A well known limitation with stretched vortex solutions of the 3D Navier–Stokes (and Euler) equations, such as those of Burgers type, is that they possess uni-directional vorticity which is stretched by a strain field that is decoupled from them. It is shown here that these drawbacks can be partially circumvented by considering a class of velocity fields of the type u =( u 1( x, y, t), u 2( x, y, t), γ( x, y, t) z+ W( x, y, t)) where u 1, u 2, γ and W are functions of x, y and t but not z. It turns out that the equations for the third component of vorticity ω 3 and W decouple. More specifically, solutions of Burgers type can be constructed by introducing a strain field into u such that u= −(γ/2)x−(γ/2)y,γz + −ψ y,ψ x,W . The strain rate, γ( t), is solely a function of time and is related to the pressure via a Riccati equation γ ̇ +γ 2+p zz(t)=0 . A constraint on p zz ( t) is that it must be spatially uniform. The decoupling of ω 3 and W allows the equation for ω 3 to be mapped to the usual general 2D problem through the use of Lundgren’s transformation, while that for W can be mapped to the equation of a 2D passive scalar. When ω 3 stretches then W compresses and vice versa. Various solutions for W are discussed and some 2 π-periodic θ-dependent solutions for W are presented which take the form of a convergent power series in a similarity variable. Hence the vorticity ω= r −1W θ,−W r,ω 3 has nonzero components in the azimuthal and radial as well as the axial directions. For the Euler problem, the equation for W can sustain a vortex sheet type of solution where jumps in W occur when θ passes through multiples of 2 π.