Questions tagged [fluid-dynamics]
The fluid-dynamics tag has no usage guidance.
107
questions
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Should water at the scale of a cell feel more like tar?
The Navier-Stokes equations are as follows,
$$\dot{u}+(u\cdot \nabla ) u +\nu \nabla^2 u =\nabla p$$
where $u$ is the velocity field, $\nu$ is the viscosity, and $p$ is the pressure.
Some elementary ...
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31
votes
5
answers
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views
Is there a mathematically precise definition of turbulence for solutions of Navier-Stokes?
Given a solution $S$ of the Navier-Stokes equations, is there a way to make mathematically precise a statement like: "$S$ is turbulent in the spacetime region $U$"?
And if such a definition ...
- 5,283
17
votes
2
answers
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views
Convergence of solutions to Navier-Stokes to Euler's equation for viscosity $\to$ zero
Let
$$
\partial_t u + \nabla_u u = - \nabla p
$$
be Euler's equation (Wikipedia) for an ideal incompressible fluid. Let
$$
\partial_t u + \nabla_u u - \nu \Delta u = - \nabla p
$$
be the Navier-...
- 1,544
15
votes
2
answers
723
views
Surjectivity of curl
Let: $\mathbb R^3\ni x\mapsto v(x)\in\mathbb R^3$ be a vector field with null divergence belonging to the Schwartz class such that
$$
\int_{\mathbb R^3} v(x) dx=0.
$$
Is it true that there exists a ...
- 15.2k
14
votes
1
answer
648
views
Riemann, fluid dynamics, and critical lines
Marcus du Sautoy, in the section Riemann's Final Twist (pp. 278-80) in his book The Music of the Primes, discusses a discovery of Jon Keating of a connection in Riemann's Nachlass between Riemann's ...
- 9,931
12
votes
0
answers
7k
views
Otelbayev's approach to Navier-Stokes [closed]
Recent news post that Mukhtarbai Otelbayev from Eurasian National University has shown existence of strong solutions of the Navier-Stokes equation in the article
"Existence of a strong solution of ...
- 12.3k
11
votes
3
answers
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views
Navier-Stokes fluid dynamics, Einstein gravity and holography
There was some activity a while ago, like 10 years ago, string theoreists try to relate
the fluid dynamics, for example, governed by Navier-Stokes equation,
to
the Einstein gravity, and its ...
- 10.3k
11
votes
2
answers
811
views
Do circular pipes maximize flow rate?
Suppose that $U \subset \mathbb{R}^2$ is nonempty, open, connected and bounded. Consider a Poisseuille flow in the pipe $U \times \mathbb{R}$. That is: a time-independent incompressible flow of the ...
- 1,451
11
votes
0
answers
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How to define (and compute) the Cartan-Killing form of the group of volume-preserving diffeomorphisms?
This question was raised a while ago in a blog post by Terry Tao on the Euler-Arnold equation and he called it "quite tricky".
Has anyone in the meantime tried to formulate this question precisely, ...
- 1,675
8
votes
3
answers
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views
Textbook suggestions for rigorous fluid dynamics
I am interested in studying fluid dynamics and am searching for a good introductory textbook. I know just the very basics of fluids on the physics side. For mathematical prerequisites, I have ...
- 421
8
votes
2
answers
318
views
Compressible Ebin-Marsden?
In Ebin and Marsden's paper Groups of Diffeomorphisms and the Motion of an Incompressible Fluid, there is a footnote on the first page indicating that non-homogeneous cases and the case of ...
- 37.6k
7
votes
0
answers
299
views
Derivation of a stochastic Navier-Stokes equation under the assumption of perturbed particle trajectories
Let
$d\in\left\{2,3\right\}$
$\mathcal V_t\subseteq\mathbb R^d$ be the bounded domain occupied by an incompressible Newtonian fluid at time $t\ge 0$
$\Phi_t:\mathcal V_0\to\mathcal V_t$ such that $\...
- 161
7
votes
0
answers
609
views
Constructing black noise with non-standard analysis
With noise in the sense of i.i.d. random sequence,
a noise is black if it is not isomorphic to standard Gaussian white noise.
