Questions tagged [fluid-dynamics]
For questions about fluid dynamics which studies the flows of fluids and involves analysis and solution of partial differential equations like Euler equations, Navier-Stokes equations, etc. Tag with [tag:mathematical-physics] if necessary.
1,149 questions
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Which orientation of a truncated cone will drain faster? [migrated]
A container shaped like a truncated cone (frustum) as in cylinder-like shape where one end has a larger radius than the other. Both orientations contain the same volume of water, and the outlet hole (...
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Bound to Solution to Steady State Navier-Stokes Equations in Sobolev Spaces
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 ...
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Difference between Helmholtz-Leray decomposition and Helmholtz decomposition.
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
$$\mathbf{u}=...
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Dynamics of Snail Balls
It looks like the standard equation of motion for a rigid body rolling without slipping down an incline of angle $\theta$ is
$$
a \;=\; \frac{g\sin\theta}{1 + I/(mR^2)},
$$
where $m$ is the mass, $R$ ...
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Volume-preserving fluid flows are incompressible. What about surface-area preserving fluid flows?
Let $\boldsymbol{u}:\mathbb R^n\to \mathbb R^n$ be a $C^1$ vector field, representing the velocity of a (steady) fluid flow. If we let $\Phi_t(\boldsymbol x)$ be the flow map for the field $\...
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What is ${\rm curl}(H^1_0 \cap [{\rm div} =0])$?
Let $\Omega$ be bounded, smooth and simply connected in $\Bbb R^3$. For simplicity let me refer to $(H^1_0(\Omega))^3$ as $H^1_0$. Same for $L^2$. The set ${\rm curl}(H^1_0 \cap [{\rm div} =0])$ ...
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Concerning the Primitive of Radon Measure
I was studying some notes on the analysis of the PDE-- Navier-Stokes-Fourier Equation. And I found an expression as follows--
$$\text{Distributional derivative:}\qquad < F'(t), \phi >\, \...
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How do I solve $y'' - \frac{1}{x}y' + \alpha^2 x^2 y =0$?
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^...
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What concise method can solve the ODE, $y'' + \left(\frac{1}{x}+2x \right)y' + 4y =0$?
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}{\...
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How sensitive is volumetric flow rate to changes in radius in Poiseuille’s law?
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 ...
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Does this proof for streamlines make sense?
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 ...
- 155
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Question about Navier Stokes Equation and boundary layers
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 ...
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Inequality $ (\nabla u)^2 \geq \left( \frac{1}{r} \partial_\theta u_r \right)^2 $ in Heywood paper
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 \...
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If the Laplacian of a function is radially symmetric, is the function radially symmetric?
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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Blasius Equation ODE
I am curious about the Blasius ODE:
$y'''(x) + y y''(x) = 0$.
It seems most of the literature in fluid dynamics relies on numerical RK methods to solve it, but it also seems some solutions are ...
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