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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Complex analysis techniques in fluid dynamics [on hold]

What are some of the areas in which complex analysis methods are applied in fluid dynamics?
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Show that $δ_{KL}$ is a Cartesian tensor

By using the definition of the Kronecker delta $δ_{KL}$, show that $δ_{KL}$ is a Cartesian tensor, that is $δ'_{MN} = L_{MK}L_{NL}δ_{KL}$ under the rotation $X_K = L_{MK}X'_M$. Solution: Using the ...
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33 views

Numerical Method for KdV travelling waves

Can someone please direct me to the best NUMERICAL method or some references for solving $$-cu_x + uu_x + u_{xxx} = 0$$ with periodic boundary conditions. This governs travelling waves of the KdV ...
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23 views

Find a limit for Doublet Stream function

In fluid Mechanics, The superimposed stream function of point source and sink is: $\psi=-\frac{Qcos\theta_1}{4\pi}+\frac{Qcos\theta_2}{4\pi}$ Graphical image of the function and for a sink - ...
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17 views

Finding streamlines

Find the streamlines, particle paths and streaklines when $$u=xe^{2t-z}, \, \, \, v=ye^{2t-z}, \, \, \, w=ze^{2t-z}$$ What is the track of the particle passing through $(1,1,0)$ at time $t = 0$? To ...
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integral of partial derivative for continuity equation

Task in question is deriving velocity parallel to z component in continuity equation. $$ \frac{\partial n}{\partial t} + \left(\nabla \cdot n \textbf{u}\right) = 0 $$ $$ \frac{\partial n}{\partial t} ...
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The Virasoro-Bott group and the KdV equations

The Euler Arnold equation expresses equations (usually from mathematical physics) as geodesic equations on a Lie group. For the famous $KdV$ equations these equations are given on the Virasoro-Bott ...
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27 views

Why is the magnitude of the curl of a vectorfield twice the angular velocity?

(if V is a vectorfield describing the velocity of a fluid or body, and $x\in R^3$) I agree that it should be when you look at the calculation, but intuitively speeking... If $\nabla \times V(x)= ...
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Derivation of the Quadrilateral Source Element used in Fluid Potentials

Background: Potential method is used to solve linear fluid dynamics problems still to this date, which is based off reducing the Navier-Stokes equation to an inviscid, irrotational, incompressible ...
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18 views

Flux of the amount of buffalo entering a square kilometer per minute

So Eq.(13) is just $\int F.n dS = \int \int div(F) dA$ So I did $div(F) = \nabla . F = y+1$, then I did $\int_2^3 \int_2^3 y + 1 dx dy = \int_2^3 y + 1 dy = .5(3^2) + 3 - (.5(2^2) + 2) = 3.5$ This ...
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13 views

Rankine ovals are oval shaped

I'm studying Rankine oval and I have a question. How do you prove the equation for its zero streamline curve is in fact oval shaped? Is there some family of oval shaped curves including Rankine Ovals? ...
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30 views

Dynamics of fluid

While reading on Wikipedia about the partial differential equations (https://en.wikipedia.org/wiki/Partial_differential_equation), I wondered how dynamics for the fluid occur in an ...
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82 views

Derivation of a stochastic Navier-Stokes equation with multiplicative noise

Most of the literature is targeting a special stochastic version of the deterministic Navier-Stokes equation without giving a derivation of the considered equation. I'm searching for such a ...
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5 views

Confusion regarding phase velocities

I am revising a waves module and a past exam paper asks to calculate the horizontal components of the phase velocity (the horizontal wavevector is (k,l)). Immediately I believe that this is ...
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16 views

how to formulate an equation for this?(modified 3D sphere)

I was reading about a situation, where a sphere is close to a solid plane boundary. whose radius is 'a' and whose centre is 'a+b' away from the solid plane boundary. so they have a function h(r,t) ...
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matlab code for finite volume method

I am trying to write a matlab code which is about applying Roe linearization to the 1D shallow water equation. In addition, I should include entropy fix. This is a function that I defined: function ...
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17 views

How did they arrive at those equations?