Tsirelson showed the existence of black noise through the scaling limit ...
user24451
6
votes
2
answers
660
views
Explanation for why an ideal fluid doesn't have increasing entropy?
The equations of motion for a very simple ideal fluid (specifically a calorically perfect, monatomic, ideal gas) are \begin{align*}\dot{\rho}+\nabla \cdot (\rho u)=0 \;&\text{(mass conservation)} \...
- 919
6
votes
2
answers
932
views
Vortex Voronoi diagram?
Suppose there are a finite number of disjoint unit-radii disks in the
plane, each spinning clockwise or counterclockwise at the same
angular velocity.
The plane is filled with a thin fluid layer,
and ...
- 149k
6
votes
1
answer
402
views
Stationary Navier-Stokes solutions
Are there known nontrivial ($u\neq0$) stationary solutions to Navier-Stokes equations in $\mathbb R^3$ ? Not square integrable of course (that's impossible), but with self-similar amplitudes of ...
- 3,055
6
votes
1
answer
428
views
Definition of the nonlinear part of the drift in a (stochastic) Navier-Stokes equation
Let
$T>0$
$d\in\mathbb N$
$\Lambda\subseteq\mathbb R^d$ be bounded and open
$\mathcal V:=\left\{v\in C_c^\infty(\Lambda)^d:\nabla\cdot v=0\right\}$, $$V:=\overline{\mathcal V}^{\left\|\;\cdot\;\...
- 161
6
votes
0
answers
179
views
How much energy will be released in the explosion when one shoots a superelastic billiard ball into a collection of still superelastic billiard balls?
Consider the following scenario. Let $\alpha>1$. Suppose whenever two superelastic balls collide at speed $\gamma$ they bounce off each other at speed $\gamma\cdot\alpha$ (i.e. $\alpha$ is the ...
- 27.8k
5
votes
3
answers
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views
Book about fluid dynamics
Next Monday, I'll have an interview at Siemens for an internship where I have to know about fluid dynamics/computational fluid dynamics. I'm not a physicist, so does somebody have a suggestion for a ...
- 77
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votes
1
answer
355
views
Incompressible Navier-Stokes equation with heat conduction
How does the incompressible Navier-Stokes system read with heat conduction?
Where can I find an existence result for its weak solutions?
user123456
5
votes
1
answer
210
views
Interpretation for a condition in fluid dynamics
I have been working with some mathematical models in biology and fluid mechanics. My problem is about
the interpretation of a condition that I found for a vector
representing the velocity of a fluid. ...
- 171
5
votes
1
answer
416
views
A moving boundary in rock mechanics
I'm concern a moving boundary problem in rock mechanics.
We consider a problem of unsaturated flow of an in-compressible fluid in a
porous medium(rock) like D. Moreover suppose that support of a ...
- 93
5
votes
0
answers
290
views
Similarity in Navier-Stokes equation and convolution in finite abelian groups?
Let $G$ be a finite abelian group, $X = (x_g)_{g \in G}$ be a vector of variables.
Set for $g \in G$:
$$\tau_g(X) := \frac{1}{|G|} \sum_{\rho \text{ irred. }} \chi_{\rho}(-g) \exp(\sum_{s \in G} \chi_{...
- 2,462
5
votes
0
answers
168
views
Estimate of $\Vert \nabla u \Vert_{L^{\infty}(\Omega)}$ of Navier Stokes equations
My question is how to estimate the term $\Vert \nabla u \Vert_{L^{\infty}(\Omega)}$. Here we consider the 2D incompressible Navier Stokes equations:$$u_t -\Delta u+u\cdot \nabla u+\nabla p=f$$ and $$\...
- 83
4
votes
1
answer
182
views
Compactly supported transverse traceless tensors
Let $(M, g)$ be a Riemanian manifold (or $\mathbb{R}^n$ if you prefer). A TT-tensor is a symmetric 2-tensor $\sigma_{ab}$ satisfying
$g^{ab} \sigma_{ab} \equiv 0$ ($\sigma$ is trace free),
$\nabla^a ...
