Forced creeping Stokes branch for pressure observations

Assume Countable Choice, meaning that every sequence of nonempty sets has a choice function. Let $D^{\ast }\subset {\mathbb{R}}^3$ be a nonempty bounded connected open set in dimensionless coordinates whose boundary has a finite rigidly moved Lipschitz graph-chart atlas. Choose $a<b$ such that $D^{\ast }\subset (a,b)\times {\mathbb{R}}^2$ and set $C_P(D^{\ast }):=b-a$. Let ${\mathrm{\ell }}_0>0$ metres be a selected reference length and $U_0>0$ metres per second a selected reference velocity. The physical domain is ${\mathrm{\Omega }}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}={\mathrm{\ell }}_0{D}^{\ast }$. Take ${\rho }_0>0$ kilograms per cubic metre and ${\mu }_0>0$ pascal seconds as primitive branch-local physical parameters: this branch adopts them as constant density and dynamic shear viscosity inputs and derives neither from another framework nor from observations. Let $b_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}:{\mathrm{\Omega }}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\to {\mathbb{R}}^3$ be a continuous body-acceleration field in metres per second squared such that $g^{\ast }(y):={\mathrm{\ell }}_0{b}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}({\mathrm{\ell }}_{0}y)\mathrm{/}{U}_0^2$ belongs to $L^2(D^{\ast };{\mathbb{R}}^3)$. The dimensionless kinematic viscosity ${\nu }^{\ast }:={\mu }_0\mathrm{/}({\rho }_0{U}_0{\mathrm{\ell }}_0)>0$, the Reynolds number $\mathrm{Re}:={\rho }_0{U}_0{\mathrm{\ell }}_0\mathrm{/}{\mu }_0=1\mathrm{/}{\nu }^{\ast }$, and the dimensional kinematic viscosity ${\nu }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}:={\mu }_0\mathrm{/}{\rho }_0$ in square metres per second are derived quantities in this branch.

Use the real spaces $V:=H_0^1(D^{\ast };{\mathbb{R}}^3)$ with gradient norm $\mathrm{\parallel }v{\mathrm{\parallel }}_V=\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_{L^2(D^{\ast })}$ and $Q:=L_0^2(D^{\ast })=\{q\in L^2(D^{\ast }):{\int }_{D^{\ast }}qdy=0\}$ with its $L^2$ norm. Here $H_0^1$ is the $H^1$ closure of compactly supported smooth vector fields, and $Q$ selects the spatial mean-zero representative of pressure. Let $u^{\ast }\in V$ be dimensionless velocity, let $q^{\ast }\in Q$ be dimensionless pressure, set $\epsilon (u^{\ast })=(\mathrm{\nabla }{u}^{\ast }+\mathrm{\nabla }{u}^{\ast \mathsf{T}})\mathrm{/}2$, and define the dimensionless load functional $F_{g^{\ast }}(v):={\int }_{D^{\ast }}{g}^{\ast }(y)\cdot v(y)dy$ for $v\in V$. The bounded-domain Poincare inequality makes $F_{g^{\ast }}$ a continuous linear functional on $V$.

The physical fields corresponding to the dimensionless solution are $u_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}({\mathrm{\ell }}_{0}y)=U_0{u}^{\ast }(y)$ and $p_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}({\mathrm{\ell }}_{0}y)={\rho }_0{U}_0^2{q}^{\ast }(y)$. Define $D_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=({\mathrm{\nabla }}_x{u}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}+{\mathrm{\nabla }}_x{u}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}^{\mathsf{T}})\mathrm{/}2$ in inverse seconds and ${\sigma }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=-p_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}{I}_3+2{\mu }_0{D}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ in pascals. The parameterized forced creeping Stokes weak formulation uses this physical stress, a divergence-free velocity, and the separately adopted steady force balance $\mathrm{div}{\sigma }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}+{\rho }_0{b}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=0$ from the current creeping-force postulate, which permits a specified body-acceleration field. It is the zero-inertia alternative branch; no material-acceleration equation is imposed on the same field. Its nondimensional weak form is the pair of equations

$2{\nu }^{\ast }{\int }_{D^{\ast }}\epsilon (u^{\ast }):\epsilon (v)dy-{\int }_{D^{\ast }}{q}^{\ast }\mathrm{div}vdy=F_{g^{\ast }}(v)(v\in V),-{\int }_{D^{\ast }}r\mathrm{div}{u}^{\ast }dy=0(r\in Q)\mathrm{.}$

The functional satisfies $\mathrm{\parallel }{F}_{g^{\ast }}{\mathrm{\parallel }}_{V^{\mathrm{\prime }}}\le C_P(D^{\ast })\mathrm{\parallel }{g}^{\ast }{\mathrm{\parallel }}_{L^2(D^{\ast })}$ by the bounded-domain Poincare inequality. The chosen stationary no-slip wall condition is the zero trace encoded by $V$. The pressure is a constraint multiplier with mean-zero gauge. This weak interface contains no material-acceleration term. It asserts no additional solution existence, uniqueness, approximation error, or empirical applicability.

Axiom of Countable Choice

Stokes spaces and weak bilinear forms

Bounded-domain Poincare inequality

Newtonian stress closure

Alternative steady creeping force balance, including body acceleration

Selected stationary no-slip boundary data

Stewart, MIT 8.09, §6.8

Tong, Fluid Mechanics, §3.4

Cambridge, Fluid Dynamics II (D), Chapter 4