The Theoretical Aspects of the Orr-Sommerfeld Methods: The Case of Two Viscoelastic Fluids

In this article, conceived for physicists and mathematicians, we describe various Orr-Sommerfeld Equations (OSEs) and stress their differences, both in modeling, justification and in the results. These equations are derived from the Poiseuille flow of two viscoelastic or Newtonian fluids. The literature proposes a link between computation and experiment which is modeled by two different equations. We reinvestigate it and stress a hidden assumption. Then, we study extensively the long wave asymptotic stability of the flow of two viscoelastic fluids and exhibit a formula for characterization of loss of stability in a new case. Some waves are found through an OSE and cannot be found through the other. We give their growth rate implicitely for some of them. Last, we prove a theorem that says whether such a wave could be unstable or not.

Keywords:

• Orr-Sommerfeld
• Stability
• Viscoelastic Fluids
• Poiseuille Flow