The Navier-Stokes Equation

Tha Navier-Stokes equation describes the motion of a parcel of fluid on Earth, either in the atmosphere or in liquid form (e.g. rivers, oceans).

Equation from Dr. Elie Bou-Zeid’s Environmental Fluid Mechanics class, Fall 2021.

\[\frac{D\vec u}{Dt} = \frac{\partial \vec u}{\partial t} + \vec u \cdot \nabla \vec u = -\frac{1}{\rho}\nabla p + \nu \nabla^2\vec u + \vec g\left(1+\frac{\rho'}{\rho}\right) - f_c \vec k \times \vec u\]

Where

  • $ \frac{D\vec u}{Dt} $ is the material derivative of velocity with respect to time
  • $\rho$ is density
  • $p$ is pressure
  • $\nu$ is viscosity
  • $g$ is gravity
  • $\rho’$ is the difference between parcel density $\rho$ and mean density $\bar \rho$: ($\rho’ = \rho - \bar \rho$)
  • $f_c$ is coriolis force: $f_c = 2\Omega \sin(\text{latitude})$
  • $k$ is unit vector in vertical direction

The Bernoulli Equation

The Bernoulli equation derives from the Navier-Stokes equation and assumptions of (1) steady-state ($\frac{\partial \vec u}{\partial t} = 0$), (2) no viscous forces ($\nu=0$), (3) no bouyant force ($\rho = \bar\rho \Rightarrow \rho’ = 0$), and (4) no coriolois $f_c = 0$.

\[\frac{U_2^2}{2} + \frac{p_2}{\rho} + gz_2 = \frac{U_1^2}{2} + \frac{p_1}{\rho} + gz_1\]

Where

  • $U_i$ is total flow speed at point $i$ (no flow normal to streamline)
  • $p$ is pressure at point $i$
  • $\rho$ is density
  • $g$ is acceleration due to gravity
  • $z_i$ is height at point $i$ relative to some baseline for all $i$

The Bernoulli equation is very useful for determining speed of flow due to pressure or elevation changes in pipes, rivers, etc.

Euler’s Formula

\[e^{i\pi} = \cos x + i \sin x\]

Euler’s Identity

A special case of Euler’s formula evaluated at $x = \pi$

\[e^{i\pi}+1 = 0\]

Newton’s Basic Equation

This equation describes the motion, displacement, and acceleration of a body due to some force (or set of forces)

\[F = ma = m\frac{\partial v}{\partial t} = m\frac{\partial x^2}{\partial^2 t}\]

Where

  • $F$ is force, in units N = kg m s$^{-2}$
  • $m$ is mass, in units kg
  • $a$ is acceleration, in units m $s^{-2}$
  • $v$ is velocity, in units m s$^{-1}$
  • $x$ is displacement, in units m
  • $t$ is time, in units s

This equation can be used in all sorts of physical systems with various zeroeth, first, second-order, and higher-order forces driving the motion of a body

Fourier Sine and Cosine Series Representations of Functions

Any function on the interval $0 \leq x \leq \ell$ can be written as a sum of sines and cosines. Even functions can be written in terms of cosines only, and odd functions can be written in terms of sines only. Both sines and cosines must be present for a function with both even and odd components (general function).

Sine (odd functions)

\[f(x) = \sum_{n=1}^{\infty}a_n\sin\left(\frac{n\pi x}{\ell}\right)\]

Where

\[a_n = \frac{2}{\ell}\int_0^\ell f(x)\sin\left(\frac{n\pi x}{\ell}\right)dx\]

Cosine (even functions)

\[f(x) = \sum_{n=0}^{\infty}b_n\cos\left(\frac{n\pi x}{\ell}\right)\]

Where

\[b_0 = \frac{1}{\ell}\int_0^\ell f(x)dx\]

and

\[b_n = \frac{2}{\ell}\int_0^\ell f(x)\cos\left(\frac{n\pi x}{\ell}\right)dx\;\;(n \geq 1)\]

Fourier Series (any function)

\[f(x) = b_0 + \sum_{n=1}^\infty\left(\underbrace{a_n\sin(n\pi x)}_{\text{odd function}}+\underbrace{b_n\cos(n\pi x)}_{\text{even function}}\right)\]

Fourier Transform and Inverse Fourier Transform

Let $f$ be the function dependent on space or time (denoted by $x$) and let $\hat f$ be the transformed function dependent on frequency (traditionally denoted by $\zeta$). Then

\[\hat f(\zeta) = \int_{-\infty}^\infty f(x)e^{-2\pi i x\zeta}dx\;\; \forall \zeta \in \mathbb{R}\]

and, inversely,

\[f(x) = \int_{-\infty}^\infty \hat f(\zeta)e^{2\pi i x\zeta}d\zeta\;\; \forall x \in \mathbb{R}\]

The Fourier transform changes a space-or-time dependent function into a frequency-dependent function. For example, if you plug an audio recording (time dependent) into a fourier transform, you would get a set of frequencies and their relative strengths in the signal back. The frequency function you get back can be inversely transformed into the original function; or, if you remove unwanted frequencies, you can inversely transform into an audio recording minus the unwanted high-pitched buzz.

Fourier transforms enable modern commincation and many forms of audio and visual data processing. Truly an amazing mathematical tool!