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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$L^2$ regularity of a convolution with newtonian potential.

I am reading Bertozzi, Majda Vorticity and incompressible flow and in page 71 72, we are concerned with recovering the velocity field of a flow from its vorticity. At some point we need to have the ...
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Showing instablity of differential equation.

Assume differential equation $$ x'=2x+y+x \cos t-y \sin t $$ $$ y'=-x+2y-x\cos t+y \sin t $$ Show that solution $(x(t),y(t))=(0,0)$ is unstable. Is there a non-trival solution such that ...
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$\int_C (\alpha x, -\alpha y) . dr = 0$ where C is the unit circle

Circulation is given by $$\int_C u . dr$$ I want to show that the circulation around the unit circle is $0$ for $u = (\alpha x, \alpha y)$. Ie. $$\int_C (\alpha x, -\alpha y) . dr = 0$$ How would ...
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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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53 views

Fluid Mechanics for Low Reynolds Number

I've tried to answer this question but I seem to get a really weird (and suggestively incorrect) answer. The question is: "Fluid is injected radially and slowly into a circle, radius R, with the ...
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35 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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25 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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61 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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16 views

diffusion equation

Kindly give me suggestions on my following assignment of Simulations in Fluid Flow: Solve the following differential equation for transport of f(x,y,z,t) by MS Excel ∂f/∂t+Ux ∂f/∂x+Uy ∂f/∂z+Uz ...
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51 views

Undergraduate Project Suggestions

A student of mine has expressed interest in doing an independent project next quarter with me. This would not be for credit and it is purely for her own educational stimulation. She wants to study ...
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1answer
35 views

differential inequality involving the square of the function

It is written in a book, (Bertozzi- Majda, vorticity and incompressible flow page 106) that given a differential inequality of the following type: $ \frac{d}{dt}\|u^{\epsilon}(t)\| \leq ...
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42 views

Fluid Dynamical problems

I'm just going through Batchelor's book on Fluid Dynamics, and I'm not too sure about a couple of questions in "Exercises for Chapter 4", which are the following: Exercise 3 for Chapter 4 "A long ...
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21 views

What would global irregularity of the Navier-Stokes Equations do?

Suppose Terrance Tao's hints at showing finite-time blowup for the true Navier-Stokes Equations prevailed, and the Navier-Stokes Problem was solved negatively (no existence and uniqueness). What good ...
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23 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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53 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 ...
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1answer
78 views

Cauchy Momentum Equation - Stress Tensor

I've been trying to understand the derivation for the Cauchy Momentum Equation for so long now, and there is one part that every derivation glides over very quickly with practically no explanation ...
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1answer
35 views

Mass Continuity Equation for Fluid - Running Into a Problem

I'm running into a problem when trying to show the mass continuity equation for a fluid, which says $$\frac{\partial \rho}{\partial t} + \left(\nabla \cdot \rho \textbf{u}\right) = 0$$ Where ...
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25 views

Lubrication Theory: Quick Question!

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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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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21 views

Convected 2nd order tensor in component form

I have a convected second order tensor that I'd like to write in component form. $\frac{D\mathbf{a}}{D t} = \frac{\partial \mathbf{a}}{\partial t} + \mathbf{v}\cdot\nabla\mathbf{a}$, where ...
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38 views

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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107 views

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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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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55 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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39 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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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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34 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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86 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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35 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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61 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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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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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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65 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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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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46 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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30 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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219 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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38 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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37 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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36 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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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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60 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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30 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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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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36 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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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 ...