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Exact viscous diffusion layer and material-circulation decay — conditional physical prediction

Physical scope

t>0,y>0 initial-wall corner excluded; exact layer bound and separately smooth periodic sine-shear material loop. Worked example is the actual explicit field/parameter selection in that section. B leaf cannot supply any later item.

Statement

Let $ρ_{0}, μ > 0$ , $ν = μ / ρ_{0}$ , and $U \in R$ . Use SI units: length metres, time seconds, velocity $U, A$ in $m s^{- 1}$ , density $k g m^{- 3}$ , viscosity $P a s$ , diffusivity $ν$ in $m^{2} s^{- 1}$ , periods $L_{x}, L_{y}, L_{z}$ in metres, and $k$ in $m^{- 1}$ . For Cartesian fields use $div u = \partial_{x} u_{x} + \partial_{y} u_{y} + \partial_{z} u_{z}$ and componentwise $Δ u$ . For $t > 0, y > 0$ and $x, z \in R$ , set $I_{0} = \int_{0}^{\infty} e^{- r^{2}} d r$ , $E (s) = I_{0}^{- 1} \int_{s}^{\infty} e^{- r^{2}} d r$ for $s \geq 0$ , and $u (t, x, y, z) = (U E (y / (2 \sqrt{ν t})), 0, 0)$ , with constant pressure. This field is smooth for positive time and positive $y$ , solves unsteady Stokes and full constant-density Navier–Stokes because its convective term vanishes, has vector wall trace $(U, 0, 0)$ for $t > 0$ , tends to zero as $y \to \infty$ , and tends to zero as $t ↓ 0$ for each fixed $y > 0$ . The corner $(t, y) = (0, 0)$ is excluded. For $0 < ε < 1$ , $y \geq 2 \sqrt{ν t} \sqrt{\log (1 / ε)} ⟹ ∣ u (t, y) ∣ \leq ε ∣ U ∣ .$

Separately let the domain be the flat three-torus $T_{L_{x}} \times T_{L_{y}} \times T_{L_{z}}$ , where $T_{L} = R / (L Z)$ and $L_{x}, L_{y}, L_{z} > 0$ . Choose an integer $m \geq 1$ , set $k = 2 π m / L_{y}$ , and take $A \in R$ . Let $π : R^{3} \to T_{L_{x}} \times T_{L_{y}} \times T_{L_{z}}$ be the quotient map. Use the flat metric and tangent-space identification induced by Euclidean translations; equations on the torus mean equations for periodic lifts on $R^{3}$ . The periodic lift $\tilde{u} (t, x, y, z) = (A e^{- ν k^{2} t} \sin (k y), 0, 0)$ defines a torus vector field by $u (t, π (q)) = d π_{q} (\tilde{u} (t, q))$ ; with constant pressure it is a smooth exact Navier–Stokes solution for all $t \in R$ . Choose $b, c > 0$ with $c < L_{x} / 2$ , $b < L_{y} / 2$ , and $α = k b \in (0, π)$ . Define the closed lifted loop $\tilde{X} (t, s) = (c \sin (2 π s) + \frac{A}{ν k^{2}} (1 - e^{- ν k^{2} t}) \sin (k b \cos (2 π s)), b \cos (2 π s), 0), 0 \leq s \leq 1,$ and put $X (t, s) = π (\tilde{X} (t, s))$ . Its material condition is $\partial_{t} \tilde{X} (t, s) = \tilde{u} (t, \tilde{X} (t, s))$ , and its flat-torus circulation is $Γ (t) : = \int_{0}^{1} \tilde{u} (t, \tilde{X} (t, s)) \cdot \partial_{s} \tilde{X} (t, s) d s$ . This value is unchanged by translating the lift by a lattice vector in $L_{x} Z \times L_{y} Z \times L_{z} Z$ . It has the value $Γ (t) = A c C_{α} e^{- ν k^{2} t}$ , where $C_{α} = \int_{0}^{2 π} \cos θ \sin (α \cos θ) d θ > 0$ ; it is nonconstant if $A \neq 0$ . These are distinct exact solutions in the selected Newtonian model, not experiments or general boundary-layer claims.

