Questions tagged [fluid-dynamics]

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Critical Reynolds numbers for turbulence in 3D and 2D planar Couette flows

In 3 spatial dimensions, the incompressible Navier-Stokes equations are: $$ \begin{split} \frac{\partial u_i}{\partial t} + \sum_{j=1}^3 u_j \frac{\partial u_i}{\partial x_j} &= - \frac{\partial p}...
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14 votes
1 answer
397 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 ...
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2 votes
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38 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 $...
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1 answer
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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 ...
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Viscous stress equation in Newtonian fluid

In this Wikipedia entry, it is said that for the incompressible isotropic case of Newtonian fluid, the viscous stress equation is $$ \tau_{i j}=\mu\left(\frac{\partial v_{i}}{\partial x_{j}}+\frac{\...
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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 ...
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5 votes
0 answers
266 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_{...
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8 votes
2 answers
223 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 ...
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1 answer
117 views

Generalising results on superfluid Kubo formulas

In a 2014 article by Chapman, Hoyos and Oz, the authors study non-equilibrium fluid dynamics and describe a method for deriving Kubo formulas for thermal transport coefficients of superfluids (the ...
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3 votes
0 answers
84 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=\...
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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 ...
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32 views

Existence and Uniqueness of lifting Hele-Shaw problem

I am researching for the existence and uniqueness of solutions for the equation in figure below enter image description here $$\nabla\cdot u = \frac{\dot b(t)}{b(t)} \text{ in }\Omega(t) \tag{1}$$ The ...
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3 votes
2 answers
157 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 ...
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3 votes
1 answer
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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 ...
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-1 votes
1 answer
72 views

Regularity of stationary incompressible Navier-Stokes equations in $\mathbb R^2$

What is the regularity of solutions for the stationary incompressible Navier-Stokes equations \begin{align*} -\Delta u +u\cdot \nabla u + \nabla p &= 0\\ \nabla \cdot u &= 0 \end{align*} in $\...
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1 vote
0 answers
128 views

Zermelo's Navigation Problem [closed]

I am trying to solve Zermelo's Navigation Problem. One of the cases I'm looking at is when the river's current is a function of the $x$-position only. From what I learned in Fluid Mechanics courses, I ...
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0 votes
1 answer
103 views

Regularity in Navier Stokes from $L^2$ bound on vorticity

How would one show that if $\omega$ is the vorticity associated to $\partial_t u+u\cdot \nabla u -\nu \Delta u +\nabla p=0$ (with smooth, compactly supported initial data) and $$\omega\in L^\infty([0,...
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2 votes
1 answer
306 views

Flow induced by differentiable velocity field is differentiable

Let $E$ be a $\mathbb R$-Banach space, $\tau>0$ and $v:[0,\tau]\times E\to E$ such that$^1$ $$x\mapsto t\mapsto v(t,x)\tag1$$ belongs to $C^{0,\:1}(E,C^0([0,\tau],E))$. This is enough to ensure ...
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5 votes
2 answers
475 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)} \...
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64 votes
3 answers
5k views

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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4 votes
0 answers
128 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 ...
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1 vote
0 answers
69 views

Modelling fluid flows with mean curvature flow

A while ago I was wondering if the displacement of fluid described in this blog post could be modelled with mean curvature flow or some other flow, but when I asked someone in Engineering they replied ...
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2 votes
0 answers
601 views

Applications of linear algebra in the design of aircraft [closed]

David Lay mentioned one application of linear algebra in the design of aircraft in the introductory part of chapter 2 of his book: [...] A computer creates a model of the surface by first ...
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0 answers
156 views

Help to understand a limit $\varepsilon\rightarrow 0$ computation on a fluid mechanic paper

In Córdoba and Gancedo - Contour dynamics of incompressible 3-D fluids in a porous medium with different densities (page 4) I read that if $$ v (x_1,x_2,x_3,t)=-\frac{\rho_2-\rho_1}{4\pi} \...
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1 vote
0 answers
53 views

Density gradient of Navier-Stokes equations in perforated domain

Question: Is it possible to bound the density gradient $\nabla \rho_\epsilon$ of strong solutions to the compressible NSE uniformly in $L^\gamma$? Preliminaries: Consider a bounded connected domain $\...
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2 votes
1 answer
322 views

Leray projector in $L^{\infty}$ and negative order Besov spaces for the Navier-Stokes equations

I was reading the paper "Norm inflation for the generalized Navier-Stokes equations" which can be found here: https://arxiv.org/abs/1212.3801. In Lemma 2.1, the authors said for any $\phi \in L^{\...
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0 answers
19 views

About parametrization of the interface of a fluid

In Navier Stokes Equation, more precisly in the evolution of a fluid interface I have an extension of the function $u$ such that: $$u(x,t)=\frac{1}{2\pi} \operatorname{P.V.}\int_\Omega\frac{(x-z(\...
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2 votes
1 answer
141 views

