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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How to prove the relation $\tan \theta = \hat{\vec{n}} \cdot \nabla h$

The relation for finding the contact angle is often given as $\tan \theta = - \hat{\vec{n}} \cdot \nabla h$ in papers such as in Sequential deposition of overlapping droplets to form a liquid line ...
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Find the mass flow rate, given a surface, density and velocity field

I have a confusion, I hope you can help me (I'd like that if you will respond, please read all my post). They ask me to find the mass flow rate passing through a surface, where the velocity field is ...
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how to calculate the pressure/velocity of a shock wave in water [on hold]

I am trying to understand the basic maths used to calculate the properties of shock waves in fluids caused by projectiles travelling through them. If a bullet impacts a tank of water (50 by 50 by ...
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1answer
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How to find the complex potential for the following flow under certain conditions?

We've used $z=i(Z+4/Z)$ as a conformal mapping to map the exterior of a circle $|Z|=2$ to the exterior of the line segment $(-4i,4i)$. We now want to write the complex potential of the uniform flow ...
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Help solving a first order non-linear differential equation derived from the navier-stokes equation

I am an engineer studying an unsteady-state flow through a pipe. The transient Bernoulli equation of this system, which I picked up from here ...
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Representation of polynomial order in CFD codes

I currently working on a CFD code over a cubic grid. Now, the number of elements used in the simulation is decomposed among the number of processors. Each of those processors (a section of the cube) ...
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1answer
21 views

Does the Laplace operator include the second derivative with respect to time variable?

Does the Laplacian of a function $f(x,z,t)$ equal $f_{xx} + f_{yy} + f_{tt}$? We aren't sure whether or not time is included in it or not.
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Is a 3D or 2D Poisson's equation separable or non sperable?

Can someone please explain to me if a 2/3D Poisson's equation is separable or non separable? Thank you
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When is shear useful?

I'd never heard of the shear of a vector field until reading this article. Shear is the symmetric, tracefree part of the gradient of a vector field. If you were to decompose the gradient of a vector ...
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Von Karman similarity solution of Navier-Stokes equations

I was wondering if anyone could help me; I've been looking at Von Karman's similarity solutions of the Nav-Stokes equations for a rotating disk in a fluid $u = r \Omega U(n)$ $v = r \Omega V(n)$ ...
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110 views

Solving a system of two second order ODEs using Runge-Kutta method (ode45) in MATLAB

I'm trying to solve the system of differential equations outlined in Von Karman's rotating disk flow. I got them into a system of ordinary differential equations: F(n), G(n), H(n) $$F'' = -G^2 + F^2 ...
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Linearisation of instability of flames

I am not sure it is a maths or physics question, kinda in between. Large-scale disturbances of a plane flame $x=\eta(y,t)$ are describe by Euler's equation and the continuity equation: $$ ...
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41 views

What is the physical meaning of $v\times n$? [migrated]

What is the physical meaning of $v\times n$, where $v$ is a velocity vector and and $n$ is a unit normal vector of a interface? Why at the free surface between two fluids, $$v^{(1)}\times ...
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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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21 views

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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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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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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25 views

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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13 views

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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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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45 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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32 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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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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19 views

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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43 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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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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23 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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25 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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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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34 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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54 views

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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34 views

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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1answer
54 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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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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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?