Statement references

Mandre, Fluid Dynamics Lecture Notes, §3.3.1, Stokes first problem

MIT 2.25 Advanced Fluid Mechanics, Solution: Stokes First Problem, all four pages; similarity and wall-stress scaling only

Facts & Assumptions

Given: $ρ_{0}, μ > 0$ , $ν = μ / ρ_{0}$ , wall speed $U$ , half-space data, positive-time restriction, torus periods, wave number and loop.

[A1]

Single-component nonpolar continuum and material balances: continuum balance laws are adopted physical assumptions; these are conditional model deductions.

[A2]

Specified wall, thermal, periodic and initial data: wall and initial values are separate inputs; the impulsive corner is outside the smooth claim.

[A3]

Newtonian stress and Fourier conductive closure: the selected mechanical closure is Newtonian in the incompressible constant-viscosity branch; heat flux is unused.

[F1]

Constant-density incompressible Newtonian model: the local equation is $div u = 0$ and $ρ_{0} D_{t} u + \nabla p = μ Δ u$ for constant $μ$ .

[F2]

Exact viscous diffusion layer and material-circulation decay: the half-space field, Gaussian tail bound, torus shear solution and specified material-loop circulation are as stated, with the corner excluded.

[F3]

Formal parameterized fluid systems and observables defines Euclidean circulation for a closed piecewise- $C^{1}$ curve $γ : [a, b] \to Ω \subseteq R^{3}$ as $\int_{a}^{b} u (γ (s)) \cdot γ^{'} (s) d s$ . For the flat quotient, Exact viscous diffusion layer and material-circulation decay supplies the periodic-lift convention: $X = π (\tilde{X})$ is material when $\partial_{t} \tilde{X} = \tilde{u} (t, \tilde{X})$ , and its circulation is $\int_{0}^{1} \tilde{u} (t, \tilde{X}) \cdot \partial_{s} \tilde{X} d s$ . Translating the lift by a period-lattice vector changes neither the periodic velocity nor the lift derivative, so the value descends to the specified torus loop.

Proof

1.1givenalgebra

Set $s = y / (2 \sqrt{ν t}) \geq 0$ . The integral $I_{0}$ is positive and finite, since its integrand is positive and $e^{- r^{2}} \leq e^{- r}$ for $r \geq 1$ . Substituting $r = s + z$ in the definition of $E$ and using $(s + z)^{2} \geq s^{2} + z^{2}$ gives $E (s) = I_{0}^{- 1} \int_{0}^{\infty} e^{- (s + z)^{2}} d z \leq e^{- s^{2}} I_{0}^{- 1} \int_{0}^{\infty} e^{- z^{2}} d z = e^{- s^{2}}$ . If $y \geq 2 \sqrt{ν t} \sqrt{\log (1 / ε)}$ , then $s^{2} \geq \log (1 / ε)$ , so $E (s) \leq ε$ and $∣ u ∣ \leq ε ∣ U ∣$ .

1.2A1A2A3F1F2F3algebra

For the periodic lift, div $\tilde{u} = 0$ , $(\tilde{u} \cdot \nabla) \tilde{u} = 0$ , and $\partial_{t} \tilde{u} = ν Δ \tilde{u}$ , so the local Newtonian equation holds. Differentiation gives $\partial_{t} \tilde{X} = (A e^{- ν k^{2} t} \sin (k b \cos 2 π s), 0, 0) = \tilde{u} (t, \tilde{X})$ , so $X = π (\tilde{X})$ is material. By [F3], the circulation is $\int_{0}^{1} \tilde{u} (t, \tilde{X} (t, s)) \cdot \partial_{s} \tilde{X} (t, s) d s$ ; parameterize this integral by $θ = 2 π s$ . The base-loop term is $A c e^{- ν k^{2} t} C_{α}$ . The extra term from the moving horizontal shift is proportional to $\int_{0}^{2 π} \sin (α \cos θ) \sin θ \cos (α \cos θ) d θ = 0$ , since its integrand changes sign under $θ \mapsto 2 π - θ$ . This proves the formula.

2.1A2F2step 1.1step 1.2∎

The first solution is asserted only for $t > 0, y > 0$ ; the second circulation is for the specified torus field and loop. They imply no general boundary-layer or Kelvin theorem.

Exact viscous diffusion layer and material-circulation decay — conditional physical prediction

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources