Spectral Theory of Non-Self-Adjoint Differential Operators and Applications to Stability Analysis in Fluid Mechanics

This dissertation develops original contributions to the spectral theory of non-self-adjoint differential operators arising in hydrodynamic stability analysis, with specific applications to the stability of viscous fluid flows in configurations relevant to petroleum pipeline transport and coastal ocean circulation in Nigerian offshore waters. The spectral analysis of non-self-adjoint operators presents fundamental mathematical challenges not present in the self-adjoint case, including the non-normality of eigenvector systems, the sensitivity of eigenvalues to perturbations, and the possible discrepancy between modal growth rates and short-time energy amplification, all of which are critical for accurate stability predictions in engineering applications. The dissertation proves novel resolvent estimate theorems for a class of non-self-adjoint Orr-Sommerfeld operators in weighted Lebesgue spaces, using pseudospectral methods to characterise the sensitivity of eigenvalues to structured and unstructured perturbations. Sharp bounds on pseudospectral radii are established, and their implications for transient energy growth in subcritical flows are derived rigorously. The Kreiss matrix theorem is generalised to the infinite-dimensional operator setting and applied to bound the maximum possible transient amplification factor for perturbation initial conditions. Numerical spectral computations using Chebyshev collocation are validated against classical benchmark results for plane Poiseuille and Couette flows and then applied to pipe flow configurations representative of Nigerian Trans-Niger pipeline operating conditions. Keywords: spectral theory, non-self-adjoint operators, hydrodynamic stability, pseudospectra, Orr-Sommerfeld