Examples from Classical Mechanics - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 105

Figure 3.5. Neutral stability diagram.
3.5.2 Borehole flow stability.
The laminar stability of drilling mud and cement flows in the borehole
annulus, important to drilling and completions, should be studied using rigorous
hydrodynamic stability methods. Instead, the critical Reynolds number of 2,300
is often blindly invoked by drilling practitioners, but this applies to parabolic
laminar profiles of Newtonian fluids in circular pipes only! This transition
number has been validated experimentally under carefully controlled conditions,
e.g., literally zero turbulence level in a smooth polished inlet.
Following Schlichting (1968), the velocity profile of interest must first be
obtained. For any eccentric annulus, this is easily accomplished using finite
difference methods applied to boundary conforming grid systems; for example,
refer to the author' s recent topic Managed Pressure Drilling (Chin, 2012) for
numerical algorithms and computed results. This calculation will have assumed
a particular rheological model, e.g., Newtonian, Bingham plastic, power law
fluid, or Herschel-Bulkley flow. Then, the viscous disturbance equations
corresponding to the constitutive model used should be derived and solved on
the same boundary-conforming mesh, with waves allowed to propagate in the
cross-plane and axial directions. An eigenvalue problem similar to the one
described in the previous section then applies whose neutral stability diagram
must be obtained. In general, the computational procedure is extremely
laborious and few exhaustive studies are available in the literature.