Fluid dynamics is a branch of physics that studies the motion of liquids or gases, which involves analysis and solution of partial differential equations like Euler equations, Navier-Stokes equations, etc.

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Connection between distributional and renormalized solutions for Boltzmann equation

I have a problem understanding the meaning of a distributional solution. Let me tell you the context the problem appeared: I read through some papers by DiPerna and Lions concerning the Cauchy Problem ...
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313 views

Modelling a Water Rocket. Requires Some Validation and Help. ( WARNING : Extremely Long but Interesting Post )

Good day people of math.stackexchange.com This is a pet project that I plan to use to convince my Prof that I would rather try something similar to this than to do the prescribed project. Edit : ...
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96 views

Question on using Leibniz formula to derive thin-film equation from Navier-Stokes

I am trying to work through the derivation in this paper by Petr Vita, which derives a thin-film simplification of the Navier-Stokes equation, similar to the Reynolds or Lubrication Equation, but ...
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284 views

Joukowski Aerofoil Plot

I've just had a go at plotting flow around aerofoils and I've come across a problem where I can't spot where I've gone wrong. I've previously worked out that the complex potential flow around a disk ...
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95 views

Non-Linear Ordinary Differential Equation in Fluid Dynamics

So while trying to model the physics of a rocket shot from the ground through the atmosphere, I came up with a second-order Non-Linear ODE of the form: $$ \ddot y + \dot y^2 e^y = f(t) $$ This is ...
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117 views

Verifying the fundamental solution of Stokes flow in R^3?

Recall that Stokes flow in $\mathbb{R}^3$ is the solution $(\textbf{u}, p)$ of $$\begin{align} -\nabla p + \mu \Delta \textbf{u} &= \textbf{f} \\ \nabla \cdot \textbf{u} &= 0, \end{align} ...
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84 views

Is this Stokes problem well-posed?

I am solving Stokes problem: $$\begin{gathered} \nabla p = \mu \Delta \vec{u} + \vec{f} \\ \nabla \cdot \vec{u} = 0 \end{gathered}$$ in a domain that's bounded by two surfaces - a cuboid and a small ...
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34 views

Objectivity or frame invariant

I am not sure it is maths or physics question. A stress tensor of a nematic liquid crystal is given as follows $$ T_{ij}=-P\delta_{ij}-n_{k,i}n_{k,j}+\widetilde{T_{ij}} $$ where ...
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33 views

Fluid Flow: lubrication, integration, ODE

Basically, I'm modelling the flow of a "coating" process -- a fluid flow between a flat moving plane and a stationary cylinder, 2D, cartesian coordinates. Subscript 0 is the at the minimum height b/w ...
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107 views

Stress Tensor and the ubiquitous cube: a misrepresentation?

So I'm just learning about Stress Tensors and have had an on going confusion and I'm wondering if the cube diagrams may to be blame. Usually when a stress tensor is introduced they show a cube and ...
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32 views

How to measure intensity of attractors

Given a dynamic system with attractors, is there any way to measure the intensity of attractors? I mean intensity by the faster one point far away from the attractor moves to it, the higher intensity ...
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91 views

Showing that pressure reaches its max on the boundary of ideal fluid in a stationary flow

The question is to show that the pressure in the stationary flow of ideal fluid achieves is maximum value on the boundary (and not at an interior point, unless the pressure is constant). I've come up ...
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146 views

Solutions of Laplace's Equation/Landau & Lifschitz Fluid Mechanics

in Fluid Mechanics by Landau and Lifschitz (a fairly well-known book - I'm just mentioning it for those who might have it) they are discussing "As we know, Laplace's equation has a solution l/r, ...
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93 views

What is $\int \frac{\delta F}{\delta u} \frac{\delta G}{\delta v} \, dx \; $?

Given $F[u]$ and $G[v]$ are functionals of a real-valued function, what is $$ \int \frac{\delta F}{\delta u} \frac{\delta G}{\delta v} \, dx \quad ? $$ I have encountered such expressions for ...
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125 views

Existence of circle cells for divergence free vector field with Dirichlet condition

Consider an open set $\Omega$ with smooth boundary and some smooth time-depending divergence-free vector field $u:[0,T]\times \Omega \rightarrow \mathbb{R}^d$, with $d=2$ or $3$, satisfying ...
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18 views

The free surface of the wave is a material surface

If we define the free surface by: $F(x,y,t)=y-h(x,t)=0$ Then for this to be a material surface $\frac{DF}{Dt}=0$ on $y=h(x,t)$ However on $y=h(x,t)$, $F=0$, so doesn't this just imply ...
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16 views

