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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Problem on Majda's Vorticity and Incompressible Flow

I'm reading Majda's book, on page 110, I cannot understand how to get $v\in C_W (0,T;H^{m})$ from the above. And he wrote "$[\phi,v^{\epsilon}]\to[\phi,v]$ uniformly on $[0,T]$ " two times, do the ...
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Evaluating vorticity as a function of velocity components.

So i have the following question.. Consider the axisymmetric flow of a viscous fluid u = ($ \frac{-\alpha r}{2} $, v(r), $\alpha z$) in cylindrical polar coordinates, where $\alpha$ is a positive ...
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Help me solve this question [on hold]

Assuming steady state and adiabatic process,The energy equation is $CpT2+\frac{V_2^2}{2}+CpT1+\frac{V_1^2}{2}=CpT3+\frac{V_3^2}{2}$,Is this right? I think it should be ...
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Show $\nabla^2 V = -\nabla \times \omega$ [on hold]

Using the index notation,show∇ ^2V=-∇ ×ω for incompressible flow,where V=velocity,ω=vorticity,please give me some hints,thank you
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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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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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13 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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Vorticity of a rigid body.

Consider a fluid in solid body roation about the z-axis with angular speed $\varOmega$ Derive an expression for the velocity field (u(x,y), v(x,y)) and show the vorticity field is the same at every ...
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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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Particle paths and standing waves

Here $x_0$ is the $x$ coordinate of a point in $x-y$ space Here $x_0$ is the $x$ coordinate of a point in $x-y$ space. I understand where the nodes and crests are on the figure. However, I don't ...
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Elementary Fluid Dynamics help!

I'm revising for my Fluids exams next month and I'm trying to understand a few definitions, and maybe grasp a physical interpretation of what exactly they are. I click on 'velocity field' on ...
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22 views

Mass conservation

I am trying to prove that $$\frac{d}{dt}\int_{a(t)}^{b(t)} \rho(x, t)g(x, t)dx = \int_{a(t)}^{b(t)} \rho(x, t)\frac{D}{Dt}g(x, t)dx$$ I have tried to evaluate the integral using Liebniz' rule, so ...
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1answer
20 views

Conservation of norms by the 2-d euler vorticity equation

In the book of Filho Lopes, Weak solutions for the equations of incompressible and inviscid Fuid dynamics. Page 59 They want to prove the following: Take $w^{\epsilon}_0$ a ...
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2answers
44 views

N2 diffusion through a vertical fluid column

Trying to figure out the mathematical model that might correlate to laboratory results. I have a cylindrical pressure vessel (picture a can) with height, h, and radius, r. It is filled with distilled ...
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1answer
28 views

Circulation of a Flow Field

Given the velocity components for a flow $$ u = 16x^2+y, \hspace{10pt} v = 10, \hspace{10pt} w = yz^2 $$ and a rectangular region $R$ in the $xy$-plane formed by the points $(0,0)$, $(10,0)$, ...
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1answer
23 views

Wall boundary condition

Why is it that at $y=0$ (at the wall), we have $v=0$ (vertical component of velocity)? Obviously $v$ cannot be negative there as there is no flow through the wall, however how do fluid particles ...
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What is mathematical definition of a fluid?

I am searching the precise and mathematical definition of a fluid for a long time but I did not find it anywhere. What I mean by precise and mathematical can be understood by the following: There is ...
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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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42 views

Continuum hypothesis confusion (fluid dynamics)

For the quantity $\rho(x,t)$ with the continuum hypothesis am I taking the average value of the density at each point in the small volume surrounding the point $x$ or am I taking the average density ...
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1answer
15 views

One dimensional flow slowly changing cross sectional area

I am rather confused by what's written in the green box. If $\frac{\partial A}{\partial x}$ was not $<<1$ would this mean that the velocity now has a vertical component and is hence not ...
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1answer
20 views

Is the Mass flow rate (Mass flux) a scalar quantity?

Wikipedia states that mass flow rate is a scalar quantity, however Mass Flow Rate= Density x Cross Sectional Area x Velocity and velocity is a vector quantity, so this would imply Mass Flow Rate is ...
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How is Euler fluids equation considered unsolved?

Apart from the Navier-Stokes equation, the Euler equation is described by Clay Math Inst. as unsolved or not well understood. My question is, is there a special case of Euler fluids equation that they ...
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Is $ \textbf{u} = y e_x − \sin x e_ y + b e_z$ a solution of the unforced incompressible Euler equations with $D = \mathbb{R}^3?

Hint; compute $∇ × (u · ∇u)$ and use to solve the problem. I dont even know how to start this problem. May you help me solve this problem?
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Describe this flow: $w=Be^{-i*pi}z^2$ using the stream function and the potential?

