Mechanical power balance under adopted wall laws

Assume Countable Choice, meaning every countable family of nonempty sets has a choice function. Let $T>0$ seconds. The spatial domain is either (B) a bounded connected nonempty open $\mathrm{\Omega }\subset {\mathbb{R}}^3$ with $C^2$ boundary, or (P) a periodic channel ${\mathrm{\Omega }}_{L_x,h,L_z}:=(\mathbb{R}\mathrm{/}{L}_x\mathbb{Z})\times (0,h)\times (\mathbb{R}\mathrm{/}{L}_z\mathbb{Z})$ with $L_x,L_z,h>0$ metres. In (P), fields are periodic in the first and third coordinates on the Euclidean covering space, and the only physical wall boundary is $\mathrm{\Gamma }=(\mathbb{R}\mathrm{/}{L}_x\mathbb{Z})\times \{0,h\}\times (\mathbb{R}\mathrm{/}{L}_z\mathbb{Z})$; identified periodic faces are not walls. In (B), set $\mathrm{\Gamma }:=\mathrm{\partial }\mathrm{\Omega }$. In either case let $n$ be the outward unit normal on $\mathrm{\Gamma }$, let $dx$ be Euclidean volume measure (or its periodic-cell quotient measure), and let $dS$ be the induced area measure on $\mathrm{\Gamma }$. Let the constant mass density be $\rho >0$ kilograms per cubic metre, and let the coefficient $\mu$ of the adopted Newtonian closure be constant and denote its value by $\eta >0$ pascal-seconds; let the body-acceleration field be $b\in C([0,T]\times \bar{\mathrm{\Omega }};{\mathbb{R}}^3)$ in metres per second squared, the velocity be $u\in C^1([0,T];C^2(\bar{\mathrm{\Omega }};{\mathbb{R}}^3))$ in metres per second, and the pressure be $p\in C([0,T];C^1(\bar{\mathrm{\Omega }}))$ in pascals. Define $D(u{)}_{ij}:=({\mathrm{\partial }}_{x_j}{u}_i+{\mathrm{\partial }}_{x_i}{u}_j)\mathrm{/}2$, $\sigma :=-pI+2\eta D(u)$, and rowwise $(\mathrm{div}\sigma {)}_i:={\sum }_j{\mathrm{\partial }}_{x_j}{\sigma }_{ij}$. For matrices $A,B\in {\mathbb{R}}^{3\times 3}$ define the Frobenius contraction $A:B:={\sum }_{i,j=1}^3{A}_{ij}{B}_{ij}$ and the squared norm $\mathrm{\mid }D(u){\mathrm{\mid }}^2:=D(u):D(u)={\sum }_{i,j=1}^{3}D(u{)}_{ij}^2$. In the proof, $D$ abbreviates $D(u)$. Put $I=(0,T)$. In case (B), put $B=\mathrm{\Omega }$ and assume $\chi \in C^2(I\times B;{\mathbb{R}}^3)$, with each ${\chi }_t:B\to B$ an orientation-preserving $C^2$ diffeomorphism, and $u(t,\chi (t,a))={\mathrm{\partial }}_t\chi (t,a)$. In case (P), put $\tilde{\mathrm{\Omega }}=\mathbb{R}\times (0,h)\times \mathbb{R}$ and $\mathrm{\Lambda }=\{(m{L}_x,0,n{L}_z):m,n\in \mathbb{Z}\}$; denote periodic lifts of $u,b,\sigma$ by $\tilde{u},\tilde{b},\tilde{\sigma }$. Assume $\tilde{\chi }\in C^2(I\times \tilde{\mathrm{\Omega }};{\mathbb{R}}^3)$, with each ${\tilde{\chi }}_t:\tilde{\mathrm{\Omega }}\to \tilde{\mathrm{\Omega }}$ an orientation-preserving $C^2$ diffeomorphism satisfying $\tilde{\chi }(t,a+\lambda )=\tilde{\chi }(t,a)+\lambda$ for every $\lambda \in \mathrm{\Lambda }$, and $\tilde{u}(t,\tilde{\chi }(t,a))={\mathrm{\partial }}_t\tilde{\chi }(t,a)$. For the material balance, write $(B,\chi ,u^{\mathrm{♯}},b^{\mathrm{♯}},{\sigma }^{\mathrm{♯}})=(\mathrm{\Omega },\chi ,u,b,\sigma )$ in case (B), and $(B,\chi ,u^{\mathrm{♯}},b^{\mathrm{♯}},{\sigma }^{\mathrm{♯}})=(\tilde{\mathrm{\Omega }},\tilde{\chi },\tilde{u},\tilde{b},\tilde{\sigma })$ in case (P). Here $\nu$ denotes the outward unit normal of ${\chi }_t(V)$; in case (P), the material-region integrals are on the Euclidean covering space. For every $t\in I$ and every nonempty bounded open $V⋐B$ with $C^1$ boundary, assume the adopted material linear-momentum balance

