# Tagged Questions

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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...