A gronwall inequality

In Majda/Bertozzi book, Incompressible flows etc.. p.118,he uses Gronwall theorem on the following inequality: $$|\nabla v(.,t)|_{L^{\infty}} \leq C\left( 1 + \int_0^t|v(.,s|_{L^{\infty}}ds ...
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33 views

Cauchy Equations and Navier Stokes

I'm attempting to take the Navier Stokes Equation and coming up with an expression that will allow me to numerically determine the velocity of non-Newtonian fluid flow. The text I'm using is Cengel ...
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29 views

Question on the local existence theory for the classical solution for the incompressible fluid dynamics equation

For the incompressible fluid dynamics system \begin{equation} \begin{split} &\nabla\cdot v=0,\\ &\dfrac{\partial v}{\partial t}+v\cdot \nabla v+\nabla p=A(S,D)v, \end{split} \end{equation} ...
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37 views

Stokes Equation

I came across the Stokes equation expressed in following form: I am trying to expand to check if it is correct but having hard time evaluating it. Can anyone give some hint on how can i expand it ...
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31 views

What is the connection (if any) between Knot Theory and Fluid Dynamics?

I've heard there is a connection to physics, but I'm unsure about any specific connection to fluid dynamics. Thanks!
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25 views

What is the “Enstrophy Miracle”?

I read that one of the main differences between the establishing global regularity for the Navier-Stokes equations for viscous and incompressible flow in 2D in 3D is that in 2D there is something ...
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101 views

Finite Difference Discretization of Darcy's law and solving with Picard method

I am trying to discretize Darcy's Law using finite differences and then solving the resulting linear system of equations with the Picard method. So far only in 1D and the steady-state (no time ...
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52 views

Navier-Stokes steady state vortex solution

For my research project I have been investigating the solution on Wikipedia. have read the paper it cites as a reference, by AM Kamchatnov, called Topological Solitons in Magnetohydrodynamics. In ...
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49 views

What is the name of such equation in fluid dynamics? How does it come out?

I read papers and see this equation. Honestly, I am not familiar with fluid dynamics. So I was wondering if this equation is common in fluid dynamics, or it is the special case in this paper. Could ...
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32 views

Spectrum of the operator of differentiation along streamlines

Suppose $\mathbb T^2$ denotes the two-torus and suppose $\psi^0$ is a steady state smooth stream function on $\mathbb T^2$ and define $\mathbf u^0 = (-\partial_y \psi^0,\partial_x \psi^0 )$ be the ...
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55 views

packing spheroids/ellipsoids

I am thinking about a problem I'm trying to formulate, from a mathematical analysis viewpoint. Suppose that you have a finite region in 3-space, and you want to populate it with spheroids or even ...
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39 views

Notation question in Majda and Bertozzi's “Vorticity and Incompressible Flow”

On pg 2, the fluid velocity in the Navier-Stokes system of equations is noted as: $v(x,t) \equiv (v^1, v^2, \ldots, v^N)^t$, where I am assuming that the velocity vector field is time-dependent. The ...
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322 views

How to plot a stream function

This question relates to fluid mechanics and I have the components in polar coordinates. The components of the velocity field are; $$v_r= \frac{-kr}{z}$$ $$v_z= kz$$ $$v_\theta= 0$$ and I have ...
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90 views

Output of wavelet transforms

I am working on a time sensitive computer science and fluid dynamics project that requires me to find applications of wavelet analysis. I know that at its core, a wavelet transform simply takes a ...
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85 views

Independency of the frame of reference of the strain rate tensor

I've got a problem regarding tensors. Premise: we are considering a fluid particle with a velocity $\mathbf{u}$ and a position vector $\mathbf{x}$; $S_{ij}$ is the strain rate tensor, defined in this ...
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45 views

improving fluid analysis on queuing problems

Discrete-queuing models are hard to solve computationally and can become easily intractable with an increased number of state-action pairs. Markov decision processes can be employed to come up with ...
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141 views

Should I get the absolute value of the result of the inverse discrete fourier transform?

The result of equation 36 can be positive and negative.And if I don't get the absolute value of it,the ocean surface tend to be very regular.But according to the paper,the author never get the ...
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106 views

Standing Wave problem-In deep water limit $h\to\infty$ show that $\omega^2=gk$

The equations I have are $\phi=(-ag/\omega)\cos(kx)\sin(\omega t)e^{kz}$ and $\eta=a\cos(kx)\cos(\omega t)$ I know that $d\phi/dz=d\eta/dt$ but when I partially differentiate and rearrange I get ...
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Elementary Fluid Dynamics help!

