Newtonian flow, pipe, circular cross-section

In order to predict Lhe transition point from stable streamline to stable turbulent flow, it is necessary to define a modified Reynolds number, though it is not clear that the same sharp transition in flow regime always occurs. Particular attention will be paid to flow in pipes of circular cross-section, but the methods are applicable to other geometries (annuli, between flat plates, and so on) as in the case of Newtonian fluids, and the methods described earlier for flow between plates, through an annulus or down a surface can be adapted to take account of non-Newtonian characteristics of the fluid. 

Thus the kinetic energy per unit mass of a Newtonian fluid in steady laminar flow through a pipe of circular cross section is u2. In terms of head this is u2/g. Therefore for laminar flow, a = i in equation 1.14. 

For the steady turbulent flow of a Newtonian fluid at high values of Re in a pipe of circular cross section, the mean velocity u is related to the maximum velocity vmix by the equation... 

A. G. Dodson, R Townsend, and K. Walters, Non-Newtonian Flow in Pipes of Non-Circular Cross Section, Comp. Fluids., 2, 317-338 (1974). 

Poiseuille flow n. Laminar flow in a pipe or tube of circular cross-section under a constant pressure gradient. If the flowing fluid is Newtonian, the flow rate will be given by the Hagen-Poiseuille equation. 

Laminar flow is the simplest of the three flows, so we discuss it first. Consider a steady laminar flow of an incompressible newtonian fluid in a horizontal circular tube or pipe. A section of the tube Ax long with inside radius is shown in Fig. 6. 4. We arbitrarily select a rod-shaped element of the fluid, symmetrical about the center, with radius r, and we compute the forces acting on it. Here it is jassumed that location 1 is well downstream from the place where the fluid pnters the tube. This analysis is not correct for the tube entrance. The flow is steady and all in the axial direction. There is no acceleration in the x direction, so the sum of the forces acting in the x direction on the rod-shaped element we have chosen must be zero. There is a pressure force acting on each end, equal to the pressure times the cross-sectional area of the end. These act in opposite directions their sum in the positive c direction is... 

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. 

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