Questions tagged [fluid-dynamics]

Question 1

Let me know if this is more on-topic for physics.se (or more generally, off-topic for mathematics.se). Okay, imagine you have a volume of space with instruments measuring air pressure (and for ...

Question 2

I'm stuck when reading Majda's Vorticity and Incompressible Flow. The whole discussion is for 2D incompressible Euler equation, meanwhile vorticity $\omega$ is concentrated on a smooth hypersurface ...

Question 3

I'm looking for a quick and dirty (citable) stability theorem for the solution to the steady state Navier-Stokes equations on a bounded, connected domain $\Omega$ (in either 2 or 3 dimensions, I'll ...

Question 4

I'm trying to understand the Leray projection $\mathbb{P}$. Here is Wikipedia's definition: One can show that a given vector field $\mathbf{u}$ on $\mathbb {R} ^{3}$ can be decomposed as $$\mathbf{u}=...

Question 5

It looks like the standard equation of motion for a rigid body rolling without slipping down an incline of angle $\theta$ is $$ a \;=\; \frac{g\sin\theta}{1 + I/(mR^2)}, $$ where $m$ is the mass, $R$ ...

Question 6

Let $\boldsymbol{u}:\mathbb R^n\to \mathbb R^n$ be a $C^1$ vector field, representing the velocity of a (steady) fluid flow. If we let $\Phi_t(\boldsymbol x)$ be the flow map for the field $\...

Question 7

Let $\Omega$ be bounded, smooth and simply connected in $\Bbb R^3$. For simplicity let me refer to $(H^1_0(\Omega))^3$ as $H^1_0$. Same for $L^2$. The set ${\rm curl}(H^1_0 \cap [{\rm div} =0])$ ...

Question 8

I was studying some notes on the analysis of the PDE-- Navier-Stokes-Fourier Equation. And I found an expression as follows-- $$\text{Distributional derivative:}\qquad < F'(t), \phi >\, \...

Question 9

I'm trying to derive the Vyas-Majdalani Vortex (2003) using a Bragg-Hawthorne PDE, $ \frac{\partial^2 \psi}{\partial r^2}-\frac{1}{r}\frac{\partial \psi}{\partial r}+\frac{\partial^2 \psi}{\partial z^...

Question 10

While solving the vorticity transport equation in cylindrical coordinates, $\frac{\partial \omega}{\partial t}=\nu \left(\frac{\partial^2 \omega}{\partial r^2}+\frac{1}{r}\frac{\partial \omega}{\...

Question 11

Poiseuille’s law describes the volumetric flow rate $Q$ of an incompressible, viscous fluid through a cylindrical pipe as: $$ Q = \frac{\pi r^4 \Delta P}{8 \mu L} $$ where: $r$ is the radius of the ...

Question 12

Two dimensional incompressible flow can be analyzed in terms of the stream function $\psi(x,y,t)$ where the velocity field is: $$ u_x(x,y,t) = \frac{\partial \psi }{\partial y} \quad \text{and} \quad ...

Question 13

I'm trying to understand the Navier Stokes Equation by solving a fluid dynamics problem. Not too sure if this is an appropriate place for my question. I have a thin layer of paint with constant ...

Question 14

I am currently trying to understand the proof of Theorem 4 in John G. Heywood's paper On Uniqueness Questions in the Theory of Viscous Flow. Near the end of the proof, the inequality $ (\nabla u)^2 \...

Question 15

This question comes from reading Majda and Bertozzi's "Vorticity and Incompressible Flow". In Example 2.1 on page 47, the authors "consider a radially symmetric smooth vorticity $\...

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Questions tagged [fluid-dynamics]

Determining the location of portals using air pressure

Setting of speed on the vortex sheet and its (possible) relation to the Rankine-Hugoniot condition

Bound to Solution to Steady State Navier-Stokes Equations in Sobolev Spaces

Difference between Helmholtz-Leray decomposition and Helmholtz decomposition.

Dynamics of Snail Balls

Volume-preserving fluid flows are incompressible. What about surface-area preserving fluid flows?

What is ${\rm curl}(H^1_0 \cap [{\rm div} =0])$?

Concerning the Primitive of Radon Measure

How do I solve $y'' - \frac{1}{x}y' + \alpha^2 x^2 y =0$?

What concise method can solve the ODE, $y'' + \left(\frac{1}{x}+2x \right)y' + 4y =0$?

How sensitive is volumetric flow rate to changes in radius in Poiseuille’s law?

Does this proof for streamlines make sense?

Question about Navier Stokes Equation and boundary layers

Inequality $ (\nabla u)^2 \geq \left( \frac{1}{r} \partial_\theta u_r \right)^2 $ in Heywood paper

If the Laplacian of a function is radially symmetric, is the function radially symmetric?

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