A new family of exact solutions to the unsteady Navier–Stokes equations using canonical transformation with complex coefficients

This paper aims to show a new family of exact solutions to the unsteady Navier–Stokes equations by using the canonical transformation with complex coefficients. This transformation transforms the system of non-linear partial differential equations to a linear system of partial differential equations with less independent variables. The present results demonstrate that the general real solutions may involve either exp, sin, cos, sinh or cosh under certain conditions depending on the type of the constants in the canonical transformation.