Questions tagged [fluid-dynamics]
For questions about fluid dynamics which studies the flows of fluids and involves analysis and solution of partial differential equations like Euler equations, Navier-Stokes equations, etc. Tag with [tag:mathematical-physics] if necessary.
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Does this equation imply "non-linear" waves?
For one-dimensional bounded domain $x \in [0, L]$ consider
$\partial_{t} v = -\frac{1}{\rho}\partial_{x}\rho - v $
$\partial_{t} \rho = -\partial_{x}(\rho v)$
with initial and boundary data as $v(0, ...
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I don't understand the notation in Mechanical engineering. Could you help me to understand it better?
Consider a particle $M$ occupying the position $\vec{a}$ at time $t = 0$ and the position $\vec{x}$ at time $t$. A function $f = f(M, t)$ associated with a particle $M$ (e.g., its velocity, ...
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Question on Continuum Mechanics: How does Eulerian derivatives of $f$ and Lagrangian derivatives of $f$ different?
Consider a particle $M$ occupying the position $\vec{a}$ at time $t = 0$ and the position $\vec{x}$ at time $t$. A function $f = f(M, t)$ associated with a particle $M$ (e.g., its velocity, ...
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Prove that $h'(t)=\ell'(t).$ How does this proof shows The vector $U$ is independent of the reference time $t_0?$
The velocity of the material point $M$ occupying the position $\vec{x}$ at time $t$ is the
vector $\vec U=\vec U(\vec x, t)=\frac{\partial \Phi(\vec a,t,t_0)}{\partial t}.$
Where $\vec x =\Phi(\vec a,...
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Physical meaning of constant boundary condition
I am studying fluid-structure interaction (so, based on Navier-Stokes equations).
In particular, i read about the Stokes Paradox, that says that there is no solution to the 2D Stokes problem outside a ...
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How do I find the y-intercepts of the vortex ring function, u(y')?
I am trying to graphically compute a vortex ring, $u(y')$ in an $x'-y'$ axis, whose vortex locations $d'$ and $-d'$ are distorted by some unknown function, $\pm y'(d')$, which are the $y-$intercepts ...
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Understanding Navier-Stokes equation. Laplacian of a vector field?
I'm reading Vorticity and incompressible flow (Majda), but I don't understand the Navier-Stokes equation:
As $v(x,t)$ is a vector field, what does $\Delta v$ means? $(\Delta v^1, ..., \Delta v^N)?$
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How do I find the absolute maximum and minimum values of the Lamb-Oseen Vortex?
I am researching alternative solutions to Stokes equations and I came across a problem with the Lamb-Oseen vortex I cannot solve that I hope will allow easier derivations of vortex functions with ...
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Exercise 7.6 of Robinson, Rodrigo, Sadowski: Smoothness of Navier-Stokes on Bounded Domains
My question is about Exercise 7.6 of the excellent book 'The Three-Dimensional Navier-Stokes Equations' by Robinson, Rodrigo and Sadowski. More generally, it is about higher regularity in space of ...
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Space of functions, Banach spaces, reference books to find basic properties of Bochner integral, Laplace and Fourier transforms.
I'm looking for references where I can find definitions and basic properties of
Bochner Integral in Banach Spaces and its basic properties, such as:
Every continuous function is integrable, ...
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2D Laplace equation analytical solution
I am trying to solve a simple Poiseulle Flow in 2D in Cartesian coordinates numerically and analytically. For the analytic part, I am stuck at the following:
Suppose we have a 2D Laplace equation
$$ \...
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Can this property that I formulated be used to find solutions to Navier Stokes in cylindrical coordinates?
Let's say I want to derive a vortex flow function $$\psi (r,t)$$ using a scalar field bell surface of the form below, where $W(t)$ controls the width (but is not actually the width) as a function of ...
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Derivation of Continuity Equation for an Incompressible flow
Good day guys,
I was playing around with the following form of the continuity equation:
$$ \frac{\partial \rho}{\partial t} - \nabla \cdot (\rho \vec{v}) = 0 $$
For an incompressible fluid: $\frac{D\...
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Complete orthonormal basis of divergence free vector fields
I'm working on a problem in fluid dynamics. I need to find a complete basis of orthonormal 3D vector fields. My "inner product" between vectors $\mathbf{v}_1$, $\mathbf{v}_2$ is a dot ...
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Evans's scalar conservation law order of operation? $\left( u_t + F(u)_x \right)$
This is a small question (taken from Evans's PDE book, page 8). I generally avoid writing parenthesis for a function operating on some character to its right. In absence of parenthesis, I read my ...
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