Creeping continuum pressure and bounded observations

Assume Countable Choice. Let $D^{\ast }\subset {\mathbb{R}}^3$ be a nonempty bounded connected Lipschitz open set whose boundary has a finite rigidly moved graph-chart atlas, contained in a slab $(a,b)\times {\mathbb{R}}^2$, and set $C_P(D^{\ast }):=b-a$. Let ${\mathrm{\ell }}_0>0$ metres and $U_0>0$ metres per second be reference length and velocity; the physical domain is ${\mathrm{\Omega }}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}={\mathrm{\ell }}_0{D}^{\ast }$. Fix constant density ${\rho }_0>0$ kilograms per cubic metre and constant dynamic viscosity ${\mu }_0>0$ pascal seconds. Write ${\nu }^{\ast }:={\mu }_0\mathrm{/}({\rho }_0{U}_0{\mathrm{\ell }}_0)>0$ for the dimensionless kinematic viscosity and ${\nu }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}:={\mu }_0\mathrm{/}{\rho }_0$ for its dimensional value in square metres per second. For each continuous body-acceleration field $b_i\in C^0({\mathrm{\Omega }}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}};{\mathbb{R}}^3)\cap L^2({\mathrm{\Omega }}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}};{\mathbb{R}}^3)$ in metres per second squared, set $g_i^{\ast }(y):={\mathrm{\ell }}_0{b}_i({\mathrm{\ell }}_{0}y)\mathrm{/}{U}_0^2$. Use the forced creeping Stokes branch defined on this page: the velocity $u_i^{\ast }\in V:=H_0^1(D^{\ast };{\mathbb{R}}^3)$ with gradient norm $\mathrm{\parallel }v{\mathrm{\parallel }}_V=\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_{L^2(D^{\ast })}$ has zero trace on the selected stationary no-slip wall, the pressure $q_i^{\ast }\in Q:=L_0^2(D^{\ast })=\{q\in L^2(D^{\ast }):{\int }_{D^{\ast }}qdy=0\}$ is the spatial mean-zero multiplier, and the weak model equation is

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

where $\epsilon (w)=(\mathrm{\nabla }w+\mathrm{\nabla }{w}^{\mathsf{T}})\mathrm{/}2$ and $F_i(v):={\int }_{D^{\ast }}{g}_i^{\ast }(y)\cdot v(y)dy$. The physical pressure is recovered from $p_{i,\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}({\mathrm{\ell }}_{0}y)={\rho }_0{U}_0^2{q}_i^{\ast }(y)$ using this mean-zero gauge. Let $m\ge 1$ be an integer and let $S_1,\dots ,S_m:Q\to \mathbb{R}$ be bounded real-linear pressure sensors with $\mathrm{\parallel }{S}_j{\mathrm{\parallel }}_{Q^{\mathrm{\prime }}}:={\sup }_{\mathrm{\parallel }q{\mathrm{\parallel }}_{L^2(D^{\ast })}\le 1}\mathrm{\mid }{S}_j(q)\mathrm{\mid }$. Define the physical pressure-output vector ${\mathcal{P}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(b_i):={\rho }_0{U}_0^2(S_1(q_i^{\ast }),\dots ,S_m(q_i^{\ast }))\in {\mathbb{R}}^m$. The earlier Stokes-load solvability and sensor-stability theorem gives the weak solution pairs and yields

$\mathrm{\parallel }{\mathcal{P}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(b_1)-{\mathcal{P}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(b_2){\mathrm{\parallel }}_{{\mathbb{R}}^m}\le 3{\rho }_0{U}_0^2{C}_{\mathrm{d}\mathrm{i}\mathrm{v}}(D^{\ast })C_P(D^{\ast }){\left({\sum }_{j=1}^m\mathrm{\parallel }{S}_j{\mathrm{\parallel }}_{Q^{\mathrm{\prime }}}^2\right)}^{1\mathrm{/}2}\mathrm{\parallel }{g}_1^{\ast }-g_2^{\ast }{\mathrm{\parallel }}_{L^2(D^{\ast })},$

where $C_{\mathrm{d}\mathrm{i}\mathrm{v}}(D^{\ast })<\mathrm{\infty }$ satisfies: there is a bounded linear right inverse $B_{D^{\ast }}:Q\to V$ with $\mathrm{div}{B}_{D^{\ast }}q=q$ and $\mathrm{\parallel }\mathrm{\nabla }{B}_{D^{\ast }}q{\mathrm{\parallel }}_{L^2(D^{\ast })}\le C_{\mathrm{d}\mathrm{i}\mathrm{v}}(D^{\ast })\mathrm{\parallel }q{\mathrm{\parallel }}_Q$. This is a conditional stability result for the adopted forced creeping model. It does not infer that inertia is negligible from $\mathrm{Re}$, estimate the error against the full Navier–Stokes model at finite Reynolds number, fix an absolute pressure gauge, or report an experiment.

Forced creeping Stokes branch and scaled weak formulation

Selected stationary no-slip boundary data

Lipschitz stability of bounded Stokes pressure observations

Bounded-domain Poincare inequality

Stewart, MIT 8.09, §6.8

Tong, Fluid Mechanics, §3.4

Cambridge, Fluid Dynamics II (D), Chapter 4

Guermond, MATH 661, Chapter 53

Boffi–Brezzi–Fortin, Mixed Finite Element Methods and Applications