I was reading this paper and I found the math behind some equations difficult to understand. can someone explain how they got the spatial dependence of the single-droplet velocity field as:(page 6) ...
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14 views

Phonon-like acoustic modes in a 1D unconfined crystal

I was studying this paper: http://www.sns.ias.edu/~tlusty/papers/PhysRep2012.pdf In page 21, they say: "To consider small fluctuations of the droplet positions around their lattice points, we define ...
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20 views

Find the pressure at a certain section of the pipe

Water flows through a sudden pipe enlargement at 0.35 m^3/s. Upstream the pipe diameter is 0.027 meters and downstream, 0.042 meters. At a point about 15 millimeters upstream of the expansion the ...
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85 views

Proof of $-\nabla\times\omega = \nabla^2 U$

What is a proof for $$ -\nabla\times\omega = \nabla^2 U $$ in the scope of fluid mechanics? I'm learning vector calculus for my project and stuck on this seemingly simple proof problem. Detailed ...
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21 views

Complex potential question

Consider the flow described by the complex potential $$w = Be^{−imπ}z^{m+1}$$ where $B > 0$ and $m ∈ (0,1]$. (i) Determine the stream function $ψ$ and the potential $φ$ in plane polar coordinates ...
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21 views

Calculating the Lagrangian map for a solenoidal vector field

I have two related questions. First what is a Lagrangian map? I've searched online but I have not been able to find an explanation that I understand. Second, for a solenoidal field $u(x,t)$ how do ...
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43 views

Solving the Navier-Stokes equations for a known density function

Consider the following form for the Navier-Stokes equations for a compressible fluid in steady-state: $$div(\rho \textbf v)=0$$ $$div(\rho \textbf{vv} )=-k^2 \nabla \rho$$ where k is a constant. If ...
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45 views

Milne-Thompson Theorem with a Vortex

I'm doing a problem related with Milne-Thompson theorem which tells that: "A cylinder of radius $a$ is immersed in a counter-clockwise whirlpool, which we model here as a potential vortex of intensity ...
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42 views

Curl of a Point Vortex Flow and its Circulation

I have the following 2D vector field $U=(u,v)=\frac{1}{x^2+y^2}(y,-x)$. When taking the curl of this field it returns zero. But when I take the circulation of the field defined as $$\Gamma=\oint_C ...
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Is there a way to relate an axisymmetric 3D flow field to cylindrical planar flow in order to determine the swirl velocity?

I have the following incompressible axisymmetric velocity field. $$u=u_r\hat e_r+u_\theta\hat e_\theta+u_z\hat e_z$$ For the planar analog to this flow (where the swirl velocity $u_\theta=0$) I know ...
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28 views

Vorticity Stretching in an Axisymmetric Flow Without Swirl

For an axisymmetric flow with zero swirl the velocity field is $$u=u_r \hat e_r+u_z\hat e_z$$ which results in the following vorticity field $$\omega=\omega_\theta\hat e_\theta$$ Here, ...
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Partial Derivatives and Operator Commutivity

I have an operator $$L\psi=\frac{1}{r^2}\partial_z^2\psi+\frac{1}{r}\partial_r(\frac{1}{r}\partial_r\psi)$$ I am interested in taking $\partial_rL\psi$ and $\partial_zL\psi$. Do the partial ...
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18 views

Nonlinear Maximum Principle estimate

Im interested in the in the 2D Boussinesq equations given by $$\begin{cases}\partial_{t}u+u\cdot\nabla u+\nabla p+\Lambda^{\alpha}u=\theta e_{2}\\ \nabla\cdot u=0\\ ...
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Why is $(\mathbf{v} \cdot \nabla)\mathbf{v} = (\nabla \times \mathbf{v}) \times \mathbf{v} + \nabla (\frac{1}{2} \mathbf{v}^2)$?

The Convective Derivative or Material Derivative is usually written as $\frac{D}{Dt}=\frac{\partial}{\partial t} + \mathbf{v} \cdot \nabla$. According to MathWorld, this equation, multiplied with ...
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Can we get a tensor out of summation of one vector with its transpose?

I don't know much about tensor calculus and here is something I'm trying to figure out. $$T=\mu({\nabla}\vec{V}+{\nabla}\vec{V}^T)$$ T is viscous stress tensor and $\vec{V}$ is the velocity vector. ...
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Euler Equation with Dirichlet condition

Some references said the incompressible Euler equation is not well posed with Dirichlet boundary condition : $$u_t+u\cdot\nabla u+\nabla p=0$$ $$\nabla\cdot u=0$$ $$u=0 \ \ \text{on}\ ...
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63 views

Proving height of water as ice melts in it is constant

edit: assuming there are no evaporative losses and the ice shrinks as a cube, it should stay constant...? I'm trying to prove mathematically that the height of water in a closed container will stay ...
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Damping in a fluid

