Solution by Quadrature

When there is only one equation, even if it is nonlinear, it may be possible to solve by quadrature. For

the problem can be separated

and integrated.

If the quadrature can be performed analytically then the exact solution has been found.

Pipe Flow

As an example, consider the flow of a non-Newtonian fluid in a pipe, as illustrated in the figure. The governing differential equation is [Bird,et al. 1960]

where r is the radial position from the center of the pipe, t is the shear stress, p is the pressure drop along the pipe, and L is the length over which the pressure drop occurs. The variables are separated once

and then integrated to give

To proceed further requires choosing a constitutive relation relating the shear stress and the velocity gradient as well as a condition specifying the constant. For a Newtonian fluid we use

Then the variables can be separated again and the result integrated to give

Now the two unknowns must be specified from the boundary conditions. This problem is a two-point boundary value problem because one of the conditions is usually specified at r = 0 and the other at r = R, the tube radius. However, the technique of separating variables and integrating works quite well.

This differential equation can also be integrated analytically using symbolic notation, as is illustrated in two examples. First and Second example.

When the fluid is non-Newtonian fluid it may not be possible to do the second step analytically. For example, for the Bird-Carreau fluid [Bird, et al.,, p. 171,1987] the stress and velocity are related by

Putting this value into the equation for stress as a function of r gives

It is not possible to solve this equation analytically for dv/dr, except for special values of n. For problems like this numerical methods must be used.

Take Home Message: Some boundary values can be solved by quadrature, and this should be tried if possible since the solution then can be written down in compact form for all parameters.