- 1,253
4
votes
1
answer
304
views
Doubt on Morrey spaces of measures according to T. Giga and Y. Miyakawa
I'm reading 'Navier-Stokes flow in $\mathbb{R}^3$ with measures as initial vorticity and Morrey spaces' (a paper of Giga and Miyakawa) but there is something that I don't understand about the ...
- 41
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votes
1
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322
views
Long wavelength instability: Linear Vs nonlinear phenomenon
I am looking into stability for certain nonlinear PDE on $\mathbb{R}$ around a specific steady solution, $f_0(x)$. The nonlinear Cauchy PDE is given by:
$\dfrac{\partial f(x,t)}{\partial t}=\mathbf{N}...
- 318
4
votes
0
answers
195
views
Schrödinger Bridge for other costs
Stochastic control formulations of the Schrödinger bridge problem between $\mu,\nu$ are well known (e.g Chen et al Eq. 4.23)
$$\inf \limits_{p_t, v_t} \int_0^T \int \frac{1}{2}\lvert v_t\rvert^2 p_t ...
- 91
4
votes
0
answers
121
views
Algebra properties regarding Gevrey spaces: closed under multiplication
In page 24 of the paper Landau Damping: Paraproducts and Gevrey Regularity, the authors claimed an algebra property of Gevrey spaces, the formula (3.14), without giving a proof. So I'm asking for a ...
- 517
4
votes
0
answers
139
views
Existence results for Lagrangian solutions to the Incompressible Euler Equation?
It is known that if a function (which we shall call the lagrangian flow, or lagrangian trajectory) $$X:(\mathbb{R}/\mathbb{Z})^3 \times [0,T] \to \mathbb{R}^3$$ with $X \in H^1_t$ (i.e. has weak time ...
- 919
3
votes
1
answer
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A solution to the Navier-Stokes equation that is defined for on $[0,T]$ with $T$ large is global?
Let $u_0 \in \dot{H}^{1/2}(\mathbb{R}^3)$. The Fujita-Kato theorem gives rise to a local unique solution $(t,x) \mapsto u(t,x)$ to the Navier-Stokes equations $$\left\{
\begin{array}{ccc}
\partial _t ...
- 105
3
votes
1
answer
531
views
Why are solenoidal fields called solenoidal?
A solenoidal tangent field, mathematically speaking, is one whose divergence vanishes. They are also called incompressible. I understand why they are called incompressible — a fluid flow is called ...
- 2,222
3
votes
2
answers
405
views
References on thin film equation: derivation and properties
Where can I find
a derivation of the thin film equation
$$u_t = - \mathrm{div} (u^m\nabla\Delta u)$$ from a physical model?
a good introduction to its properties (e.g. conserved quantities and ...
user121481
3
votes
3
answers
413
views
A question about intuition of fluid limit in queuing system
This is a question about intuition in understanding the fluid limit queuing system.
Assume we have a sequence of queuing systems $\{S^N\}_{N=1}^{\infty}$ with N servers and each server has unit ...
- 191
3
votes
1
answer
908
views
Solving Stokes Equations using 3D Fourier transforms
How do you calculate the inverse Fourier transform of $\frac{k_ik_j}{k^4}$. I know it has to be a matrix of the form $=δ_{ij}A(r)+r_ir_jB(r)$, but how do you calculate the functions A(r) and B(r)?
I ...
- 31
3
votes
2
answers
176
views
Variational principle for relativistic gas dynamics
I know quite a lot of Variational principles (VP) yielding systems of classical mechanics. By a VP, I mean something like
$$\delta{\cal L}[U]=0$$
where ${\cal L}$ is a functional and the field belongs ...
- 51.6k
3
votes
1
answer
492
views
Hadamard-Rybczynski problem
HR problem deals with a spherical fluid viscous drop falling in a different fluid under influence of gravity. The outer fluid is of uniform speed $U$ in direction of gravity, far away from the drop. ...