Consider $w=Be^{-i\pi}z^2$. Is it right that I've determined the streamfunction to be $\Phi=-Br^2\cos(2\theta)$ and the potential $\Psi = Br^2\sin(2\theta)$? then $u_r=-2B\cos(\theta)$ and $u_\theta = ...
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Maximum of a Geostrophic Wind

Consider a low-pressure system centered on 45 degrees South, whose sea-level pressure field is given by p = p$_{0}$ - $\triangle p$ e$^{\frac{-r^{2}}{R^{2}}}$ , (2) where r is the radial distance from ...
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22 views

Horizontal Pressure gradient.

The dynamics in the ocean can be described by the equation of motion $\frac{Du}{Dt}=-{\nabla}{\Phi}- \frac{1}{p}{\nabla}p- f\cdot u$ . Consider the motion of water in a full kitchen sink, with the ...
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velocity potential from flow

i have a complex potential $ w = Be^{-im\pi}z^{m+1} $, and have found the velocity potential and the stream function to be $ \phi = Brcos(\theta(m+1) + \pi(m+2)) $ and $ \psi = Brsin(\theta(m+1) ...
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23 views

continuity equation for measures from a purely mathematical point view

I'm looking for some sources on the derivation of the continuity equation, I'd like to show that if I have certain initial mass distribution (let's say probabilistic) $\rho$ and if particles move ...
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31 views

vorticity flux conservation for NS equation in 2D

Can someone explain to me why the vorticity flux is conserved for a solution to navier stokes equation in 2D ? Ie why $\int_{\mathbb{R}^2} w(x,t) dx =cst$ if $w$ satisfy the vorticity equation for ...
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Flow past a moving sphere

When the air passes over a moving sphere the boundary layer separates opposite to the direction of travel. The separation occurs at different positions to the back of the moving sphere. If separation ...
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complex potentials in plane polar coordinates - stream function

Determine the stream function and the potential in plane polar coordinates and sketching streamlines We need to take the value of m=1. I have an idea on how to do the parts and i know what a ...
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1answer
22 views

How do i correctly go from a two variable function to a function of difference?

I would like to know how I can go from a two argument function $g(x_1,x_2)$ formally correct to a function of the difference of the parameters $g(x_1-x_2)=g(x)$ this seems to involve integration over ...
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How to visualise the rate of strain of a fluid

I was wondering if anyone has any experience of how to visualise the rate of strain (tensor) of a fluid. I have computed the separate components but am not not sure how to interpret the data, I tried ...
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Why multiply the Rayleigh equation by the complex conjugate of the streamfunction to get Rayleigh's stability criterion?

In order to establish the stability criteria for Rayleigh's equaiton, we first write this equation as $$ \psi_{yy} - k^2 \psi + \frac{\beta - U_{yy}}{U-c} \psi = 0 $$ and then multiply it by ...
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Why is the difference of a stream function at two points the flow rate PER UNIT WIDTH?

For a velocity vector field $\bf{u}$$ = (u,w)$ in two dimensions ($x$ and $z$), we define the stream function $\psi$ to be; $$\psi(P_1) = \int_{P_0}^{P_1} \mathbf{u}\cdot\mathbf{n}dl$$ where $P_0$ is ...
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Have I simplified this partial differential as much as possible?

Let $$\psi = 2\left(\frac{xy}{(x^2+y^2)^2} - \frac{xy}{x^2+y^2}\right)$$ Not very good with the software as you can see so help with that would be great... Am I correct when I say "partial $\psi$ by ...
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52 views

Quasiconformal Mappings in Fluid Dynamics

I know that conformal mappings can be used to study 2 dimensional fluid flows. But I was wondering how quasiconformal mapping have been applied in this respect?
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35 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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1answer
36 views

Euler equations

What's the relationship between the incompressible, free surface euler equations and the euler equations? Are the latter just the former when the free surface is identically zero?
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29 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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Importance of Initial Guess in the numerical solution to the following fluid flow problem

Greetings Stackexchange community. Forgive me if the question is repetitive and/or answered before. I am currently working on a simple fluid flow problem, 'Heated laminar vertical Jet'/Brand and Lahey ...
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37 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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1answer
32 views

well-posedness of a mathematical model

what is the meaning of Well-posedness of a mathematical model of a physical phenomena for example stokes equation in fluid dynamics ? what is the necessity to prove that a model is well-posed? how ...
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What should be the exponent on the parameter to get the same solution?

Regarding the Boussinesq equations of motion: I was reading a paper which stated the following: Shouldn't the $\alpha$ before the $t$ be $\alpha^{-1}$ instead? Could be a typo or maybe I am not ...
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Prove an identity using differential calculus to a problem connected to fluids

Euler's equation for a incompressible inviscid fluid is $\displaystyle \frac{\partial \textbf{v}_t}{\partial t}+(\textbf{v}_t \cdot \nabla)\textbf{v}_t=-\nabla p_t$ where ...
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$L^2$ regularity of a convolution with Newtonian potential

I am reading Vorticity and incompressible flow (Bertozzi, Majda) and on 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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67 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 ...