Articles of fluid dynamics

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 $i$-component of the force per unit surface that is exerted at an element of the surface perpendiculart to the direction $j$. Is this definition of […]

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 $\mathcal D(\Omega,\mathbb R^d):=C_c^\infty(\Omega,\mathbb R^d)$ $H^{-1}(\Omega):=H_0^1(\Omega)’$ and $H^{-1}(\Omega,\mathbb R^d):=H_0^1(\Omega,\mathbb R^d)’$ Let me cite a well-known theorem: Let $u\in\mathcal D'(\Omega,\mathbb R^d)$ $\Rightarrow$ $$\left.u\right|_{\mathfrak D(\Omega)}=0\;\Leftrightarrow\;\exists p\in\mathcal D'(\Omega):u=\nabla p\tag 1$$ where $$\mathfrak D(\Omega):=\left\{\phi\in\mathcal D(\Omega,\mathbb R^d):\nabla\cdot\phi=0\right\}\;.$$ Now, if $F\in H^{-1}(\Omega,\mathbb R^d)$, then $$f:=\left.F\right|_{\mathcal D(\Omega,\:\mathbb R^d)}\in\mathcal D'(\Omega,\mathbb R^d)\;.$$ And since $\mathfrak […]

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 up to time $T>0$ of such a fluid is given by the instationary Navier-Stokes equations $$\left\{\begin{matrix}\displaystyle\left(\frac\partial{\partial t}+\boldsymbol u\cdot\nabla\right)\boldsymbol u&=&\displaystyle\nu\Delta\boldsymbol u-\frac 1\rho\nabla p+\boldsymbol f&&\text{in }\Omega\times (0,T)\\\nabla\cdot \boldsymbol u&=&0&&\text{in }\Omega\times (0,T)\end{matrix}\right.\;,\tag […]

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, $$ \mathbf{u} \cdot (\mathbf{\nabla} \cdot \mathbf{u}) = \frac{-1}{\rho} \mathbf{\nabla}p – g\mathbf{\hat{z}} $$ that is, $$ \frac{\partial p}{\partial x} = \frac{\rho}{2} \sin(2x) $$ $$ \frac{\partial p}{\partial y} = \frac{\rho}{2} \sin(2y) $$ […]

Differentiation under the integral sign – line integral?

In the proof of Kelvin’s circulation theorem it is common to do the following manipulation: $$\newcommand{\p}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\f}[2]{\frac{ #1}{ #2}} \newcommand{\l}[0]{\left(} \newcommand{\r}[0]{\right)} \f{D \Gamma}{Dt}=\f{D}{Dt} \oint_{C(t)} \vec u \cdot d\vec x$$ $$=\oint_{c(t)} \f{D\vec u}{Dt}\cdot d\vec x+\oint_{c(t)}\vec u\cdot d\l \f{D\vec x}{Dt}\r$$ I am uneasy about the validity of this expression. Please can someone explain the logic […]

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 $$l_{ip}=\frac{\partial \bar x_i}{\partial x_p}=\frac{\partial x_p}{\partial \bar x_i}$$ Hint Use this property of the rotation tensor $$l{l^T} = I$$

water wave and fluids dispersion relation

Small-amplitude water waves travel on the free surface $y = \eta(x, t)$ of an incompressible inviscid fluid of uniform depth $h$. Derive the linearised boundary conditions $$\text{ at }y=0\quad \frac{\partial\varphi}{\partial t}=\frac{\partial\eta}{\partial t},\quad \frac{\partial\varphi}{\partial t}+g\eta=0 $$ and write down the boundary condition satisfied by the velocity potential $\varphi$ at the rigid boundary $y = −h$. Show […]

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 also provide some articles or books about the topic? Most physics books seem not to care about these details.

Solving for streamlines from numerical velocity field

Say I have a given numerical velocity field in two dimensions, (u,v). I am trying to find the streamlines from this data set at a particular contour level and I thus have to solve the differential equation $$ dy/dx = v/u=g(x,y) $$ I can rewrite the equation to $$ dy = g(x_i,y_i)dx $$ The subscript […]

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 t=T, where T is greater than zero and less than infinity. I think it’s […]