Questions tagged [fluid-dynamics]

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3
votes
1answer
65 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 ...
2
votes
1answer
229 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^{\...
2
votes
1answer
225 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 ...
-1
votes
1answer
64 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 $\...
1
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0answers
79 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 ...
0
votes
1answer
92 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,...
1
vote
3answers
971 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 ...
2
votes
2answers
123 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 ...
5
votes
2answers
352 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)} \...
64
votes
3answers
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 ...
4
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0answers
122 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 ...
1
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0answers
55 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 ...
1
vote
1answer
208 views

Reformulation of the classical Navier-Stokes equation as a semilinear evolution equation and corresponding mild solutions

Let $d\in\mathbb N$ $\lambda^d$ denote the Lebesgue measure on $\mathbb R^d$ $\Lambda\subseteq\mathbb R^d$ be open $\mathcal V:=\left\{v\in C_c^\infty(\Lambda)^d:\nabla\cdot v=0\right\}$, $$V:=\...
1
vote
1answer
127 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 ...
2
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0answers
416 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 ...
0
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0answers
154 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} \...
1
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0answers
43 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 $\...
11
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2answers
778 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 ...
0
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0answers
17 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(\...
10
votes
3answers
784 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 ...
3
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0answers
322 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 ...
2
votes
1answer
103 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^...
1
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0answers
80 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(\...
16
votes
2answers
2k 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-...
0
votes
0answers
82 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, $X$ ...
1
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0answers
84 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 ...
4
votes
1answer
289 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 ...
0
votes
1answer
60 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 ...
5
votes
1answer
183 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?
3
votes
1answer
253 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). ...
2
votes
1answer
453 views

Stokes operator without Dirichlet boundary condition

Let $\Omega$ be a domain, then the following stokes operator is quite well known: $\mathcal{H} \rightarrow \mathcal{V}_{\sigma} $ $f \rightarrow u$ such that $ - \Delta u = f $ where $\mathcal{...
10
votes
0answers
462 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, ...
5
votes
0answers
142 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 $$\...
3
votes
1answer
123 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 ...
6
votes
1answer
390 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\;\...
0
votes
1answer
99 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 ...
3
votes
2answers
309 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 ...
3
votes
1answer
302 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. ...
1
vote
1answer
302 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 ...
6
votes
2answers
879 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 ...
4
votes
1answer
238 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}...
0
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0answers
126 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{...
1
vote
0answers
58 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 ...
0
votes
0answers
82 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\...
0
votes
0answers
103 views

Is $(u\cdot\nabla)v\in H^1$, if $u,v\in H^2$?

Let $d\in\left\{2,3\right\}$ with $\Lambda\subseteq\mathbb R^d$ be bounded and open with $\partial\Lambda\in C^1$ In Lemma 6.1 of Navier-Stokes Equations and Nonlinear Functional Analysis by Roger ...
3
votes
0answers
74 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 ...
1
vote
1answer
212 views

Showing nonlinearity of PDEs arising from physics by mathematical argument alone

Roger Temam writes in SOME DEVELOPMENTS ON NAVIER-STOKES EQUATIONS IN THE SECOND HALF OF THE 20th CENTURY: A remarkable property of the Navier-Stokes equations is that they are one of the very ...
0
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1answer
305 views

Domain of the Stokes operator

Let $\Omega\subseteq\mathbb R^d$ be open ($d\in\mathbb N$) $\mathcal D:=C_c^\infty(\Omega)^d$ and $$\mathfrak D:=\left\{\phi\in\mathcal D:\nabla\cdot\phi=0\right\}$$ $\mathcal H:=\overline{\mathfrak ...
3
votes
0answers
83 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 ...
1
vote
0answers
101 views

Derive a SPDE of evolutionary type for $u$ from ${\rm d}X(t)=u(t,X(t)){\rm d}t+\xi(t,X(t)){\rm d}W(t)$

Let $U$ and $V$ be separable $\mathbb R$-Hilbert spaces $\iota:U\to V$ be a Hilbert-Schmidt embedding $Q:=\iota\iota^\ast$ $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $U$ $(\Omega,\mathcal A,\...