- 319
3
votes
1
answer
452
views
Difference between linear and parabolic velocity profiles in Stokes flows of two fluids
For droplet interactions in low-Reynolds number flow, solutions are available when the underlying flow can be written as linear compositions of strain and rotation, see Batchelor & Green (1972a).
...
- 41
3
votes
1
answer
353
views
First integrals of a 3D incompressible flow
Let $\Omega$ be an unbounded periodic smooth domain of $\mathbb{R}^3$. We are Given an incompressible vector field $q:\Omega\subset\mathbb{R}^3\rightarrow \mathbb{R}^3$ (i.e. $\nabla\cdot q\equiv 0$ ...
- 41
3
votes
1
answer
197
views
The discrete theory of compressible fluids dynamics
I am working on the discrete theory of compressible fluids dynamics, i.e., numerically solving and simulating the compressible fluids , we are interested in the way using discrete exterior calculus, ...
- 1,499
3
votes
0
answers
66
views
Does the incompressible Navier-Stokes equation have a smooth solution if the initial vorticity is smooth and $p$ integrable for any $p$?
Consider the incompressible Navier-Stokes equation on $(0,T)\times \mathbb{R}^3$ for fixed $T>0$.
If the sequence of mollified initial vorticities $(\omega_0^{\nu})_{\nu}$ is uniformly bounded in $...
- 173
3
votes
0
answers
61
views
How I can distibute values over the computational cells?
I am an engineering student and I try to solve the fluid equations over a given set of computational cells. I have a mathematical question about a field I am currently studying, precisely the ...
3
votes
0
answers
162
views
Radial-energy decomposition of a velocity field in 2D: is anyone able to show that the lemma below is true (…or false)?
In the book “Vorticity and Incompressible Flow” by Majda and Bertozzi, there is the following lemma.
Lemma 3.2. Any smooth incompressible vector field $v$ in $\mathbb{R}^2$ with vorticity
$$\omega=\...
- 1,055
3
votes
0
answers
345
views
On Solving a Fourth-Order Non-Linear PDE
I am presently working on a problem in fluid dynamics where our group is investigating the behavior of temperature and velocity at the leading edge of a flat plate when fluid flows past it. The ...
- 419
3
votes
0
answers
81
views
Books on turbulent compressible fluid (gas) in heated channel
It's been a while I got my hands dirty with simulation of hydrodynamics and it was mostly incompressible and laminar. Now, I need to model turbulent flow in channel with additional external heating ...
- 355
3
votes
0
answers
89
views
Lattice Boltzman derivation for vorticity eqn $\omega_{t}+ v\cdot \nabla \omega=\mu \Delta \omega$
So as showed by Frisch et al. (a), the 2D Euler equation $$v_{t}+ v\cdot \nabla v=\mu \Delta v$$ can be derived by the Hexagonal-placed automaton (for low velocity).
I am curious about the existence ...
- 395
3
votes
0
answers
103
views
Question on the local existence theory for the classical solution for the incompressible fluid dynamics equation
For the incompressible fluid dynamics system
\begin{equation}
\begin{split}
&\nabla\cdot v=0,\\
&\dfrac{\partial v}{\partial t}+v\cdot \nabla v+\nabla p=A(S,D)v,
\end{split}
\end{equation}
...
- 31
3
votes
0
answers
176
views
Is a certain set of periodic solutions of the 2D Navier-Stokes equations closed generically?
I would be interested to know if a certain set of periodic solutions for
the two-dimensional Navier-Stokes equations is closed generically.
Many similar (yet not identical) set-ups can be found in the ...
3
votes
0
answers
174
views
What does the renormalization group flow corresponding to a turbulent subrange with a broad band forcing look like?
In a renormalization group analysis of turbulent flows, such as for example done by Barbi and Münster here who derive an action for the Navier-Stokes equations, insert it into the Wilson equation, and ...
- 408
3
votes
0
answers
143
views
What is the relationship between complex time singularities and UV fixed points?
In this paper it is described how the turbulent kinetic energy spectrum and the flatness (a measure for intermittency) are governed by the position of the (dominant) singularities of the solutions of ...
- 408