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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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
$$\mathbf{u}=...
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Volume-preserving fluid flows are incompressible. What about surface-area preserving fluid flows?
Let $\boldsymbol{u}:\mathbb R^n\to \mathbb R^n$ be a $C^1$ vector field, representing the velocity of a (steady) fluid flow. If we let $\Phi_t(\boldsymbol x)$ be the flow map for the field $\...
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Concerning the Primitive of Radon Measure
I was studying some notes on the analysis of the PDE-- Navier-Stokes-Fourier Equation. And I found an expression as follows--
$$\text{Distributional derivative:}\qquad < F'(t), \phi >\, \...
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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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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:
$$
u_x(x,y,t) = \frac{\partial \psi }{\partial y} \quad \text{and} \quad ...
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Inequality $ (\nabla u)^2 \geq \left( \frac{1}{r} \partial_\theta u_r \right)^2 $ in Heywood paper
I am currently trying to understand the proof of Theorem 4 in John G. Heywood's paper On Uniqueness Questions in the Theory of Viscous Flow. Near the end of the proof, the inequality $ (\nabla u)^2 \...
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Blasius Equation ODE
I am curious about the Blasius ODE:
$y'''(x) + y y''(x) = 0$.
It seems most of the literature in fluid dynamics relies on numerical RK methods to solve it, but it also seems some solutions are ...
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Does a stream function exist for $\bf{v} = \bf{i}_{\phi}\sin(3\theta)$?
I am looking at an exercise where the winds on the Earth are modelled (very simplified) as the following velocity field in a spherical coordinate system (where $\phi$ is the azimuthal angle, and $\bf{...
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Please help me understand the extension of the weak formulation Navier Stokes
In the book Franck Boyer , Pierre Fabrie Mathematical Tools for the Navier-Stokes Equations and Related Models Study of the Incompressible the following is stated at page 360.
Because:
$$u'∈L^{4/d} ((...
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Pressure field from a sound source located exactly at the interface between two media
Background
I have asked this question on physics stack exchange, but was encouraged to migrate it over here.
There are loosely related questions here, here, here and others, but I did not understand ...
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How to Use Quaternions for Integrals, Differential Equations, Physics and Robotics?
For the case of translation, it is very simple to use Euler's method (among other methods that are far more efficient) to interpolate between positions with increments for time dt while applying ...
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eigenfunction of the Stokes operator and Laplacian are the same on bounded sets?
Consider the eigenfunction of the Stokes operator $w_i$. I want to prove that $w_i$ is an eigenfunction of Laplacian with divergence zero. Here is my attempt, please correct me if my statements are ...
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How to prove existence of radial-energy decomposition?
I meet a problem in Majda's Vorticity and Incompressible Flow. On page 93 he introduced the following radial-energy decomposition for 2D flow:
Definition. A smooth incompressible velocity field $v(x)$ ...
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How to find a variational structure of an elliptic system of equations?
I have a system of elliptic equations now. How can I find the corresponding functional so that the solutions of the system of equations are the minima of the functional? Are there any theorem or ...
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Lemma II.6.3 in Galdi's Introduction to the Mathematical Theory of Navier-Stokes
Lemma: Let $\Omega \subseteq \mathbb{R}^{n}$, $n\geq 2$, be an exterior domain and let
$u\in D^{1,q}(\Omega)$ with $1\leq q<n$
Then, there exists a unique $u_{0} \in \mathbb{R}$ such that, for all ...
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Axisymmetric Stokes flow field in cylindrical coordinates in a semi infinite domain
I am trying to solve the following simultaneous second-order PDEs for deriving the 2D and axisymmetric Stokes flow field (in cylindrical coordinates) $\mathbf{u}(r,z) = u_r(r,z)\hat{\mathbf{e}}_r + ...
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Problem on proof of conservation law of total flux of velocity in Majda's book (about vorticity)
I meet a problem when reading the proof of proposition 1.12 in Majda's book Vorticity and incompressble flow. The proposition says:
Let $v$ (and $\omega=\mathrm{curl}\ v$) be a smooth solution to the ...
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Can the sum of integrals over infinitesimal intervals be made into an integral of an integral?
I am investigating the relationship between Hamilton's principle in physics and the Monge-Kantorovich theory of optimal transport in economics. I derived the action density formula in $R^2$ for a ...
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How does the expression of $x_t$ and $y_t$ coming?
Consider a simple, smooth, closed initial curve $\gamma(0)$ in $\mathbb{R}^2$, let $\gamma(t)$ be the one-parameter family of curves, where $t \in [0, \infty)$ is time, generated by moving the initial ...
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Reynolds transport theorem: formal statement?
Reynolds transport theorem gives a relation:
$$\frac{d}{dt}\int_{\Omega(t)}fdV = \int_{\Omega(t)} \frac{\partial f}{\partial t} dV + \int_{\partial\Omega(t)}(\mathbf{u} \cdot \mathbf{n})fdA,$$
for an ...
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Trying a change of variable like a reciprocal $z=\dfrac{v}{\|v\|^2}$ on the Navier-Stokes Equation
Trying a change of variable like a reciprocal $z=\dfrac{v}{\|v\|^2}$ on the Navier-Stokes Equation
I would like to know how the Navier-Stokes equation would look like if is applied the following ...
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How do we know when total variation diminishing (TVD) schemes are appropriate to apply to a given PDE?
I am working on the research of hydrodynamics and see some researchers applying the TVD numerical scheme to flow equations such as shallow water equations or kinematic wave equations. But in their ...
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Potential function of a flow around a stagnation point
Following the Steve Brunton lecture about the potential flow. It is possible de find the potential function by solving the Laplace's equation. In his first example (the same as the following picture) ...
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Math_Physics - Why $\Sigma$ in $\frac{d}{dt}q(t) + \oint_S j\cdot dS = \Sigma$?
I'm working on understanding continuity equations, which look like:
$$\frac{d}{dt}q(t) + \oint_S j\cdot dS = \Sigma.$$
The first two terms are time variance of a quantity and the associated flux of a ...
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