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Exact viscous diffusion layer and material-circulation decay — worked interpretation

Physical scope

Numerical substitutions in two exact Newtonian model solutions; not an observation or a general boundary-layer or Kelvin result.

Example

For the half-space Stokes first-problem model, choose $ν = 1 m^{2} s^{- 1}$ , $U = 1 m s^{- 1}$ , and $t = 1 s$ . At $y = 2 \sqrt{\log 2} m$ , the exact Gaussian-tail estimate gives $∣ u ∣ \leq \frac{1}{2} m s^{- 1}$ . For the separate torus shear choose $ν = 1 m^{2} s^{- 1}$ , $L_{x} = L_{y} = L_{z} = 2 π m$ , $m = 1$ , $k = 1 m^{- 1}$ , $A = 2 m s^{- 1}$ , $b = π / 3 m$ , and $c = 1 m$ . Then $α = π / 3$ and $Γ (t) = 2 C_{π / 3} e^{- t / (1 s)} m^{2} s^{- 1}, C_{π / 3} = \int_{0}^{2 π} \cos θ \sin ((π / 3) \cos θ) d θ > 0.$ These are exact model values, not measured velocity or circulation.

Example references

Mandre, Fluid Dynamics Lecture Notes, §3.3.1

MIT 2.25 Advanced Fluid Mechanics, Solution: Stokes First Problem, similarity scaling

Facts & Assumptions

Given: The positive-time half-space and torus solutions, parameters above, and specified loop.

[F1]

Exact viscous diffusion layer and material-circulation decay — conditional physical prediction: the half-space field obeys $∣ u ∣ \leq ∣ U ∣ e^{- y^{2} / (4 ν t)}$ , and the torus material-loop circulation is $A c C_{α} e^{- ν k^{2} t}$ .

[F2]

Formal parameterized fluid systems and observables defines Euclidean circulation for closed piecewise- $C^{1}$ curves as $\int u (γ (s)) \cdot γ^{'} (s) d s$ . For the torus loop, the physical supplier [F1] instead specifies the flat-quotient lift $X = π (\tilde{X})$ , the periodic velocity $\tilde{u}$ , and the circulation $\int_{0}^{1} \tilde{u} (t, \tilde{X} (t, s)) \cdot \partial_{s} \tilde{X} (t, s) d s$ . The given lift is closed, and period-lattice translations preserve this integral; this is the convention used for the torus value below.

Proof

1.1givenF1algebra

At the selected height $y^{2} / (4 ν t) = \log 2$ , so $∣ u ∣ \leq 1 \cdot e^{- \log 2} = \frac{1}{2} m s^{- 1}$ .

2.1givenF1F2step 1.1algebra

Interpret the torus circulation as the periodic-lift integral in [F2], whose value is supplied by [F1]. For the torus parameters, $ν k^{2} = 1 s^{- 1}$ , $A c = 2 m^{2} s^{- 1}$ , and $α = k b = π / 3$ ; substitution gives the displayed circulation. Positivity of $C_{π / 3}$ follows from the theorem's stated range.

3.1F1step 1.1step 2.1∎

Both outputs are deductions for their specified exact solutions and make no general empirical or boundary-layer claim.

Exact viscous diffusion layer and material-circulation decay — worked interpretation

Example

Example references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources