1 The Vorticity equation • Use of the streamfunction allows us to automatically satisfy the continuity equation. Now we will try to transform the momentum equation @u 1 + (u · r)u = − rp + νr2u @t ρ into an equation for the vorticity !! = r × u. • Here are a few results that we will use in the derivation: 1 r(u · u) = (u · r)u + u × (r × u); (1) 2 r × rφ = 0; (2) and r × × · r − · r r · − r · (u !) = (! )u (u )! + u | {z !} ! | {z u}; (3) 0 0 where the last two terms vanish because r · ! = r · (r × u) = div curl u = 0 and r · u = 0: 2 • First, we use (1) in the momentum equation to obtain @u 1 1 + r(u · u) − u × (r × u) = − rp + νr2u | {z } @t 2 ! ρ i.e. @u 1 1 2 + r(u · u) − u × ! = − rp + νr u @t 2 ρ | {z } | {z } LHS RHS • Now take the curl of the LHS: @ 1 r×LHS = (r × u) + r × r(u · u) − r × (u × !) @t | {z } 2 | {z } | {z } ! 0 because of (2) see (3) i.e. @! r × LHS = − (! · r)u + (u · r)! @t • ...and the RHS: 1 r × RHS = − r × rp +νr2 (r × u) ρ | {z } | {z } 0 because of (2) ! • Now combine the remaining non-zero terms @! 2 + (u · r)! −(! · r)u = νr !: |@t {z } D!=Dt 3 • The resulting equation is the vorticity transport equation D! 2 = (! · r)u + νr ! (4) Dt which shows that the rate of change of the vorticity of ma- terial particles, D!=Dt, is controlled by `vortex stretching' (described by (! · r)u; this is a familiar result from inviscid fluid mechanics) and by diffusion (described by νr2!). The diffusion of vorticity only occurs in viscous flows. • For 3D flows, the first term on the RHS in (4) represents vortex stretching: velocity gradients lead to a change in the rate of rotation of material particles. • Note that for 2D flows, vortex stretching is absent since u = u(x; y) ex + v(x; y) ey and ! = !(x; y) ez and therefore (! · r)u = 0. • The vorticity transport equation provides an interesting inter- pretation of the kinematic viscosity ν: The kinematic viscosity is the diffusion coefficient for the diffusion of vorticity. • Many phenomena in viscous fluid mechanics can be interpreted in terms of the diffusion of vorticity but this is (unfortunately) beyond the scope of this course. 4 • For 2D flows, the vorticity transport equation D! 2 = νr ! Dt together with the equation for the vorticity in terms of the streamfunction ! = −∇2 and u = @ =@y and v = −@ =@x provide the streamfunction-vorticity formulation of the Navier- Stokes equations, which consists of only two PDEs for the scalars ! and rather than the three equations for u; v and p in the `primitive variable' form. • Scaling arguments show that in the limit of zero Reynolds number, only one fourth-order PDE for the streamfunction needs to be solved, namely the biharmonic equation r4 = 0; where @4 @4 @4 r4 = + 2 + : @x4 @x2@y2 @y4 • This can also be shown directly by taking the curl of the Stokes equations..