Examples from Classical Mechanics - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 87

Newtonian, are characterized by nonlinear laws. Their analogous right sides are
considerably more complicated, the form depending on the “rheological model”
used, e.g., Bingham plastic, Herschel-Bulkley, power law, viscoelastic, etc.

3.2.2 Simple hydraulic flows.
When hydraulic (principally incompressible ) flowing effects are to be
modeled, the convective left sides of Equations 3.16 to 3.18, which describe
acceleration, must be retained. These terms sometimes vanish, e.g., for straight
ducts with pressure constant across the cross-section. The details of the stress
tensor must also be preserved, and the complete nonlinear equations must be
solved together with “no-slip” velocity boundary conditions. An exposition of
the state-of-the-art in non-Newtonian flow modeling is given in the drilling and
cementing textbooks by this author (Chin, 1992, 2001, 2012).

3.2.3 Acoustic simplifications.
For acoustic (or sound) motions, two simplifications are possible. We
assume zero mean background velocity, so that the quadratic velocity products
on the left sides of Equations 3.16 to 3.18 remain small during the passage of
disturbance waves; nonzero velocities are considered in later in this topic in the
context of ocean waves. If the resultant wave motions are not critically damped,
the rheology of the fluid can also be neglected, in a crude first approximation.
Rheological effects, as they affect attenuation, are typically reintroduced after-
the-fact using imaginary frequencies, as indicated in Chapter 2.
More exact procedures and solutions dealing with the effects of flow
velocity, pipe diameter and fluid type on attenuation are given in the literature.
Basic analysis techniques are described in the classic work of Lamb (1945);
these specialized results, though, fall beyond the scope of this topic. When fluid
rheology is neglected, the right side terms of Equations 3.16 to 3.18 (or their
counterparts in non-Newtonian flow) vanish. If fluid speeds are small, the
nonlinear convective terms on the left sides can also be ignored, leaving

u/ t = - p/ x (3.20)

m

v/ t = - p/ y (3.21)

m

m w/ t = - p/ z (3.22)
where m is a mean density (formally, we set = m + ' where ' is the
disturbance part). Similarly, Equation 3.19 can be linearized to give

'/ t + m ( u/ x + v/ y + w/ z) = 0 (3.23)

We can write Equations 3.20 to 3.23 in vector form as

q / t = - p (3.24)

m

'/ t + q = 0 (3.25)

m