2 - Equations of Fluid Motion

Summary

Introduction

A fluid is a substance that yields to applied shear stress; i.e., it is a substance that continuously deforms under an applied shear stress. Fluids comprise both liquids and gases. While in the macroscopic scale, fluids are made up of molecules, it is possible to disregard the molecular viewpoint while discussing fluid properties by invoking the concept of “continuum.” The “continuum” hypothesis assumes that fluid is continuous and made up of a very large number of fluid elements. The advantage of the “continuum” hypothesis is that it provides for continuously distributed body of matter in which fluid properties vary smoothly. A fluid element is said to be composed of several millions of molecules within it and has a scale for which macroscopic fluid properties such as density and temperature can be prescribed. Hence in a “continuum” hypothesis approach, the fluid element is the smallest scale of analysis. The scale of the fluid element cannot be too small as such an element would not provide a robust average of fluid properties. Moreover, the scale of fluid element cannot be too large, as this would contribute to smoothing over and ignoring relevant and important scales of variability.

An obvious length scale for gases is the “mean free path” (λ), which is defined as the average distance traversed by a gaseous molecule between two successive collisions. The mean free path for liquid molecules assume typically much smaller values than that for gas. For the atmosphere, the “mean free path” at sea level is 10^-7m, at 50 km, it is 10^-4m, and at 150 km, it is 1 m. The “characteristic length scale” (L) for a typical fluid situation can be defined as the most important length scale component of the fluid motion. For example, for a fluid flowing in a long tube of circular cross section, the radius or diameter of the circular cross section is the “characteristic length scale.” The “continuum” hypothesis is applicable for flow situations in which the “characteristic length scale” L of the flow is much larger than the “mean free path” ƛ. For the atmosphere, the “continuum” hypothesis is applicable for fluid motion up to the mesospheric height of 90 km.