The Orr–Sommerfeld equation is a fundamental equation in fluid dynamics that describes the stability of an incompressible flow, particularly in the context of boundary layer theory. It is named after William Richard Orr and Arnold Sommerfeld, who contributed to its development. The equation arises when analyzing small disturbances or perturbations in a basic flow profile. It is particularly important in studying the stability of laminar flows and understanding transition to turbulence.
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The Orr-Sommerfeld equation governs the wall-normal velocity of normal mode disturbances to an incompressible viscous parallel shear flow. Here , , and is the Reynolds number. It follows by eliminating pressure from the linearized Navier-Stokes equations. At a rigid no-slip wall, . In the inviscid limit it reduces to the Rayleigh equation for inviscid shear flow.