  In this paper we consider smooth solutions of the Navier--Stokes equations
with a linear dependence on the spatial variable. We reduce the evolution of
these solutions to a matrix ODE, and show that there are such solutions that
blowup in finite-time. Note that because these solutions have linear growth at
infinity, this blowup is not a counterexample disproving the global regularity
of strong solutions of the Navier--Stokes equations, as strong solutions must
have sufficient decay at infinity. This paper does not resolve the Millennium
Problem. Nonetheless, these solutions do exhibit several properties that are
closely related to the problem of blowup for strong solutions of Navier--Stokes
equations, including the presence of unbounded planar stretching, and the
alignment of the vorticity with the middle eigenvector of the strain matrix.
