Fluid Friction in One-Directional Flow and Other Geometries

11 FLUID FRICTION IN ONE-DIRECTIONAL FLOW AND OTHER GEOMETRIES 

Steady flow in a circular pipe is one of the simplest flow problems in laminar flow, the velocity is one-directional, independent of time, and dependent on only one dimension, the radius. 

A somewhat harder problem is steady flow of an incompressible newto-nian fluid in sonJe duct or pipe which is of constant cross section but not circular, such asja rectangular duct or an open channel. The problem of laminar flow of a newtonian fluid can be solved analytically for several shapes. Generally the velocity depends on two dimensions. In several cases of interest, the problems can jbe solved by the same method we used to find Eq. 6.8, i.e., setting up a force balance around some properly chosen section of the flow, solving for the sh r stress, introducing the newtonian law of viscosity for the shear stress, and integrating to find the velocity distribution. From the velocity distribution the flow rate-pressure-drop relation is found. 

The fact that these are all similar to the solution for laminar flow in a horizontal circular tube may be seen by comparing the horizontal, steady flow solutions with that for a circular tube. For a circular tube 

We are no more able to calculate the pressure drop in steady, turbulent flow in a noncircular conduit than we are in a circular one. However, it seems reasonable to expect that we could use the friction-loss results for circular pipes to estimate the results for other shapes. Let us assume that the shear stress at the wall of any conduit is the same for a given average fluid flow velocity independent of the shape of the conduit. Then, from a force balance on a horizontal section like that leading to Eq. 6.3, we conclude that in steady flow 

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