I am trying to understand damping in a fluid.Take, for instance, water flowing down a surface. I know a damping term $-\alpha\mathbf{u}$, where $\mathbf{u}$ is the velocity field, is added to the sum ...
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33 views

Curl of a vector in the complex plane

Let there be a vector $u(z)$ in the complex plane. Are these two statements equivalent? $$\nabla\times\overline u=\overline{\nabla\times u}$$ If not, why? I think they should be equal since $\nabla$ ...
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53 views

Showing Bernoulli function is constant on streamlines

An incompressible inviscid fluid, under the influence of gravity, has the velocity field $$\textbf u = (− \cos(x)\sin(y), \, \, \sin(x)\cos(y), \, \, 0)$$ with the $z$-axis vertically upwards, where $g$ ...
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Using Euler's equation

An incompressible inviscid fluid, under the influence of gravity, has the velocity field $$\textbf u = (− \cos(x)\sin(y), \, \, \sin(x)\cos(y), \, \, 0)$$ with the $z$-axis vertically upwards, where $g$ ...
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54 views

Explain a physical problem by mathematics

I am not very good at physics, so I don't understand how to solve the following problem with vector calculus. A perfect incompressible fluid moves steadily under gravity around the outside of a fixed ...
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Prove Bernoulli Function is Constant on Streamline

I have an incompressible, inviscid fluid, under the influence of gravity, with a velocity potential: $$ \mathbf{u} = (-\cos(x)\sin(y), \sin(x)\cos(y), 0) $$ Using Euler's equations, $$ \mathbf{u} ...
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63 views

How do I determine if the equation is a conservation law?

We have the PDE $\frac{\partial u}{\partial t}+a(x,y)\frac{\partial u}{\partial x}+b(x,y)\frac{\partial u}{\partial y}=0$. What would be conditions on $a$ and $b$ for the equation to constitute a ...
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Linearizing Euler's continuity equations

I have three Euler equations (for a polytropic gas) and I'd like to linearize them to get 2 other equations. Any help would be appreciated! The 3 initial equations are: 1) ...
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42 views

Derivation with Euler's Equations

I have three equations as follows (for a polytropic gas): 1) $\displaystyle\quad\frac{\partial\rho}{\partial t}+\nabla\cdot\rho \mathbf{u} = 0$ 2) $\displaystyle\quad\rho \left( \frac{\partial ...
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Calculate volume using flow rate and time difference in a container

So bare with me, I am a computer scientist undergrad taking part in an engineering week and this is a calculation we require for our report. So I'd like you to imagine I am filling a cylindrical ...
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Physical difference between $\nabla^\bot\cdot u=0$, and $\nabla\cdot u^\bot=0$ and the existence of a scalar potential

If there exists a $2D$ vector field $u=u(x)=(u_1,u_2)$ such that $\nabla\cdot u=0$ is it equivalent to saying following? $$\nabla\cdot u=?(\nabla\cdot u)^\bot=\nabla^\bot\cdot u^\bot=\nabla\times ...
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Material Derivative of the Gradient of a Scalar Field

Let $f$ be a scalar field that is continuous and does not vary along the flow, that is $D_t(f)=0$ where $D_t=\partial_t+\vec u\cdot\nabla$ where $\vec u$ is the incompressible velocity field (i.e ...
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A model for constant temperature of water in a container

I put some water in a container with initial temperature $T_0$ in a room, and the room's initial temperature is $T_a$. Now the container is filled to the maximum, so any more water coming in will ...
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How to Derive Material Derivative in Rotating Frame

I am currently working through "An Introduction to Dynamic Meteorology (5th Ed.)" by Holton and Hakim, and am looking for more detail on material derivatives in rotating reference frames than what ...
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Suitable change of variable to transform ODE into the standard form of Bessel's equation

I've been stuck on this for a while now. I need to use a suitable change of variable $r \rightarrow \xi$ to transform this ODE: $r^2V'' + rV' - (\frac{i\Omega r^2}{\nu}+1)V = 0$ into the standard ...
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fluid flow velocity , boundary conditions

can someone please help me with par(c)? $ u = U(1-\frac{a^{3}}{2r^{3}}) cos \theta $ - $ U(1+ \frac{a^{3}}{2r^{3}}) sin \theta $ and the boundary conditions are that r = a, u(r) =0, When sin ...
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fluid between two fixed walls equation help

I am not to sure, how to answer this question, but is it cause v = 0? how to show u is independent of x?