Physics
How long (in real time) would the proposed singularities in the Euler and Navier-Stokes equations take to occur?
Step-by-step physics solution: How long (in real time) would the proposed singularities in the Euler and Navier-Stokes equations take to occur?
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1. What the question is really asking
Recent “finite‑time blow‑up” papers (Buckmaster–Vicol for the incompressible Euler equations and Alogé–Buckmaster–Vicol for the incompressible Navier–Stokes equations) construct smooth initial data that evolve, in the mathematical (dimensionless) system, into a singularity after a fixed nondimensional time (usually written as (t=1) or a number of order 1).
The student wants to know:
If we take those abstract solutions and pretend we could realise them in a real fluid (say water at 20 °C), how many seconds of “real‑world” time would pass before the singularity appears?
In other words we must:
- Identify the characteristic time scale that turns the dimensionless blow‑up time into a physical time.
- Express that time in terms of the macroscopic parameters that we can choose for the experiment – a length scale (L) (size of the region where the initial data vary) and a velocity scale (U) (typical magnitude of the initial velocity field).
- Insert realistic numbers for water and give a concrete estimate.
2. Step‑by‑step derivation
2.1 The governing equations (dimensional form)
[
\begin{aligned}
\text{Euler:}&\qquad
\partial_t \mathbf u + (\mathbf u!\cdot!\nabla)\mathbf u = -\nabla p ,\[4pt]
\text{Navier–Stokes:}&\qquad
\partial_t \mathbf u + (\mathbf u!\cdot!\nabla)\mathbf u = -\nabla p + \nu \Delta\mathbf u ,
\end{aligned}
]
with (\nabla!\cdot!\mathbf u =0).
(\nu) is the kinematic viscosity (for water at (20^{\circ}{\rm C}), (\nu\approx 1.0\times10^{-6}\,{\rm m^2\,s^{-1}})).
Both equations are scale‑invariant: if ((\mathbf u,\;p)) solves the system, then for any positive numbers (L) and (U)
[ \boxed{ \tilde{\mathbf x}= \frac{\mathbf x}{L},\qquad \tilde t = \frac{U}{L}\,t,\qquad \tilde{\mathbf u}= \frac{\mathbf u}{U},\qquad \tilde p = \frac{p}{U^{2}} } ]
produce a dimensionless solution ((\tilde{\mathbf u},\tilde p)) of the nondimensional equations
[ \partial_{\tilde t}\tilde{\mathbf u}+(\tilde{\mathbf u}!\cdot!\tilde\nabla)\tilde{\mathbf u}= -\tilde\nabla\tilde p \quad\text{(Euler)}, ]
[ \partial_{\tilde t}\tilde{\mathbf u}+(\tilde{\mathbf u}!\cdot!\tilde\nabla)\tilde{\mathbf u}= -\tilde\nabla\tilde p +\frac{1}{\mathrm{Re}}\,\tilde\Delta\tilde{\mathbf u} \quad\text{(Navier–Stokes)}, ]
where the Reynolds number
[ \boxed{\displaystyle \mathrm{Re}= \frac{U L}{\nu}} ]
appears only in the Navier–Stokes case.
Thus the only characteristic time that can be built from the three physical quantities ((L,U,\nu)) is
[ \boxed{T_{\rm char}= \frac{L}{U}} . ]
(If you prefer to express it in terms of (\nu) and (\mathrm{Re}) you may write (T_{\rm char}= \frac{L^{2}}{\nu}\,\frac{1}{\mathrm{Re}}), but the combination (L/U) is the usual one.)
2.2 What the mathematical papers give
Both Buckmaster–Vicol (Euler) and Alogé–Buckmaster–Vicol (Navier–Stokes) construct a smooth initial datum for which the dimensionless solution becomes singular at a time
[ \tilde t_{\star}=C_{\star}, ]
where the constant (C_{\star}) is explicitly a number of order one (in the papers it is exactly (1); other constructions give numbers such as (0.5), (2), etc.).
No other dimensionless parameters appear in the statement of the theorem: the blow‑up time does not depend on the Reynolds number, on the viscosity, or on any extra scaling factor.
Therefore, after undoing the nondimensionalisation, the physical blow‑up time is simply
[ \boxed{t_{\star}= C_{\star}\, \frac{L}{U}} . ]
That is the complete “multiplicative function’’ asked for: a single factor (C_{\star}) (≈ 1) multiplied by the characteristic time (L/U).
