) + {\partial \over \partial z} (\phi {\partial c_l\over \partial z})\),but \(c_l\) is meaningful in x axis,
\[\nabla \cdot (\phi \nabla c_l)={\partial \over \partial x} (\phi {\partial c_l\over \partial x}) =-{1\over Da} {\partial \over \partial x} (\phi {\partial \chi\over \partial x})\]So the equation becomes,
\[\phi \mathbf{v_l} \cdot [\hat z-{\nabla\chi \over Da}] = -{1 \over Da Pe } {\partial \over \partial x} (\phi {\partial \chi\over \partial x}) + \chi\] \[\phi \mathbf {v_l} \cdot [{\nabla\chi \over Da}- \hat z] = {1 \over Da Pe } {\partial \over \partial x} (\phi {\partial \chi\over \partial x}) - \chi\]5. The dimensionless Chemical component conservation
eq (5) \({\partial \over \partial t} ((1-\phi) c_s) + \nabla \cdot ((1-\phi) \mathbf{v_s} c_s ) =- c_\Gamma \Gamma\),
\[c_s{\partial (1-\phi) \over \partial t} +(1-\phi) {\partial c_s \over \partial t} + (1-\phi) \mathbf{v_s} \cdot \nabla c_s + c_s \nabla \cdot ((1-\phi) \mathbf{v_s}) =- c_\Gamma \Gamma\]Use eq(1) \({\partial (1-\phi) \over \partial t} + \nabla \cdot ((1-\phi) \mathbf{v_s})=-\Gamma\) to simply eq(35),
\[(1-\phi) {\partial c_s \over \partial t} + (1-\phi) \mathbf{v_s} \cdot \nabla c_s =(c_s- c_\Gamma) \Gamma\]Scale eq(36),
\[(1-\phi_0\phi) {\partial c_{s0} c_s \over \partial (\alpha /w_0 \beta)t} + (1-\phi_0 \phi) \phi_0 w_0 {c_{s0} \over H} \mathbf{v_s} \cdot \nabla c_s =(c_{s0} c_s- c_\Gamma') (\phi_0 w_0 \beta /\alpha) \Gamma\] \[{w_0 \beta c_{s0} \over \alpha}(1-\phi_0\phi) {\partial c_s \over \partial t} + {\phi_0 w_0 c_{s0} (1-\phi_0 \phi) \over H} \mathbf{v_s} \cdot \nabla c_s ={\phi_0 w_0 \beta \over \alpha}(c_{s0} c_s- c_\Gamma') \Gamma\]Divided by \(w_0 \beta /\alpha\),
\[c_{s0}(1-\phi_0\phi) {\partial c_s \over \partial t} + {\phi_0 c_{s0}\alpha \over\beta H} (1-\phi_0 \phi) \mathbf{v_s} \cdot \nabla c_s ={\phi_0}(c_{s0} c_s- c_\Gamma') \Gamma\] \[{\partial c_s \over \partial t} +{\phi_0 \over M} \mathbf{v_s}\cdot \nabla c_s = {\phi_0 \over (1-\phi_0 \phi) c_{s0}} (c_{s0} c_s-c_\Gamma') \Gamma\]If we take the limit \(\phi_0 <<M<<1\),
\[{\partial c_s \over \partial t} = {\phi_0 \over (1-\phi_0 \phi) c_{s0}} (c_{s0} c_s-\beta H (z-{\chi \over Da})-\alpha) \chi\]Dimensionless equations
So the five governing equations are,
\[{\partial \phi \over \partial t} + {\phi_0 \over M} \mathbf{v_s} \cdot \nabla \phi= (1-\phi_0 \phi) C+\chi\] \[M (1- \phi_0 \phi) C + \phi \cdot \nabla v_l+ (v_l - \phi_0 v_s) \cdot \nabla \phi = 0\] \[\phi(\mathbf{v_l} -\phi_0 \mathbf{v_s}) = k((1-\phi_0 \phi)\mathbf{\hat z}- S\nabla \mathcal{P})\] \[{M \over Da}\phi{\partial \chi\over \partial t} +\phi v_l \cdot [{\nabla\chi \over Da}-z] = {1 \over Pe} \nabla \cdot (\phi({\nabla \chi \over Da}-z)) - \chi\] \[{\partial c_s \over \partial t} +{\phi_0 \over M} \mathbf{v_s}\cdot \nabla c_s = {\phi_0 \over (1-\phi_0 \phi) c_{s0}} (c_{s0} c_s-(\beta H + \alpha) \Gamma\]Taking the limit of small porosity, the five dimensionless governing equations (1)-(4) become,
\[{\partial \phi \over \partial t} = \mathcal{P} + \chi\] \[M{\partial \phi \over \partial t}+\nabla \cdot (\phi \mathbf {v_l}) =M\chi\] \[\phi \mathbf{v_l}=k (\hat z-S \nabla \mathcal P)\] \[\phi \mathbf {v_l} \cdot ({\nabla \chi \over Da} - \hat z)= {1\over DaPe} {\partial \over \partial x}(\phi{\partial \chi \over \partial x})-\chi\] \[{\partial c_s \over \partial t} = {\phi_0 \over (1-\phi_0 \phi) c_{s0}} (c_{s0} c_s-\beta H (z-{\chi \over Da})-\alpha) \chi\]