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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If $\mathcal H$ is the closure of the set $D$ of divergence-free smooth functions in $L^2$, then $H_0^1∩\mathcal H$ is the closure of $D$ in $H_0^1$

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D:=C_c^\infty(\Omega,\mathbb R^d)$ and $$\mathfrak D:=\left\{u\in\mathcal D:\nabla\cdot u=0\right\}$$ $H:=H_0^1(\Omega,\mathbb R^d)$...
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14 views

Working out Stokes' law

A couple of weeks ago my professor gave me a paper on Stokes' law and how to derive it from the equations of motion and continuity. (http://www.ux.uis.no/~finjord/pdf/flu/stokes.pdf) In one of the ...
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Prove that $\left.F\right|_{\left\{ϕ∈C_c^∞(Ω,ℝ^d):∇⋅ϕ=0\right\}}=0⇔∃p∈C_c^∞(Ω)$ with $F=∇p$, for all $F∈H_0^1(Ω,ℝ^d)'$

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\langle\;\cdot\;,\;\cdot\;\rangle$ denote the inner product on $L^2(\Omega,\mathbb R^d)$ $\mathcal D:=C_c^\infty(\Omega,\mathbb R^d)$, $$H:=\...
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Estakhr's relativistic correction & Navier-Stokes existence and smoothness problem

Why does this physical relativistic correction does not seem to have an affect in solving the mathematical problem of Navier-Stokes existence and smoothness? or it does!? $$\bbox[#AFA,5px,border:2px ...
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101 views

If $F∈H^{-1}(Ω,ℝ^d)$ and $∃p∈\mathcal D'(Ω):\left.F\right|_{\mathcal D(Ω,ℝ^d)}=∇p$, then $∃\overline p∈H^{-1}(Ω):F=∇\overline p$

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D(\Omega):=C_c^\infty(\Omega)$ and $\mathcal D(\Omega,\mathbb R^d):=C_c^\infty(\Omega,\mathbb R^d)$ $H^{-1}(\Omega):=H_0^1(\Omega)'$...
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I don't understand De Rham's theorem about the gradient of a distribution

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D(\Omega):=C_c^\infty(\Omega)$ and $$\mathfrak D(\Omega):=\left\{\Phi\in\mathcal D(\Omega)^d:\nabla\cdot\Phi=0\right\}$$ In a ...
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21 views

Divergence of Material Derivative

Let $u : \Bbb{R}^n\times \Bbb{R} \to \Bbb{R}^n\times \Bbb{R} $ be a divergence free vector field. Then the material derivative $D $ is given by: $$ \frac { \partial u_j}{\partial t}+\sum_{i=1}^{n} ...
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31 views

Computing the Jacobian of the Euler equations

Given the Euler equations $$ \frac{\partial q}{\partial t}+\frac{\partial f(q)}{\partial x}=0,\qquad q=\begin{pmatrix}\rho\\\rho u\\\rho e\end{pmatrix}, \qquad f(q)=\begin{pmatrix} \rho u\\\rho u^2+...
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Computational Topology

So I've revently gotten an opportunity to begin independent study in a field of my interest. As it turns out a close friend of mine is a post doctoral researcher who's working on essentially ...
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15 views

Maxwell-type system: how to solve?

$\DeclareMathOperator{\curl}{curl}\renewcommand{\div}{\mathop{\mathrm{div}}}$ Consider the following system of equations for the unknown vector field $A$ in the unit 3d ball $B$ with boundary $S$: $$ ...
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1answer
47 views

Velocity profile in a triangular duct (Derivation)

How do I find a velocity profile of an incompressible fluid in a triangular duct. Can someone point me to a step-by-step solution so that I could understand the process of derivation of the final form ...
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66 views

Blasius Theorem why do we always take complex conjugate to represent the net force?

