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Mathematics Stack Exchange

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.

Fluid dynamics is a branch of physics that studies the the flows of fluids-liquids and gases, which involves analysis and solution of partial differential equations like Euler equations, Navier-Stokes equations, etc.

1103 questions
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Did I just solve the Navier-Stokes Millennium Problem?

I think I may have just solved a Millennium Problem. I found an exact 3D solution to Navier-Stokes equations that has a finite time singularity. The velocity, pressure, and force are all spatially periodic. The solution has a time singularity at…
A.G.
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10
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Stirring a square cup results in a strange pattern. What's the math behind it?

I came across this picture earlier today: https://i.stack.imgur.com/lMrkU.jpg and it left me kind of baffled. Can anyone explain the mathematics behind the reason why this is happening?
Michael Smith
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4
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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 given by $F_x + iF_y = -\frac{i \rho}{2}…
Gvxfjørt
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3
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Viscous Burgers Equation

I don't understand the bit of the solution of highlighted in green. Up to this point I've been using $U'=\frac{dU}{d\xi}$. Why can I know interchange that with $U'=\frac{dU}{dx}$? For the viscous Burgers equation $$u_t+uu_x=\nu u_{xx},\quad…
user144895
3
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1 answer

Newton's Second Law

I don't follow the part of the solution, which I have underlined in green. Which equation would I get this from (if any)?
usainlightning
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3
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Stream function, velocity potential/field from complex potential

Have this question, but have no answers in the book, so just wanted to check my progress. Consider the following complex potential: $$\Omega(z) = -\frac{k}{2\pi}\frac{1}{z}$$ Calculate the corresponding velocity potential, stream function and…
Mike Miller
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3
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Burgers vortex: finding the angular velocity

I'm trying to solve exercise 5.19 from Elementary Fluid Dynamics by D. J. Acheson. The exercise is as follows: Seek an exact, steady solution to the Navier-Stokes equations of the form $$ \mathbf{u} = -\frac{1}{2}\alpha r \mathbf{e}_r + u_\theta (r)…
Supercell
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Trouble with fluid mechanics question (Bernoulli equation for streamlines)

Here is the problem statement: A liquid is in the annular space between two vertical cylinders of radii $\kappa R$, $R$, and the liquid is open to the atmosphere at the top. Show that when the inner cylinder rotates with angular velocity $\Omega_i$…
Enforce
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3
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Mathematics of Geophysical Fluid Dynamics

Does anyone know some of the contemporary mathematical challenges associated with the fluid dynamics behind climate modelling? Of course the Navier-Stokes equations are relevant, but even with these I am unsure of what approaches mathematicians who…
David F
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Vorticity constant along a streamline

In the case of a 2D incompressible flow $\omega = - \nabla^2 \psi$ where $\psi$ is the stream function. From the vorticity equation of motion I am able to find the following: $$ \frac{\mathrm{d}\omega}{\mathrm{d}t} = \frac{\partial\omega}{\partial…
Partey5
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Water explosion and Navier-Stokes global regularity problem

In this article, Terence Tao is talking about a water explosion thought experiment that can lead to the solution of Navier-­Stokes global regularity problem. Can anybody explain this in more mathematical terms?
Adit Gupta
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Fluid dynamics question, related to the derivation of Bernoulli's Theorem for steady flow

I'm finding it very challenging to understand a step in the derivation of the 'moment equation', found in these notes http://www.maths.ox.ac.uk/system/files/coursematerial/2011/989/66/fluids1.pdf I am struggling with the transition of (1.33) to…
Freeman
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Derivation of the energy equation in fluid dynamics

I'm working through Acheson's 'Elementary Fluid Dynamics' and i'm having trouble deriving the conservation of energy equation (exercise 1.4) $\frac{d}{dt} \int_V \frac{\rho \vert u\vert^2}{2} dV = -\int_{\partial V}(p +\rho \phi + \frac{\rho \vert…
user78655
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2
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1 answer

Continuity equation including mass generated

I don't really conceptually understand why you integrate the generated mass from $a$ to $b$. I understand that you have to take account that it's in within $\left[a,b\right]$, but not why you integrate it.
user144895
2
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2 answers

The matrix A in the system of Euler Equations

I am running a simulation on a 1D Euler equations: $$\frac{\partial \rho}{\partial t}+ \frac{\partial (\rho u)}{\partial x}=0$$ $$\frac{\partial (\rho u)}{\partial t}+ \frac{\partial (\rho u^2+\rho)}{\partial x}=0$$$$\frac{\partial E}{\partial t}+…
Novice
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