$\frac{d}{dt}{\int }_{{\chi }_t(V)}\rho u^{\mathrm{♯}}dx={\int }_{{\chi }_t(V)}\rho b^{\mathrm{♯}}dx+{\int }_{\mathrm{\partial }{\chi }_t(V)}{\sigma }^{\mathrm{♯}}\nu dS,$

Assume incompressibility as a model constraint, $\mathrm{div}u=0$ throughout the fluid domain (and $\mathrm{div}\tilde{u}=0$ on its lift in case (P)). At every time impose one of the following wall laws: (N) no-slip, $u=g$ on $\mathrm{\Gamma }$ for a prescribed tangential wall-velocity field $g\in C^1([0,T];C^2(\mathrm{\Gamma };{\mathbb{R}}^3))$ in metres per second with $g\cdot n=0$; or (S) stationary Navier friction on branch (B) only, $u\cdot n=0$ and $(\sigma n{)}_{\tau }=-\beta u_{\tau }$ on $\mathrm{\Gamma }$, where $a_{\tau }:=a-(a\cdot n)n$ and $\beta \ge 0$ is the adopted constant wall coefficient in pascal-seconds per metre. Then the local momentum equation

$\rho ({\mathrm{\partial }}_{t}u+(u\cdot \mathrm{\nabla })u)=\mathrm{div}\sigma +\rho b$

holds on the fluid domain by the local momentum result from the adopted material balance; incompressibility is the separate model constraint stated above, not a conclusion of that result. Let $K(t):={\int }_{\mathrm{\Omega }}\rho \mathrm{\mid }u(t,x){\mathrm{\mid }}^2\mathrm{/}2dx$. Then for every $t\in (0,T)$ in case (N),

$\frac{dK}{dt}={\int }_{\mathrm{\Omega }}\rho b\cdot udx-2\eta {\int }_{\mathrm{\Omega }}\mathrm{\mid }D(u){\mathrm{\mid }}^{2}dx+{\int }_{\mathrm{\Gamma }}g\cdot \sigma ndS,$

and for every $t\in (0,T)$ in case (S),

$\frac{dK}{dt}={\int }_{\mathrm{\Omega }}\rho b\cdot udx-2\eta {\int }_{\mathrm{\Omega }}\mathrm{\mid }D(u){\mathrm{\mid }}^{2}dx-\beta {\int }_{\mathrm{\Gamma }}\mathrm{\mid }{u}_{\tau }{\mathrm{\mid }}^{2}dS\mathrm{.}$

Every term has units of watts. In branch (B) only, in either stationary homogeneous case, (N) with $g=0$ or (S) with a fixed $\beta$, there is at most one velocity field $u$ in this smooth class for a given body force and initial velocity. No pressure uniqueness is asserted. These are deductions conditional on the stated model and wall assumptions; no solution-existence or empirical-applicability claim is made.

David Tong, Fluid Mechanics, Chapter 2, §§3.1.1–3.1.2 and §§3.2.1–3.2.3

Joseph M. Powers, Lecture Notes on Intermediate Fluid Mechanics, §§4.4.5.2 and 5.4.1

Chen and Qian, A Study of the Navier–Stokes Equations with the Kinematic and Navier Boundary Conditions, §§1–2

David Tong, Fluid Mechanics, Chapter 2, §§3.1.1–3.1.2

Joseph M. Powers, Lecture Notes on Intermediate Fluid Mechanics, §4.4.5.2