Parallel shear flow - Knowlpedia

A planar velocity field called a parallel shear flow has the form

u(x_1,x_2,x_3)=(U(x_2),0,0),

where the velocity is parallel to a fixed direction and its speed varies transversely. It is incompressible, and $U'(x_2)$ measures its shear. Its vorticity is $(0,0,-U'(x_2))$ .

Advection and evolution

For this profile, $(u\cdot\nabla)u=0$ , since the field is independent of $x_1$ . Allowing $U=U(t,x_2)$ , the unforced Navier–Stokes equation with constant pressure reduces to $\partial_tU=\nu\partial_{x_2}^2U$ .

Terminology

Shear can also describe differential motion in more general geometries. In a rotating flow, variation of angular velocity with radius is differential rotation, and its shear expressions include the cylindrical geometry.

References

Lenya Ryzhik, Lecture Notes for Math 256B (2024), Sections 1–3.