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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Is the problem that Prof Otelbaev proved exactly the one stated by Clay Mathematics Institute?

Recently, mathematician Mukhtarbay Otelbaev published a paper Existence of a strong solution of the Navier-Stokes equations, in which he claim that he solved one of the Millennium Problems: existence ...
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References for the Navier-Stokes equations

For understanding the Navier-Stokes equations, are there any references which may include one or more of the followings: mathematical rigorousness motivation preliminaries introduction etc.
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Solving a system of two second order ODEs using Runge-Kutta method (ode45) in MATLAB

I'm trying to solve the system of differential equations outlined in Von Karman's rotating disk flow. I got them into a system of ordinary differential equations: F(n), G(n), H(n) $$F'' = -G^2 + F^2 ...
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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 ...
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398 views

Complex parametrization of Airplane wing?

I read once about complex parametrization with fluid-dynamics objects such as airplane wings, something related Rieman Zeta function. What are the mathematical models this kind of things such as ...
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Modelling a Water Rocket. Requires Some Validation and Help. ( WARNING : Extremely Long but Interesting Post )

Good day people of math.stackexchange.com This is a pet project that I plan to use to convince my Prof that I would rather try something similar to this than to do the prescribed project. Edit : ...
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Existence and uniqueness of Stokes flow

What are the solution existence and uniqueness conditions for Stokes' flow? $$\begin{gathered} \nabla p = \mu \Delta \vec{u} + \vec{f} \\ \nabla \cdot \vec{u} = 0 \end{gathered}$$ Maybe you could ...
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Is Euler's lemma of fluid mechanics a nonlinear version of Liouville's theorem of ODEs?

Liouville's Theorem Consider the following linear system of ordinary differential equations: $$\tag{1} \dot{\mathbf{x}}=A(t)\mathbf{x}(t).$$ Let $\mathbf{x}_1, \mathbf{x}_2, \ldots, ...
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Prove $l_{ip}=\frac{\partial \bar x_i}{\partial x_p}=\frac{\partial x_p}{\partial \bar x_i}$

Let $l$ be a rotation tensor such that $$\bar x_i=l_{ip}x_p$$ where $l_{ip}$ is the direction cosine between the unit vectors in the component directions $x_p$ and $\bar x_i$. Prove that ...
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How can we describe the diffusion of “things” injected into a fluid?

Let $d\in\left\{2,3\right\}$ and $\Omega_t$ be the bounded set occupied by a fluid at time $t\ge 0$. Let $c\in\Omega_0$ be a particle and $$[0,\infty)\to\mathbb R^d\;,\;\;\;t\mapsto ...
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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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Streamlines - Pathlines

Construct and draw the streamlines of the velocity field $u=az-bt, v=\frac{b}{4}z-cy, w=2(a-1)$. Calculate $c$ (as a function of the constants $a$, $b$) such that the flow field ...
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Implementation of a simulation of an incompressible Newtonian fluid with uniform density

Let $d\in\left\{2,3\right\}$ and $\Omega\subseteq\mathbb R^d$ be a bounded domain. I want to simulate an incompressible Newtonian fluid with uniform density $\rho$ and viscosity $\nu$. The evolution ...
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Stress vector - Stress tensor

Is the definition of the stress vector the following? The stress vector is the force per unit surface. The stress tensor is the matrix $\{\sigma_{ij}(x,t)\}$ and its $(i,j)$-component is the ...
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Fluid mechanics resources for pure mathematicians?

I'm currently taking a course on fluid mechanics, and I'm finding it very difficult to become motivated and interested. I've always been more interested in the pure math side of my courses, and love ...
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Continuity equation on manifolds

Mass conservation is usually written as $$\frac{\partial \rho}{\partial t} + \operatorname{div}(\rho \boldsymbol v) = 0$$ $\rho$ is the density and $\boldsymbol v$ is the fluid velocity. My attempt ...
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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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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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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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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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Intuition behind divergence?

$\overrightarrow f = 3x\overrightarrow i - 3y\overrightarrow j$ $\overrightarrow g = 3x\overrightarrow i + 3y\overrightarrow j$ If I calculate the divergence of $f$ I get $0$. If I calculate the ...
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How to differentiate Complex Fluid Potential

I have $$F(z) = \phi + i\psi$$ Also, it is given that $$u = \frac{\delta \phi}{\delta x}, \ \ v = \frac{\delta\phi}{\delta y}$$ and $$u = \frac{\delta \psi}{\delta x}, \ \ v = ...
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Plotting Streamlines Maple

I wonder if anybody could help with this. I've been asked to plot the streamlines of the complex potential $\Omega(z)=Uz + \frac{m}{2\pi}ln(z)$ to which I get the stream function ...
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fluid dynamics in polar coordinates

On page 12 of Malham's fluid dynamics notes the following flow field is considered: $\boldsymbol u= (u,v) = (kx, -ky)$. It's easy to see in these Cartesian coordinates that this is solenoidal: ...
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Fluid Forces using Calculus (Find Work Done)

A tank in the shape of an inverted right circular cone has height 12 meters and radius 11 meters. It is filled with 6 meters of hot chocolate. Find the work required to empty the tank by pumping the ...
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How to integrate twice of this viscous term?

I am reading a paper, and I do not understand why the author said the following term when integrated twice will become, $\int\limits_\Omega {{\rm{d}}\Omega {{\bf{\psi }}^{\bf{u}}}\cdot\nabla ...