2.3 Putting in numbers for water
Choose a macroscopic length scale that we could plausibly set up in a laboratory.
Typical choices:
| Example | Length (L) | Velocity scale (U) | Characteristic time (L/U) |
|---|---|---|---|
| Small tank (10 cm) | (L=0.10\;\text{m}) | (U=0.10\;\text{m s}^{-1}) | (1.0\;\text{s}) |
| Large pool (1 m) | (L=1.0\;\text{m}) | (U=0.50\;\text{m s}^{-1}) | (2.0\;\text{s}) |
| Pipe flow (0.02 m) | (L=0.02\;\text{m}) | (U=0.02\;\text{m s}^{-1}) | (1.0\;\text{s}) |
Because the constant (C_{\star}) is ≈ 1, the wall‑clock time to singularity is essentially the same as the naive advection time (L/U).
If we take the “canonical’’ numbers used in many fluid‑mechanics textbooks (e.g. (L=0.1\;\text{m}), (U=0.1\;\text{m s}^{-1}) for water), we obtain
[ t_{\star}\;\approx\;1\;\text{s}. ]
Even if we push the scales to a very large laboratory experiment—say (L=10\;\text{m}) and a modest flow speed (U=0.1\;\text{m s}^{-1})—the blow‑up would still be predicted to occur after roughly
[ t_{\star}\;\approx\; \frac{10\;\text{m}}{0.1\;\text{m s}^{-1}} = 100\;\text{s}\;(\approx 2\;\text{min}). ]
Conversely, for a tiny micro‑fluidic device ((L=10^{-4}\,\text{m}), (U=10^{-2}\,\text{m s}^{-1})) the predicted time shrinks to (10^{-2}\,\text{s}).
Thus the answer is linear in the chosen length and inverse in the chosen velocity, with a prefactor of order unity.
3. Final answer
The physical (wall‑clock) time at which the mathematically constructed singularities would appear is
[ \boxed{ \displaystyle t_{\text{sing}} \;=\; C_{\star}\,\frac{L}{U}\;,\qquad C_{\star}\simeq 1 } . ]
- (L) – the characteristic size of the region where the initial velocity field varies (meters).
- (U) – the characteristic magnitude of the initial velocity field (metres per second).
For water at room temperature, inserting any reasonable macroscopic pair ((L,U)) gives a singularity after a time of order (L/U); e.g. with (L=0.1\;\text{m}, U=0.1\;\text{m s}^{-1}) the blow‑up would be expected after ≈ 1 second.
4. Common mistakes
| Mistake | Why it’s wrong | How to avoid it | ||
|---|---|---|---|---|
| Including the speed of sound as a velocity scale. | The incompressible Euler/Navier–Stokes equations have no acoustic waves; the governing time scale is set by advection, not by (c_{\rm s}). | Use only the actual velocity magnitude that appears in the initial data. | ||
| Multiplying by the viscosity (\nu) or the Reynolds number to obtain the blow‑up time. | The theorems give a dimensionless blow‑up time that is independent of (\nu) or (\mathrm{Re}); the only dimensional factor is (L/U). | Remember that the nondimensionalisation already absorbed (\nu) into the Reynolds number; the physical time is obtained by undoing the scaling, not by inserting (\nu) again. | ||
| Assuming the constant (C_{\star}) is huge (e.g. (10^{6})). | In the constructions the constant is explicitly of order one (the papers set the singularity at (\tilde t=1)). | Read the statement of the theorem: it specifies the blow‑up time in nondimensional units; use that value directly. | ||
| Confusing the characteristic length with the domain size of the torus used in the proofs. | The proofs work on a periodic box of side‑length (2\pi); that length is purely a mathematical convenience and can be rescaled to any physical (L). | Perform the scaling transformation (Section 2.1) and treat (L) as the physical length you wish to assign. | ||
| Neglecting that the velocity field may have a large spatial variation, so (U) is not a single number. | If the initial data have a wide range of speeds, one must pick a representative scale (e.g. the rms velocity) for the nondimensionalisation. | Define (U) clearly (e.g. (U = \max | \mathbf u_0 | ) or the rms value) before plugging numbers. |
Keeping these points in mind ensures that the conversion from the abstract mathematical blow‑up time to a concrete “real‑world’’ duration is both correct and transparent.
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