# What causes fluid rotation in the Navier--Stokes equation?

It is known that rotation in the flow results from the viscous terms in the Navier-Stokes (N-S) equation.

However, when deriving the N-S equation from the general principle of linear momentum in Continuum Mechanics, we use the constitutive relation for isotropic Newtonian fluids which states that the deviatoric part of the stress tensor is proportional to the deviatoric part of the rate of deformation tensor. Since the rate of deformation tensor is the symmetric part of the velocity gradient tensor and the vorticity tensor is the skew-symmetric part, how is the vorticity generated when we assume that there is no vorticity produced by the stress?

• – Qmechanic Aug 31 '17 at 13:51

## 1 Answer

Just because the stress tensor is determined by the (symmetric) rate of deformation tensor does not mean that the flow cannot generate vorticity. Certainly, at a no-slip boundary, the vorticity at the wall will not be equal to zero. For example, in 2D rectilinear flow, the vorticity at the wall will be equal to the second partial derivative of the stream function with respect distance in the normal direction, subject to the constant that the first partial derivative of the stream function in the normal direction is equal to zero.

Or better yet. Consider the case of simple shear flow between two parallel plates. What are the components of the velocity gradient tensor, the rate of deformation tensor, the vorticity tensor, and the stress tensor?