The Korteweg–de Vries equation modified by viscosity for waves in a two-layer fluid in a channel of arbitrary cross section

Using a two-layer fluid model, Korteweg–de Vries (KdV) equations modified by viscosity are derived that describe weakly nonlinear long waves propagating along a channel of uniform but arbitrary cross section. Equations are deduced for both surface waves and internal waves. The case of high Reynolds number is considered, and the method of matched asymptotic expansion is employed. The coefficients of the KdV equation, which depend on the geometry of the channel cross section, are determined exactly for a rectangular cross section. Some particular cases including the Boussinesq limit are considered.