4.3 Fluid Dynamics
105
4.3.12 Vorticity
The very names divergence and curl used in mathematics originated from fluid flow
relations. Divergence of fluid in a small volume will exist if ∇ · v is not zero within
that volume. If a fluid has a curl within a small volume, then there is a ‘vortex’ in
the fluid creating circular motion within that volume. With Stokes’ integral theorem,
∇ × v · dS =
v · dr, we know that vorticity exists in a fluid wherever there is a
net circular motion around a point in the fluid. The vector field = ∇ × v defines
the ‘vorticity’ of the fluid near the point where the curl is calculated. Fluid motion
without vorticity is called ‘irrotational’.
We met the curl of material displacements before when we considered displacements which represent rotations rather than strain. (See Chap. 3, footnote 32.)
Dividing the relation δθ = (1/2)∇ × ξ by an interval of time, we find the local
angular rotation of the fluid is given by
ω =
1
2
∇ × v .
(4.55)
From the identity
(v · ∇) v =
1
2
∇v
2
− v × (∇ × v) ,
(4.56)
the curl of the terms in Navier–Stokes equation gives, for an incompressible fluid,
ρ
∂
∂t
− ρ∇ × (v × ) = −η∇
2 .
(4.57)
Equation (4.57), together with the incompressibility condition ∇ · v = 0 and
boundary conditions, mathematically determine the velocity field of the fluid. We
see that once the initial conditions are fixed, the later velocities of the fluid elements
do not depend directly on the fluid pressure or the presence of gravity.
Moreover, if the fluid had no viscosity, and the initial vorticity vanished
everywhere in a region, then ∂ /∂t would vanish, so it must vanish at any time
later. A non-viscous fluid remains irrotational if it starts that way. Vortices cannot
be created in such a fluid through fluid element interactions. In contrast, bumble
bees can create useful vortices in air because the air is viscous.
For a fluid under irrotational conditions, the velocity can always be expressed
as the divergence of some scalar function φ. The solutions for incompressible and
irrotational fluids can then be found with standard techniques for solving Laplace’s
equation ∇ 2 φ = 0. In fact, since static electric potentials also satisfy Laplace’s
equation, if you know the potential around positive and negative charges, you also
know how fluid will flow in analogous geometries. Mathematically, a point positive
charge becomes a ‘source’ of fluid. A point negative charge becomes a ‘sink’ of
fluid. The electric field lines become fluid flow lines.
105
4.3.12 Vorticity
The very names divergence and curl used in mathematics originated from fluid flow
relations. Divergence of fluid in a small volume will exist if ∇ · v is not zero within
that volume. If a fluid has a curl within a small volume, then there is a ‘vortex’ in
the fluid creating circular motion within that volume. With Stokes’ integral theorem,
∇ × v · dS =
v · dr, we know that vorticity exists in a fluid wherever there is a
net circular motion around a point in the fluid. The vector field = ∇ × v defines
the ‘vorticity’ of the fluid near the point where the curl is calculated. Fluid motion
without vorticity is called ‘irrotational’.
We met the curl of material displacements before when we considered displacements which represent rotations rather than strain. (See Chap. 3, footnote 32.)
Dividing the relation δθ = (1/2)∇ × ξ by an interval of time, we find the local
angular rotation of the fluid is given by
ω =
1
2
∇ × v .
(4.55)
From the identity
(v · ∇) v =
1
2
∇v
2
− v × (∇ × v) ,
(4.56)
the curl of the terms in Navier–Stokes equation gives, for an incompressible fluid,
ρ
∂
∂t
− ρ∇ × (v × ) = −η∇
2 .
(4.57)
Equation (4.57), together with the incompressibility condition ∇ · v = 0 and
boundary conditions, mathematically determine the velocity field of the fluid. We
see that once the initial conditions are fixed, the later velocities of the fluid elements
do not depend directly on the fluid pressure or the presence of gravity.
Moreover, if the fluid had no viscosity, and the initial vorticity vanished
everywhere in a region, then ∂ /∂t would vanish, so it must vanish at any time
later. A non-viscous fluid remains irrotational if it starts that way. Vortices cannot
be created in such a fluid through fluid element interactions. In contrast, bumble
bees can create useful vortices in air because the air is viscous.
For a fluid under irrotational conditions, the velocity can always be expressed
as the divergence of some scalar function φ. The solutions for incompressible and
irrotational fluids can then be found with standard techniques for solving Laplace’s
equation ∇ 2 φ = 0. In fact, since static electric potentials also satisfy Laplace’s
equation, if you know the potential around positive and negative charges, you also
know how fluid will flow in analogous geometries. Mathematically, a point positive
charge becomes a ‘source’ of fluid. A point negative charge becomes a ‘sink’ of
fluid. The electric field lines become fluid flow lines.