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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Solving differential equation for an expanding bubble

I need to solve the equation \begin{eqnarray} R^3 \frac{d } {dt} \left [ \frac{4}{3} \rho_{\rm ext} \left ( \frac{dR}{dt} \right )^2 \right ]+ 4 p R^2 \frac{d R} {dt} =\frac{F_E}{4\pi} ...
3
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253 views

Newton's Second Law

I don't follow the part of the solution, which I have underlined in green. Which equation would I get this from (if any)?
2
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1answer
20 views

Continuity equation including mass generated

I don't really conceptually understand why you integrate the generated mass from $a$ to $b$. I understand that you have to take account that it's in within $\left[a,b\right]$, but not why you ...
6
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242 views

Idea behind distributional solutions

I have a problem understanding the meaning of a distributional solution. Let me tell you the context the problem appeared: I read thorugh some papers by DiPerna and Lions concerning the Cauchy Problem ...
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224 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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67 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 ...
4
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123 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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79 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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35 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 ...
2
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44 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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83 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 ...
2
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28 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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84 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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103 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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78 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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117 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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12 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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18 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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23 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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51 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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33 views

Complex Velocity on a solid body

I wish to find the complex velocity on a solid body in the flow (to find stagnation points). I have started with angular flow past a cylinder ...
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31 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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153 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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43 views

Aerofoil Theory Project

I'm doing a project (dissertation) on the mathematics of Aerofoil Theory. I wonder if I could get some advice on a possible structure. I'm new to fluid dynamics, so it's quite hard to know where to ...
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67 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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73 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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69 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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41 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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127 views

Should I get the absolute value of the result of the inverse discrete fourier transform´╝č

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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98 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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+50

Sketching the shock path

I can do the vast majority of this question except the bit underlined in green at the bottom. I don't really understand what it is asking. By 'some large t' does it mean 'at large values of t'. To get ...
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19 views

Partial Differential Equation with a flux term

I don't understand why $\phi=\frac{1}{2}u^2$ is the flux in this case?
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16 views

How to design Boundary condition for Euler equations (CFD)?

I'm developing on the calculation of the euler equations using the finte volume method. As you may know each cell is calculated by the incoming and outgoing flux. That means I need in a 1D System the ...
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20 views

How do the fundamental solutions for pressure and stress in Stokes flow define flows themselves?

This questions is related to section 3.2 Pozrikidis' "Boundary integral and singularity methods for linearized viscous flow" book. If $\mathbf{p}$ and $\mathbf{T}$ are the pressure vector and stress ...
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22 views

Spectral interpolation - Rotation equivalent to translation properties of Fourier transform?

I am using a spectral code for flow simulations. My aim is to obtain flow field data from points which do not coincide with the simulation grid without using inaccurate interpolation schemes in real ...
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21 views

Is it possible to simulate fluid dynamics in a time-based and deterministic manner?

The Problem Domain I have a number of network-connected PCs. I want to be able to simulate and replicate the same simple fluid dynamics simulation (Eg Navier-Stokes), in real-time, between them. That ...
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30 views

Change in momentum

I have tried this problem via units but I think i'm getting confused as to the difference between momentum flux and momentum. I'm not sure where to begin with the solution stated either.
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35 views

Fluid Dynamics Flow Conditions

Above is my question (a past paper question). I am struggling with the two final paragraphs: I cannot see any reason why $H_m$ needs to not exceed a specific value $H_c$. I don't know of any ...
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14 views

Deriving a conservative form of the Cauchy Equation?

I am asked given an inviscid fluid, to determine the Cauchy stress tensor and show that the balance of linear momentum and the conservation of mass together imply that $${\partial\over\partial ...
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43 views

Conformal Mappings- Fluid Dynamics

a)Show that the transformation z=F(Z) where F(Z)=Z+$a^2\over Z$, a is real positive constant, z=x+iy and Z=X+iY, maps the exterior of the circle |Z|=a to the exterior of the plate Z=0, -2a b)Write ...
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35 views

Induced Velocity- motion of a vortex

A vortex of strength $\Gamma$>0 is placed initially at z=0 beneath a solid flat wall y=d, where z=x+iy and d>0. Then the vortex moves due the induced velocity. The position of the vortex is described ...
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51 views

continuum mechanics: deriving eulerian 'conservation of mass' and 'balance of linear momentum' equations for cylinder with variable cross-section

I know the equations for conservation of mass and balance of linear momentum for a 1 dimensional cylinder with fixed cross-sectional area, A. Is there a way of deriving these equations in fluid ...
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27 views

Solve 3 coupled first order differential equations (boundary layer)

I'm solving boundary layer problem on the following image : I've got the governing equations : Only U, delta_B and delta_T are functions of x. Density, viscosity, H, L are contants. Does any one ...
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0answers
14 views

Estimate for Leray Operator acting on Gaussian Kernel

I just wonder if someone could give me an estimate for: $$ |{(\Pi \Phi)(x)}| \leq c(1+|x|)^{-n} $$ where $\Phi(x)= \pi^{ -n/2}e^{-|{x}^ {2}}|$ and $\Pi$ is the Leray operator $\Pi:L^{2}\to L^{2}$ ...
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18 views

find power developed by the turbine

Air enters a gas turbine at a mass flow rate of 15 Kg/s, the velocity and temperature at inlet are 150 m/s and 800 K and those at exit are 250 m/s and 600 K. Neglect Heat losses, find power developed ...
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20 views

How can I prove that assuming a power law forcing to drive incompressible turbulent flows leads to a scale invariant turbulent energy spectrum?

To maintain incompressible turbulence, the incormpressible Navier Stokes equations $$ \frac{\partial u_i}{\partial x_i} = 0 $$ $$ \frac{\partial u_i}{\partial t} + u_j\frac{\partial u_i}{\partial ...
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37 views

What is singular about the “singular” volume fraction advection equation in two-phase flow?

In studying about a subject in two-phase flow, i have encountered the following Partial differential equation: $$\frac{\partial a }{\partial t}+u_i \frac{\partial a }{\partial x}= r(p_g - p_l)$$ ...
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111 views

Shear stress in directions other than the flow direction

I already asked this question here http://physics.stackexchange.com/questions/74310/shear-stress-in-directions-other-than-the-flow-direction, but I am not getting a response. I think applied ...
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72 views

A problem about emptying a conical shaped tank with vertex up

A tank has the shape of a right circular cone with its vertex up. How to find the time (expressed in terms of the dimensions of the cone and the area of the orifice) required to empty a liquid from ...
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100 views

space discretization of convection term using first order upwind

Please help me discretize this term using finite difference or finite volume methods: $\int_{\Gamma}\rho\underline{v}.\underline{n}d\Gamma$ where $\rho$ and $\underline{v}$ both vary. I know that ...