Suppose fluid flows steadily past an obstacle $B$ with simple closed boundary $∂B$. If gravity is neglected, the net force $(F_x,F_y)$ exerted on $B$ by the fluid (per unit length out of the plane) is ...
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2answers
88 views

Linearising Wave Equations

We are given the equations $$\frac{\partial{u}}{\partial{t}}+u\frac{\partial{u}}{\partial{x}}+g\frac{\partial{h}}{\partial{x}}=0$$ and $$\frac{\partial{h}}{\partial{t}}+\frac{\partial{(hu)}}{\...
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23 views

Neumann boundary condition, spherical shell.

The velocity of a fluid $\mathbf{u}$ is assumed to have the velocity potential $\Phi$ such that $\mathbf{u}= \nabla \Phi$. The fluid is contained in a rigid shell, of radius $a$, which is moving with ...
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23 views

Minimal time to steady-state flow (1st Stokes Problem)

How to obtain a minimal time to steady-state flow? This is related to the Stokes 1st Problem. $$\frac{u}{U}=0.05=1 - \operatorname{erf}(\eta)$$ $$\eta = \frac{y}{2\sqrt{vt}} \approx 1.4$$ $$y=\...
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Help with integral from Boltzmann equation

I have a function $$ g\left(x,v,t\right) = u\left(x,t\right)\cdot v + \theta\left(x,t\right)\frac{1}{2}\left(\left\lvert v\right\rvert^2 - 5\right) $$ where $g(x,v,t),\theta(x,t)$ are scalars and $u(...
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9 views

Convective operator: Distributive with superposition of vector fields?

I'm unable to find a great deal of information on this. I'm mostly sure that the convective operator over a vector field $A$ acting on a function $f$: $$ (A \cdot \nabla )f$$ is distributive, i.e. ...
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24 views

Divergence theorem not consistent when calculating the flux of $F= (x^k,y^k,z^k)$ on $S^2$?

Divergence theorem not consistent when calculating the flux of $F= (x^k,y^k,z^k)$ on $S^2$? ($ k > -2$). Using the divergence theorem, I got that the flux is: $\frac{3\pi}{k}(1-(-1)^k)$ and ...
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43 views

Solve $\frac {\partial^2 v}{\partial r^2}+\frac 1r \frac {\partial v}{\partial r}-\frac v{r^2} =0$

$$\frac {\partial^2 v}{\partial r^2}+\frac 1r \frac {\partial v}{\partial r}-\frac v{r^2} =0$$ i did $$\frac {1}{r}\frac {\partial }{\partial r}\bigg( r \frac {\partial v}{\partial r} \bigg) =\frac v{...
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42 views

Finding $x$ and $y$ components of Navier Stokes

An incompressible viscous fluid of constant densite and kinematic viscosity occupies the space between porous walls at $y=0$ and $y=d$. The steady two dimensional flow is subject to a constant ...
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92 views

Integrate second order DE once

Given the vorticity equation $$\frac{D \omega}{Dt}=(\omega \cdot \nabla)\textbf{u}+ν\nabla^2ω$$ and $\textbf{u} = (−αr/2,v(r),αz) $ in cylindrical polars where alpha is positive constant. Find $\...
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31 views

solution of small perturbation in fluid dynamics

Suppose we have a 2D flow in a narrow gap as shown in the picture The flow is governed by Navier-Stokes equation $\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u} = - \frac{\...
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41 views

Stream function and Vorticity relationship with Streamlines

For a two-dimensional flow, the velocity is given by $\textbf u= (u(x,y,t),v(x,y,t),0)$. Define the stream function $ψ (x,y,t)$. Evaluate $(\textbf u·∇) ψ$ and deduce a relationship between the stream ...
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1answer
20 views

Derivative of the stress tensor

Let $\partial u_i/\partial x_i=0$ then given that $$\sigma_{ij} = -p\delta_{ij}+\mu\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)$$ Show that $$\frac{\partial \...
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1answer
47 views

