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This paper concerns the large Reynold number limits and asymptotic behaviorsof solutions to the 2D steady Navier-Stokes equations in an infinitely longconvergent channel. It is shown that for a general convergent infinitely longnozzle whose boundary curves satisfy curvature-decreasing and any given finitenegative mass flux, the Prandtl's viscous boundary layer theory holds in thesense that there exists a Navier-Stokes flow with no-slip boundary conditionfor small viscosity, which is approximated uniformly by the superposition of anEuler flow and a Prandtl flow. Moreover, the singular asymptotic behaviors ofthe solution to the Navier-Stokes equations near the vertex of the nozzle andat infinity are determined by the given mass flux, which is also important forthe constructions of the Prandtl approximation solution due to the possiblesingularities at the vertex and non-compactness of the nozzle. One of the keyingredients in our analysis is that the curvature-decreasing condition onboundary curves of the convergent nozzle ensures that the limiting inviscidflow is pressure favourable and plays crucial roles in both the Prandtlexpansion and the stability analysis.

Prandtl Boundary Layers in An Infinitely Long Convergent Channel