I'm revising for my Fluids exams next month and I'm trying to understand a few definitions, and maybe grasp a physical interpretation of what exactly they are. I click on 'velocity field' on ...
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94 views

How is Euler fluids equation considered unsolved?

Apart from the Navier-Stokes equation, the Euler equation is described by Clay Math Inst. as unsolved or not well understood. My question is, is there a special case of Euler fluids equation that they ...
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12 views

Is $ \textbf{u} = y e_x − \sin x e_ y + b e_z$ a solution of the unforced incompressible Euler equations with $D = \mathbb{R}^3?

Hint; compute $∇ × (u · ∇u)$ and use to solve the problem. I dont even know how to start this problem. May you help me solve this problem?
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9 views

Describe this flow: $w=Be^{-i*pi}z^2$ using the stream function and the potential?

Consider $w=Be^{-i\pi}z^2$. Is it right that I've determined the streamfunction to be $\Phi=-Br^2\cos(2\theta)$ and the potential $\Psi = Br^2\sin(2\theta)$? then $u_r=-2B\cos(\theta)$ and $u_\theta = ...
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Maximum of a Geostrophic Wind

Consider a low-pressure system centered on 45 degrees South, whose sea-level pressure field is given by p = p$_{0}$ - $\triangle p$ e$^{\frac{-r^{2}}{R^{2}}}$ , (2) where r is the radial distance from ...
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15 views

velocity potential from flow

i have a complex potential $ w = Be^{-im\pi}z^{m+1} $, and have found the velocity potential and the stream function to be $ \phi = Brcos(\theta(m+1) + \pi(m+2)) $ and $ \psi = Brsin(\theta(m+1) ...
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31 views

vorticity flux conservation for NS equation in 2D

Can someone explain to me why the vorticity flux is conserved for a solution to navier stokes equation in 2D ? Ie why $\int_{\mathbb{R}^2} w(x,t) dx =cst$ if $w$ satisfy the vorticity equation for ...
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40 views

Flow past a moving sphere

When the air passes over a moving sphere the boundary layer separates opposite to the direction of travel. The separation occurs at different positions to the back of the moving sphere. If separation ...
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45 views

complex potentials in plane polar coordinates - stream function

Determine the stream function and the potential in plane polar coordinates and sketching streamlines We need to take the value of m=1. I have an idea on how to do the parts and i know what a ...
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17 views

How to visualise the rate of strain of a fluid

I was wondering if anyone has any experience of how to visualise the rate of strain (tensor) of a fluid. I have computed the separate components but am not not sure how to interpret the data, I tried ...
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11 views

Why multiply the Rayleigh equation by the complex conjugate of the streamfunction to get Rayleigh's stability criterion?

In order to establish the stability criteria for Rayleigh's equaiton, we first write this equation as $$ \psi_{yy} - k^2 \psi + \frac{\beta - U_{yy}}{U-c} \psi = 0 $$ and then multiply it by ...
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Why is the difference of a stream function at two points the flow rate PER UNIT WIDTH?

For a velocity vector field $\bf{u}$$ = (u,w)$ in two dimensions ($x$ and $z$), we define the stream function $\psi$ to be; $$\psi(P_1) = \int_{P_0}^{P_1} \mathbf{u}\cdot\mathbf{n}dl$$ where $P_0$ is ...
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19 views

Importance of Initial Guess in the numerical solution to the following fluid flow problem

Greetings Stackexchange community. Forgive me if the question is repetitive and/or answered before. I am currently working on a simple fluid flow problem, 'Heated laminar vertical Jet'/Brand and Lahey ...
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60 views

Calculating force per unit width

Question: A line source of strength $2πm$ is located a distance $a$ above a horizontal plate. Find the force per unit width on the plate, ignoring gravity and taking the pressure below the plate to be ...
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75 views

problem with deriving continuity equation

I am studying Aerodynamics, to be more precise, the fundamentals of Aerodynamics. The first law is the continuity equation, for which it is explained in the book that I am using. However, I wished to ...
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62 views

Would a Counterexample to Navier-Stokes Problem be Sufficient?

Given the problem statement for the Navier-Stokes Existence & Smoothness problem from the Clay Institute website, wouldn't one need to show only one counterexample to the conjecture of global ...