In this paper we extend the plane blow-up results of Grundy & McLaughlin (1
997) to the three-dimensional Navier-Stokes equations. Using a solution str
ucture originally due to Lin we first provide numerical evidence for the ex
istence of blow-up solutions on -infinity < x, z < infinity, 0 less than or
equal to y less than or equal to 1 with boundary conditions on y = 0 and y
= 1 involving derivatives of the velocity components. The formulation enab
les us to consider plane and radial flow as special cases. Various features
of the computations are isolated and are used to construct a formal asympt
otic solution close to blow-up. We show that the numerical and asymptotic a
nalyses provide a mutually consistent global picture which supports the con
clusion that, for the family of problems we consider here, blow-up in fact
can take place in three dimensions but at an inverse linear rate rather tha
n the faster inverse square of the plane case.