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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If $\operatorname{Def} X = \frac{1}{2} \mathcal L_X g$ then what is $\operatorname{Def}^* \operatorname{Def} X$?

Let us define a deformation operator $\operatorname{Def}$ on a Riemannian manifold $(M,g)$, acting on divergence-free vector fields as $$ \operatorname{Def} X = \frac{1}{2} \mathcal L_X g, $$ where ...
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Fluid Dynamics Problem

A vertical circular stack 100 ft high converges uniformly from a diameter of 20 ft at the bottom to 16 ft at the top. Coal gas with a unit weight of 0.030 pcf enters the bottom of the stack with a ...
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23 views

Solve for Pressure Drop across and orifice

I'm running 1/2" pipe but the only way to connect a 9/16" flow switch is with a 1/4" adapter. I'd have to go from 1/2" down to 1/4" to the 9/16" flow switch down to 1/4" back to 1/2". I was ...
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20 views

Why is the Galerkin-Method not optimal for non-self-adjoint equations

often i read phrases that explain the bad behavior of standard Galerkin-FEM for convection dominated problems by the equations beeing non-self-adjoint. Examples: Zienkiewicz, The Finite Element ...
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1answer
34 views

Velocity of fluid in presence of Sphere

I am struggling with the following problem: A rigid sphere of radius $a$ is placed in a stream of fluid whose velocity in the undisturbed state is $V$. Determine the velocity of fluid at any point of ...
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1answer
31 views

Intuition behind divergence?

$\overrightarrow f = 3x\overrightarrow i - 3y\overrightarrow j$ $\overrightarrow g = 3x\overrightarrow i + 3y\overrightarrow j$ If I calculate the divergence of $f$ I get $0$. If I calculate the ...
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2answers
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Given the velocity field $\ (u,v) = UL( \frac{-y}{x^2+y^2}, \frac{x}{x^2+y^2}$), how can we find the Legrangian coordinates?

I'm currently taking Graduate Fluid Dynamics and I'm using Stephen Childress' book, An intro to theoretical fluid mechanics. I've been stuck on one problem for quite some time and, so I'd like to see ...
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1answer
25 views

Notation regarding the continuity equation for conservation of mass

I have the following equation for the net mass flow out of a control volume through a surface $S$ - $$\int \int_S p \overrightarrow{V} \cdot \overrightarrow{d}S$$ (Actually there should be an ellipse ...
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32 views

Liouville's theorem (Hamiltonian) [closed]

can some one give me a link for a rigorous proof for Liouville's theorem (Hamiltonian) thanks
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58 views

Joukowski transformation of streamlines around cilinder in mathematica

I have problem transforming streamlines around cilinder, which is in fact simple circle that rotates, to a airfoil. It is done using Joukowski transformation $z = z+\frac{c}{z}$. The circle transforms ...
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1answer
32 views

Inviscid Shallow Water Equation

Aside from wikipedia where might I find a fairly comprehensive, yet simple to read, piece of literature on the inviscid shallow water equation? Can you recommend any texts? I don't want literature ...
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1answer
53 views

Solution to second order differential equation

I'm reading a paper in which the authors solve the following equation: $\frac{d^{2}}{dz^{2}}\hat{p}$($\bf{q}$$,z)$-$q^{2}\hat{p}$($\bf{q}$$,z)$-$\frac{iq_{y}}{(2\pi)^{2}}\delta(z-z_{2})$=0 here ...
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viscous burgers equation physical meaning

The viscous Burgers' equation : $$ q_{t} + qq_{x} = vq_{xx}, \:\:\: \mbox{where} \:\:\: v > 0, $$ combines the nonlinear propagation of $q(x,t)$ and the diffusion. What is this equation for? (in ...
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44 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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70 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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1answer
58 views

Normal mode solutions

Why does the solution choose those highlighted in green. Don't you normally have to pick the more general $\displaystyle\eta=\hat\eta (e^{ik(x-ct)}+$complex conjugate$)$ and then consider $\hat\eta ...
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1answer
41 views

Linear Water Waves

When solving , $\displaystyle {\partial^2 \tilde\phi\over\partial x^2}+{\partial^2 \tilde\phi\over\partial y^2}=0$ Why is it that $-h<y<0$ as opposed to ...
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43 views

Surface tension

Consider deep water gravity waves but assume now that there are two fluids separated by the interface at $y=\eta(x,t)$ and suppose that the upper fluid extends to $y\to\infty$. Let the fluid in the ...
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1answer
26 views

Finite element convergence rates for mixed problems

I've coded up a Stokes Flow problem using finite elements and am in the process of verifying that it works. I'm just not sure what convergence rate I should be expecting as I globally refine the mesh. ...
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64 views

