Common-wall residual and capillary boundary model

Let $I\subseteq \mathbb{R}$ be an open time interval and let $D\subseteq {\mathbb{R}}^3$ be a bounded open $C^2$ domain. Here a $C^2$ domain is a nonempty connected open set whose boundary near each of its points, after a rigid Cartesian change of coordinates, is the graph $s_3=b(s_1,s_2)$ of a $C^2$ function on an open subset of ${\mathbb{R}}^2$, with the domain locally on one side of that graph. Use the Cartesian gradient, divergence, row-component Jacobian, and componentwise Laplacian on $D$. For constants ${\rho }_0,\nu >0$, a body acceleration $f\in C(I\times D;{\mathbb{R}}^3)$, a velocity $w\in C^1(I;C^2(D;{\mathbb{R}}^3))$, and a scalar pressure $q\in C(I;C^1(D))$, define the full momentum residual

$\mathcal{R}(w,q):={\mathrm{\partial }}_{t}w+(w\cdot \mathrm{\nabla })w+{\rho }_0^{-1}\mathrm{\nabla }q-\nu \mathrm{\Delta }w-f\mathrm{.}$

The forced constant-density incompressible model is $\mathcal{R}(u,p)=0$ with $\mathrm{div}u=0$. For boundary statements, $C^k(\bar{D})$ means restrictions of $C^k$ functions on an open neighborhood of $\bar{D}$, with all derivatives through order $k$ restricted continuously to the closure. A classical boundary trace is the restriction of this continuous extension to $\mathrm{\partial }D$; it is independent of the extension. A common-wall comparison requires $u,v\in C^1(I;C^2(\bar{D};{\mathbb{R}}^3))$ in this convention, with $\mathrm{div}u=\mathrm{div}v=0$, and their classical traces agree with the same prescribed boundary velocity $\gamma (t,\cdot )$. The interior residual above remains defined for the stated interior regularity even when no wall trace is supplied. For a divergence-free comparison field $w\in C^1(I;C^2(\bar{D};{\mathbb{R}}^3))$ in this same boundary regularity class with a different trace, a divergence-free wall lift is a field $\mathrm{\ell }\in C^1(I;C^2(\bar{D};{\mathbb{R}}^3))$ in the same extension convention with $\mathrm{div}\mathrm{\ell }=0$ and $\mathrm{\ell }{\mathrm{\mid }}_{\mathrm{\partial }D}=w{\mathrm{\mid }}_{\mathrm{\partial }D}-\gamma$; whenever such a lift is supplied, the corrected field is $v=w-\mathrm{\ell }$ and its residual is $\mathcal{R}(v,q)$. This definition asserts no lift exists for arbitrary boundary data.

For a flat periodic channel $D_{\mathrm{c}\mathrm{h}}={\mathbb{T}}_{L_x}\times (0,h)\times {\mathbb{T}}_{L_z}$, where ${\mathbb{T}}_L=\mathbb{R}\mathrm{/}(L\mathbb{Z})$ and $L_x,L_z,h>0$, a specific tangential lift for lower and upper wall defects $a(t)e_x$ and $b(t)e_x$ is

$\mathrm{\ell }(t,x,y,z)=(a(t)(1-y\mathrm{/}h)+b(t)y\mathrm{/}h,0,0)\mathrm{.}$

Here $e_x=(1,0,0)$, $a,b\in C^1(I,\mathbb{R})$, and the channel is periodic in $x,z$ with physical walls $y=0,h$. This lift is defined only for the stated tangential wall-defect data.

For the capillary-gravity model, let $h,\rho ,g,L_x,L_y>0$, $\sigma \ge 0$, use horizontal coordinates $(x,y)\in {\mathbb{T}}_{L_x}\times {\mathbb{T}}_{L_y}$ and vertical coordinate $z\in (-h,0)$, positive upward, with the flat mean surface at $z=0$. Let $\eta \in C^1(I;C^2({\mathbb{T}}_{L_x}\times {\mathbb{T}}_{L_y}))$ be surface elevation and $\varphi \in C^1(I;C^2({\mathbb{T}}_{L_x}\times {\mathbb{T}}_{L_y}\times [-h,0]))$ a real scalar velocity potential: the represented velocity is $u:=\mathrm{\nabla }\varphi =({\varphi }_x,{\varphi }_y,{\varphi }_z)$, with no added uniform component. Functions on a horizontal torus are identified with their periodic lifts to ${\mathbb{R}}^2$, and their spatial derivatives are the derivatives of these lifts. In this closed-slab regularity notation the fields extend to a neighborhood of the slab with the indicated derivatives continuous, so all displayed derivatives extend continuously to the top and bottom. Set $\mathrm{\Delta }={\mathrm{\partial }}_x^2+{\mathrm{\partial }}_y^2+{\mathrm{\partial }}_z^2$ and ${\mathrm{\Delta }}_H={\mathrm{\partial }}_x^2+{\mathrm{\partial }}_y^2$. The capillary-gravity potential model is

$\mathrm{\Delta }\varphi =0(-h<z<0),{\varphi }_z(t,x,y,-h)=0,$

${\eta }_t(t,x,y)={\varphi }_z(t,x,y,0),{\varphi }_t(t,x,y,0)+g\eta (t,x,y)-\frac{\sigma }{\rho }{\mathrm{\Delta }}_H\eta (t,x,y)=0.$

These are definitions of mathematical model interfaces. They assert neither existence of a solution for general data nor nonlinear free-surface or sharp-interface convergence. In the optional SI interpretation, $[x]=[y]=[z]=[h]=\mathrm{m}$, $[t]=\mathrm{s}$, $[\eta ]=\mathrm{m}$, $[\varphi ]={\mathrm{m}}^2{\mathrm{s}}^{-1}$, $[\rho ]=\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$, $[g]=\mathrm{m}{\mathrm{s}}^{-2}$, $[\sigma ]=\mathrm{N}{\mathrm{m}}^{-1}$, $[\nu ]={\mathrm{m}}^2{\mathrm{s}}^{-1}$, and $[f]=[\mathcal{R}]=\mathrm{m}{\mathrm{s}}^{-2}$.

Cartesian fluid operators and the formal constant-density incompressible system: Formal parameterized fluid systems and observables.

Bounded-domain Navier–Stokes function spaces and energy estimates: Temam, §§2.5, 3.1, printed pp. 15–18 (PDF pp. 33–36).

Periodic relative-energy estimate and bounded-domain no-slip qualifications: Wiedemann, §§5.1–5.2, printed pp. 38–43.

Finite-depth capillary-gravity boundary model: MIT 18.357, Chapter 19, printed pp. 78–79.

Independent deep-water capillary-wave derivation and surface-tension pressure: ETH Zürich, lecture 23, printed pp. 185–187 (PDF pp. 5–7).