Fluid Statics I

Learning outcomes

  • Derive Pascal's Law
  • Learn about how pressure is distributed in volumes of fluid at rest
  • Learn about hydrostatic pressure
  • Explore how pressure and temperature is distributed in the Earth's atmosphere
  • Compare different units of pressure measurement

Introduction

Herein we will describe hydrostatics in some detail. This, and many of the other topics we will cover, will involve some mathematics, but I will endeavour to help your understanding of this with diagrams, animations and interactive python code snippets. As with all problems in engineering and physics we attempt to gain an understanding of the world by imagining a simpler version of reality ignoring all wrinkles and disturbances and focusing on the general nature of the system. In the example below we take the ocean floor and imagine a cube of sea water and the sand beneath it. We simplify this as a perfectly still region of fluid so that the upper and lower surface are perfectly flat. We assume that the sea bed is completely rigid and does not move. We ignore and influences from the side and assume that invisible force fields are preventing the water escaping – or simply that the side walls are periodic so that any water exiting one side is transported into the opposite side. Any disturbance will eventually dissipate and we are left with a perfectly still box of fluid. Now within this volume of fluid we can consider a finite element of fluid.

It is important to be able to look at the world in this manner and reduce complex systems to these kinds of cartoons. As we learn more about the system we can add complexity.

Fluid Statics and fluid domains

When performing calculations in engineering we often need to take a complex snapshot of the real world, for example a section of the sea bed (left) and simplify it to a more straightforward approximation, in this case a stationary incompressible fluid on a rigid flat floor (middle) and further consider a fluid element in this 2.5D model of reality.

Pascal's Law

We touched on Pascal's in the previous notebook. Here it is again.

\begin{equation*} \Delta P = \rho g \Delta h \end{equation*}

We've previously looked at Boyles's Law for gases and how this lead to the Ideal Gas law. Pascal's Law deals with incompressible fluids. This is a limiting case as we know that air can be readily compressed with enough effort by an air compressor or a fighter jet flying at supersonic speeds. Water, like most dense liquids, on the other hand can be considered incompressible. So the Pascal's Law is applicable in cases where we can consider the fluid to be incompressible and the density $\rho$ doesn't change. Surprisingly this covers a lot of engineering problems. A cylinder of compressed air at rest is subject to Pascal's law because in that state it's density is constant.

Lets begin by considering the fluid element shown above.

Note, if you are using Google Colaboratory or Jupyter's nbviewer, the interactive 3D viewer does not yet work. You can use this video as a workaround if required. The HTML version should work fine.

In [1]:
import sys

if sys.platform != 'linux': 
    from PyGEL3D import gel
    from PyGEL3D import js

    js.set_export_mode()

    m = gel.obj_load("media/2.1/pascal.obj")
    js.display(m, smooth=False, data=None, wireframe=True)
        
else:
    print('play video in previous cell')