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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1answer
139 views

Rankine Hugoniot Jump Condition Derivation

I follow the majority of this derivation I just do not understand where the two terms highlighted in green come from.
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36 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? Recall how we derived all our equations: Take an interval $[a,b]$ and consider $$\dfrac{\mathrm d}{\mathrm ...
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1answer
36 views

Proving that pressure at a point does not depend on orientation

In a) the solution states that $dS_1=$cos$\theta dS_2$, in other words it considers the surface area to be equivalent to the length of a line, in order to use basic trigonometry. I understand we are ...
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30 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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1answer
33 views

Special form of the Divergence theorem

It states in my notes relating to the derivation of the euler equations that a 'special form' of the divergence theorem is: $\iint{\phi\hat{n}}\space{dS}=\iiint\nabla\phi\space{dV}$ with $\phi$ a ...
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1answer
45 views

The acceleration in terms of the eulerian velocity

I'm having trouble in deriving the the acceleration in terms of the eulerian velocity. How do I apply the chain rule for partial derivatives to achieve the result highlighted in green?
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26 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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1answer
41 views

Derivation of the energy equation in fluid dynamics

I'm working through Acheson's 'Elementary Fluid Dynamics' and i'm having trouble deriving the conservation of energy equation (exercise 1.4) $\frac{d}{dt} \int_V \frac{\rho \vert u\vert^2}{2} dV = ...
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31 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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32 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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48 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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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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31 views

Numerical Hodge decomposition with boundaries

For a fluid simulation, I'm using the algorithm proposed by Jos Stam (http://www.intpowertechcorp.com/GDC03.pdf). One step of the algorithm is a routine that projects the velocity vector field so ...
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1answer
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definition of a smooth scalar potential

Have been asked to show that any flow described by a smooth scalar potential is irrotational. I know to show if a flow is irrotational curl of q = 0. But not too sure what is meant by smooth scalar ...
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Show that the velocity of the particle is… (Newtons Law question)

A particle of mass m is released from rest from height of ten metres above a body of viscous fluid. Show that the velocity of the particle at the moment of impact with the fluid is $14\text{m ...
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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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1answer
50 views

Integration by parts (fluids question)

I can't quite follow the solution for the part highlighted. Writing $U(x,t)=\phi(x)e^{\lambda t}$ casts $(2)$ into $$\lambda\phi=\phi^{\prime\prime}-(\overline u\phi)^\prime,\tag3 \\ ...
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2answers
162 views

Viscous Burgers Equation

I don't understand the bit of the solution of highlighted in green. Up to this point I've been using $U'=\frac{dU}{d\xi}$. Why can I know interchange that with $U'=\frac{dU}{dx}$? For the viscous ...
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1answer
44 views

Shock formation in nonlinear transport equation

The solution implies that $u_0(0-)$ is equal to $\alpha$ . However the question states that $u_0(x)=\alpha$ for $x>0$. Wouldn't $u_0(0-)$ be equal to the "smooth function for $x<0$"? I have ...
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1answer
35 views

Taylor expansion of characteristics

I am unable to follow the section of the solution I have underlined in green. Let us revisit the calculation in your notes that shows that a shock can form in finite time starting from appropriate ...
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1answer
43 views

Euler equation derivation

I am trying to follow the solution for i) , but i'm stuck on the parts I have underlined in green and orange.
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1answer
21 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 ...
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1answer
275 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)?
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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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1answer
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Structure of level sets of a noncritical point of a smooth function on a two dimensional domain

Let $\psi$ be a smooth function on a two dimensional simply connected domain $\Omega$ such that $\psi=0$ on the boundary $\partial \Omega$. Suppose $\rho$ is not a critical value of $\psi$ then it is ...
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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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2answers
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The matrix A in the system of Euler Equations

I am running a simulation on a 1D Euler equations: $$\frac{\partial \rho}{\partial t}+ \frac{\partial (\rho u)}{\partial x}=0$$ $$\frac{\partial (\rho u)}{\partial t}+ \frac{\partial (\rho ...
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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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1answer
133 views

fluid dynamics: sketching streamlines of velocity field when there is only one non-zero velocity component

I have been asked to sketch the streamlines in the $x_2$$x_2$-plane for the two-dimensional field: $$v=(x_1x_2,0,0)$$ All the examples I have seen of this kind of question use the $v_1$ and $v_2$ ...
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1answer
28 views

Doublet derivation

I'm just following these notes. I am not sure how they went from the third line to the fourth line where it is expanded. Any chance someone could point me in the right direction? Thanks :-)