Week 8: Fluids

Fluids Summary

•Fluids are states of matter characterized by a lack of long range order. They are characterized by their density ρ and their compressibility. Liquids such as water are (typically) relatively incompressible; gases can be significantly compressed. Fluids have other characteristics, for example viscosity (how “sticky” the fluid is). We will ignore these in this course.

•Pressure is the force per unit area exerted by a fluid on its surroundings:

Its SI units are pascals where 1 pascal = 1 newton/meter squared. Pressure is also measured in “atmospheres” (the pressure of air at or near sea level) where 1 atmosphere ≈ 105 pascals. The pressure in an incompressible fluid varies with depth according to:

where P0 is the pressure at the top and D is the depth.

•Pascal’s Principle Pressure applied to a fluid is transmitted undiminished to all points of the fluid.

•Archimedes’ Principle The buoyant force on an object

where frequency Vdisp is the volume of fluid displaced by an object.

•Conservation of Flow We will study only steady/laminar flow in the absence of turbulence and viscosity.

where I is the flow, the volume per unit time that passes a given point in e.g. a pipe.

•For a circular smooth round pipe of length L and radius r carrying a fluid in laminar flow with dynamical viscosity126 µ, the flow is related to the pressure di erence across the pipe by the resistance R:

126Wikipedia: http://www.wikipedia.org/wiki/viscosity. We will defer any actual statement of how viscosity is related to forces until we cover shear stress in a couple of weeks. It’s just too much for now. Oh, and sorry about the symbol. Yes, we already have used µ for e.g. static and kinetic friction. Alas, we will use µ for still more things later. Even with both greek and roman characters to draw on, there just aren’t enough characters to cover all of the quantities we want to algebraically work with, so you have to get used to their reuse in di erent contexts that hopefully make them easy enough to keep straight. I decided that it is better to use the accepted symbol in this textbook rather than make one up myself or steal a character from, say, Urdu or a rune from Ancient Norse.

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