Cylindrical coordinates in axis symmetric flow

I am getting stuck in a detail in a paper. It's about the axi symmetric Navier Stokes equations $$u_t - \nu\,\Delta u + u\cdot \nabla u + \nabla p=0$$ We consider in cylindrical coordinates $u=(u^r, u^...
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3 votes
0 answers
329 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 ...
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1 vote
0 answers
87 views

On self-similar methods of transforming the momentum equation to an ode

I have used the steam functions $u = \psi_{y}$ and $v = -\psi_{x}$ to transform the momentum equations to the following form $$\rho\left(\psi_{y}\psi_{xy} - \psi_{x}\psi_{yy}\right)=-p_{x}+\mu\left(\...
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0 votes
0 answers
103 views

Deformation gradient conservation law from Lagrangian to Eulerian formulation

In the following, I use the standard notation for (solid) mechanics and conservation laws, i.e. $F$ the formation gradient, $H$ the cofactor, $v$ the velocity field and $J$ the Jacobian. Moreover, $X$ ...
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1 vote
0 answers
90 views

2D Stochastic Navier Stokes equations with Navier boundary condition

For the 2D Stochastic Navier Stokes equations with Navier boundary condition $$du = (\Delta u - u\cdot \nabla u - \nabla p)dt + \Phi dW$$ where we consider additive white noise here. I want to use the ...
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0 votes
1 answer
61 views

Finding Free surface elevation in semi-infinite channel

A semi-infinite channel of finite depth is occupied by an ideal fluid layer initially at rest . the vertical finite end of the channel is fixed and only a part of the horizontal bottom , with finite ...
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5 votes
1 answer
219 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?
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11 votes
3 answers
942 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 ...
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3 votes
1 answer
334 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). ...
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5 votes
1 answer
336 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 ...
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5 votes
0 answers
149 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 $$\...
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3 votes
1 answer
131 views

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 ...
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10 votes
0 answers
511 views

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, ...
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1 vote
1 answer
130 views

Separation property for non-injective flows

Let us consider a non-injective flow $X$ on $\mathbb{R}^d$, i.e. a continuous map $X:\mathbb{R}_+\times \mathbb{R}^d \to \mathbb{R}^d$ with $X(0,\cdot)=\mathrm{id}$ and satisfying the semigroup ...
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0 votes
1 answer
102 views

Does Helmholtz's decomposition give an over-determined rotational flow?

From Helmholtz's decomposition, $v=v_{\scriptscriptstyle IR} +v_{\scriptscriptstyle R} $ where $\nabla\times v_{IR} =0$ and $\nabla\cdot v_R=0$ when apply this to the linearized Navier-Stokes ...
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3 votes
2 answers
359 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 ...
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3 votes
1 answer
386 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. ...
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1 vote
1 answer
312 views

Steady Euler flows with compact support?

What is known about (3D) steady incompressible Euler flows with compact support? (Looking for results in a field you are not familiar with sure is tough. I had a hope to find clues ...
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2 votes
2 answers
130 views

Torque term coming from added mass effects

I'm studying a quasi-steady force model (for a 2D problem) published in a fluids journal, and one of the torque terms is a bit perplexing: the term is a product of a difference of added mass ...
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4 votes
1 answer
267 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}...
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0 votes
0 answers
131 views

Time discretization of the variational formulation of the Navier-Stokes equation

Let $T>0$ $I:=(0,T]$ $d\in\mathbb N$ $\Lambda\subseteq\mathbb R^d$ be nonempty and open, $$\mathcal V:=\left\{\phi\in C_c^\infty(\Lambda,\mathbb R^d):\nabla\cdot\phi=0\right\}$$ and $$V:=\overline{...
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1 vote
0 answers
60 views

Time discretization of the (stochastic) Navier-Stokes equation

Let $d\in\mathbb N$ $\Lambda\subseteq\mathbb R^d$ be nonnempty and open $\langle\;\cdot\;,\;\cdot\;\rangle:=\langle\;\cdot\;,\;\cdot\;\rangle_{L^2(\Lambda,\:\mathbb R^d)}$ I've found a thesis where ...
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0 votes
0 answers
85 views

If $H$ is the closure of the set of solenoidal smooth vecor fields in $L^2$ and $P_H$ denote the orthogonal projection onto $H$, then $P_HH_0^1⊆H_0^1$

Let $d\in\mathbb N$ $\Lambda\subseteq\mathbb R^d$ be open $\mathcal V:=\left\{\phi\in C_c^\infty(\Lambda,\mathbb R^d):\nabla\cdot\phi=0\right\}$ and $$H:=\overline{\mathcal V}^{\left\|\;\cdot\;\right\...
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