# Questions tagged [fluid-dynamics]

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### 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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### 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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### 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 ...
1answer
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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 ...
1answer
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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?
3answers
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### 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 ...
1answer
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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, ...
1answer
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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 ...
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### 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. ...
1answer
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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 ...
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### 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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### 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 ...
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### 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 ...
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### 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 ...