Finding pressure using Bernoulli's Theorem

The inviscid irrotational flow around a circular cylinder of radius $a$ is described by the complex potential $$w =Uz+ \frac{Ua^2}{z}$$ where $U$ is a positive constant. I found $\psi=U \cos \theta (...
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66 views

Finding stagnation points and stream function

Sorry for the lack of latex. The question I want to ask would need all this info and it would take very long to write it. (a) Irrotational flow means $\nabla \times \textbf u =0$ so we can define ...
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33 views

Using Euler's equation and vector identity

An unsteady incompressible inviscid fluid flow satisfies the continuity equation $∇·\textbf u = 0$ and Euler’s equation $$\frac{∂ \textbf u }{∂ t} +(\textbf u·∇)\textbf u = − \frac1ρ ∇p$$ where $\textbf ...
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18 views

Weak form of PDE for 2-phase-flow

I want to simulate the flow of 2 incompressible and immiscible fluids in porous media. It is assumed that the conditions are satisfied for the Darcy equation to be valid. The combination of mass ...
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32 views

Solve $f'' - \frac{iw}{\upsilon}f=0$

We are given $u=u(y,t)$ and $$\frac{\partial u }{\partial t}= \upsilon \frac{\partial ^2 u }{\partial y^2}$$where $\upsilon$ is the viscosity. Look for solutions of the form $u=Re\{Ue^{iwt}f(y)\}$ ...
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38 views

Stochastically perturbed fluid flow map ${\rm d}Φ_t(x_0)=u_t(Φ_t(x_0)){\rm d}t+ξ_t(Φ_t(x_0)){\rm d}W_t$

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $U$ be a separable Hilbert space $Q$ be a bounded, linear, nonnegative and symmetric operator on $U$ with finite trace $(W_t)_{t\ge 0}...
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22 views

Viscous fluid boundary condition

Consider an incompressible viscous fluid of kinematic viscosity $ν$ , dynamic viscosity $µ$ and density $ρ$ . A viscous boundary layer is located over a solid surface at $y = 0$ and $x > 0$. The flow ...
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25 views

What does $p_b \propto ρ^n _b $ mean

$p_b \propto ρ^n _b $, in fluid mechanics, where $p_b$ is the pressure inside a bubble and $\rho$ is the density. What does that symbol looking like alpha mean?
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Manipulating vorticity equation

We have $\omega = (0,0, \xi(x,y,t))$ and $\textbf u =(u(x,y,t),v(x,y,t),0)$ and that $$\frac{\partial \xi}{\partial t} +u \frac{\partial \xi}{\partial x} +v\frac{\partial \xi}{\partial y}=0$$ is a ...
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${\partial\over{\partial x_j}}\left(\partial x_i\over\partial t\right)\ne{\partial\over{\partial t}}\left(\partial x_i\over\partial x_j\right)$?

$ \boldsymbol x = f(\boldsymbol X,t)$ is the position of a particle in an instant of time $\boldsymbol X$ is the initial position $t$ time $\boldsymbol u$ velocity In my opnion $f$ is continuos... ...
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48 views

Finding pdes of velocity component and pressure

An incompressible viscous fluid of constant density $ρ$ and kinematic viscosity $ν$ occupies the space above a solid boundary at $y = 0$ in two-dimensional Cartesian coordinates $(x,y)$. For time $t &...
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25 views

Finding an expression for velocity [closed]

Consider an annulus formed by two circular cylinders, with one cylinder inside the other. The inner cylinder has radius $a$ and the outer cylinder has radius $b$. The cylinders have a common axis, and ...
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Taking curl of Euler equation

Consider an inviscid incompressible flow. Euler’s equation can be written as $$\frac{\partial \textbf u}{\partial t} + \textbf ω × \textbf u = −\textbf∇\bigg( \frac pρ + \frac 12 \textbf u^2 + V \bigg)$...
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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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35 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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29 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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23 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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38 views

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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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)= curl(...
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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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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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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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36 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 infinite-...
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99 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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19 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) $...