{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Derivation of Bernoulli's Equations\n", "\n", "\n", "## Learning outcomes\n", "\n", "* Learn how Bernoulli's equations are derived.\n", "* Work in curvilinear coordinates\n", "* Learn about some applications of Bernoulli's equations.\n", "\n", "## Recap of Vector fields\n", "\n", "As we saw in the previous notebook, the velocity of a fluid flow at each point in an Eulerian sense can be represented by a vector field $\\vec{V}(\\vec{x},t)$ where the position vector $\\vec{x}$ denotes the position of each velocity vector relative to the origin, $\\vec{x}(x,y,z) = x\\mathbf{\\hat{i}}+y\\mathbf{\\hat{j}}+z\\mathbf{\\hat{k}}$ where $\\mathbf{\\hat{i}}$, $\\mathbf{\\hat{j}}$ and $\\mathbf{\\hat{k}}$ are unit basis vectors, and the current time is given by $t$. Any position vector can be described extending from the origin to some point $(x,y,z)$. Similarly $\\vec{V}({\\vec{x}},t) = u\\mathbf{\\hat{i}}+v\\mathbf{\\hat{j}}+w\\mathbf{\\hat{k}}$ where $u={u({\\vec{x}}},t)$, $v={v({\\vec{x}}},t)$ and $w={w({\\vec{x}}},t)$ are the velocity magnitudes in each Cartesian direction for the current time, $t$. An example 2D flow field is shown below. Each vector's tail is positioned on the Cartesian grid according to its position vector $\\vec{{}x}$ and the velocity vector field, which in this case is time invariant, is given by:\n", "\n", "\\begin{equation}\n", "\\vec{{}V}(x,y) = (\\sin(y)+\\pi) \\mathbf{\\hat{{}i}} + 1.2\\cos(x) \\mathbf{\\hat{{}j}},\n", "\\end{equation}\n", "\n", "which results in a flow from left to right with a slight sinuous meander in the $y$ direction.\n", "We can plot this and also include some streamlines.\n", " \n", "\n", "