# Questions tagged [fluid-dynamics]

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85
questions

**3**

votes

**1**answer

50 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**

votes

**1**answer

62 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**

vote

**0**answers

78 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

**1**answer

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,...

**2**

votes

**1**answer

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 ...

**5**

votes

**2**answers

341 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

**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 ...

**4**

votes

**0**answers

120 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**

vote

**0**answers

53 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 ...

**2**

votes

**0**answers

404 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**

votes

**0**answers

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**

vote

**0**answers

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 $\...

**2**

votes

**1**answer

224 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^{\...

**0**

votes

**0**answers

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(\...

**2**

votes

**1**answer

102 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^...

**3**

votes

**0**answers

321 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 ...

**1**

vote

**0**answers

79 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(\...

**0**

votes

**0**answers

81 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**

vote

**0**answers

83 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 ...

**0**

votes

**1**answer

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

**1**answer

182 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?

**10**

votes

**3**answers

767 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**

votes

**1**answer

249 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).
...

**4**

votes

**1**answer

285 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 ...

**5**

votes

**0**answers

139 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

**1**answer

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 ...

**10**

votes

**0**answers

452 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, ...

**1**

vote

**1**answer

126 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 ...

**0**

votes

**1**answer

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

**2**answers

303 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

**1**answer

294 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

**1**answer

301 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 ...

**2**

votes

**2**answers

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 ...

**4**

votes

**1**answer

237 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**

votes

**0**answers

125 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

**0**answers

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

**0**answers

81 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

**0**answers

101 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 ...

**6**

votes

**1**answer

389 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\;\...

**1**

vote

**1**answer

207 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:=\...

**3**

votes

**0**answers

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

**1**answer

209 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 ...

**3**

votes

**0**answers

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

**0**answers

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,\...

**1**

vote

**0**answers

90 views

### Recast a finite-dimensional multiparameter SDE as an infinite dimensional SDE

In another question, I've asked how we can derive a stochastic Navier-Stokes equation under the assumption of perturbed particle trajectories.
More concretely, I want to obtain a SDE of type as ...

**0**

votes

**1**answer

302 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

**3**answers

365 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 ...

**7**

votes

**0**answers

270 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 $\...

**1**

vote

**0**answers

75 views

### Hyperbolic PDE from total derivative?

Given a density function $p(t, \boldsymbol{x})$, where $t$ is time and the vector $\boldsymbol{x}$ represents a point in $n$ dimensional space, a hyperbolic PDE describing the time evolution of the ...

**1**

vote

**0**answers

102 views

### Fluid dynamics of a rotating liquid droplet

I'm looking for an analytical solution of the Navier-Stokes equation with the following boundary conditions: a liquid is held inside a spherical shell, which is rotating at a constant rate, and is ...