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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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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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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44 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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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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36 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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26 views

Surface tension

Am I able to tell from this question that there is no surface tension? It's just the solution assumes this but it is not stated explicitly in the question.
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25 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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60 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
70 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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21 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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32 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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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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1answer
28 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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24 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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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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1answer
27 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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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

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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46 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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22 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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28 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
17 views

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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26 views

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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41 views

Integration by parts (fluids question)

I can't quite follow the solution for the part highlighted.
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139 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 $U'=\frac{dU}{dx}$?
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33 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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31 views

Taylor expansion of characteristics

I am unable to follow the section of the solution I have underlined in green.
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39 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
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 ...
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255 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
43 views

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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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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2answers
40 views

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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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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83 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
21 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 :-)
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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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36 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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1answer
15 views

Point source derivation

I've just been reading this on a website to derive a point source. All makes sense, but I am not sure why it is $(2\pi r)v_r$. Thanks!
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43 views

fluid dynamics- is this flow incompressible?

I have been given a flow with Lagrange path trajectories: $$x(\alpha,t)=(\alpha_1\cos(t)+\alpha_2\sin(t),\alpha_2\cos(t)-\alpha_1\sin(t),\alpha_3)$$ and I have to determine whether it is an ...
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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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1answer
55 views

Flow of a particle in dynamical system

Suppose I have the linear system $\dot{x}=Ax$, with $A=\left[ \begin{array}{cc} -1 & 0 \\ 0 & 2\\ \end{array}\right]$. I know that the phase portrait of the linear system has a saddle in ...
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classification of PDEs

What type of PDEs (partial differential equations) are the following: $\frac{\mu}{K}\textbf{u} + \frac{\partial \textbf{u}}{\partial t} = -\nabla p $ (Darcy's law), $\frac{\partial c} {\partial ...
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52 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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37 views

Continuum mechanics: Find the material (Lagrange) particle trajectories using the (known) Eulerian density and velocity functions

I have been given the Eulerian density function for a one-dimensional flow in the region $x\ge0$,$t\ge0$ to be: $$\rho(x,t)=(t+1)e^{-(t+1)x}$$ and have used the given fact that $v(0,t)=0$ and the ...
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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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46 views

Continuum mechanics: Finding the material derivative of plasma subjected to a decaying electric field

I have been given the fact that a fluid is subjected to a decaying electric field of (scalar) magnitude: $$ e(\boldsymbol{x},t)=r^{-1}e^{-At},$$$$r^2=x_1^2+x_2^2+x_3^2$$ where $A$ is a positive ...