# Questions tagged [fluid-dynamics]

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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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### 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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### 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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### 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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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), $\nabla^a ... -1 votes 1 answer 72 views ### Regularity of stationary incompressible Navier-Stokes equations in$\mathbb R^2What 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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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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### 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
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### 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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### 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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### 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? 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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### 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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### 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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### 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
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### 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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### 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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### 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 ... 386 views

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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### 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 ...