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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363 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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101 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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347 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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27 views

Helicity is Conserved

In fluid mechanics, the helicity is defined as $$\int_{R^3} u(x,t)\cdot \omega(x,t),$$ where $u(x,t)$ is a smooth solution of the Euler equations $$\partial_tu + (u \cdot \nabla) u = -\nabla p$$ ...
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98 views

Water explosion and Navier-Stokes global regularity problem

In this article, Terence Tao is talking about a water explosion thought experiment that can lead to the solution of Navier-­Stokes global regularity problem. Can anybody explain this in more ...
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29 views

Presentation of Navier-Stokes eqns

This may be a trivial point, perhaps it's a lack of understanding on my part? When I was first introduced to fluid mechanics I was instructed to write the continuity and (generalized) Navier-Stokes ...
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101 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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131 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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88 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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24 views

Two methods of calculating a Jacobian determinant

Suppose you have two fluid bodies, one described by a set of vectors $V$, and a perturbation of $V$ given by $V+\Delta V$. Suppose that the two regions are related by the transformation $\mathbf ...
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19 views

Velocity potential of flow under rigid disk

Determine velocity potential of the flow in this system: Rigid disk of radius R at a heigh h(t) above horizontal plane z=0 with incompressible, inviscid flow between them, and h< The flow is ...
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51 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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39 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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117 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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93 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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160 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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96 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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126 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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49 views

Inverse Laplace Transform using Hetnarski's Algorithm

I'm trying to find the velocity component of an MHD flow using Laplace transforms. R.B. Hetnarski's algorithm for inverting the laplace transforms of some exponential functions was recommended to me ...
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22 views

Show the relation $W$ is constant

If the space $W$ is constant (doesn't move with the flow), show that $$\frac{d}{dt}\int_{W}\left (\frac{1}{2}\rho |\overrightarrow{u}|^2+\rho \epsilon\right )dV=-\int_{\partial{W}}\rho \left ...
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11 views

Visco-elastic fluid reference

What is a good book on visco-elastic fluids for self-teaching after one has studied Gurtin's Intro to Continuum mechanics? Thanks!
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37 views

Find the mass flow rate, given a surface, density and velocity field

I have a confusion, I hope you can help me (I'd like that if you will respond, please read all my post). They ask me to find the mass flow rate passing through a surface, where the velocity field is ...
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19 views

Linearising equations about a base state.

Consider a shallow-water system with mean depth H, where the base state consists of the flow (u,v)=($u_{0},$0), with a sloped water surface $\eta_{0}$(x,y) = - $\gamma y$, where u$_{0}$ and $\gamma$ ...
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29 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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21 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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57 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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37 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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40 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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38 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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136 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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60 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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53 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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34 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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56 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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41 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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370 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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104 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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87 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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150 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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108 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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How much water could be stored?

We have two water storage tanks--- Tank A and Tank B --- on the roof of the upper storey of our two-storey house. The tanks are cylindrical in shape. Each of the two tanks have a circular opening ...
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7 views

using complex potential to calculate stream function

The answer is $ \psi = Br^{12} sin12 \theta $ I dont know how to get this? say the complex potential equals $ Bz^{n} $ = $ Bre^{i\theta} $ how do i get the above answer? thank you
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55 views

Why we have to take the vector $\overrightarrow{e}$ ?

The differential equation of the balance of the momentum is $$\rho \frac{\partial{\overrightarrow{u}}}{\partial{t}}=-\rho (\overrightarrow{u} \cdot \nabla )\overrightarrow{u}-\nabla p+\rho ...
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25 views

Law of hydrostatic pressure

For a calm fluid of uniform density $\rho_0$, that occupies the space $W \subset \mathbb{R}^3$, and is subject to massive forces (per unit of mass) $\overrightarrow{b}(\overrightarrow{x})$, write the ...
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24 views

Find the streamlines of the velocity field

I have to find the streamlines of the following velocity fields: $$u=x(1+2t), v=y$$ $$u=xy, v=0$$ I have done the following: $$\frac{dx}{u}=\frac{dy}{v} \Rightarrow ...
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42 views

Streamlines - Pathlines

Construct and draw the streamlines of the velocity field $u=az-bt, v=\frac{b}{4}z-cy, w=2(a-1)$. Calculate $c$ (as a function of the constants $a$, $b$) such that the flow field ...