fluid dynamics In physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids – liquids and gases. It has several subdisciplines, including (the study of air and other gases in motion ...

, the enstrophy

\mathcal

can be interpreted as another type of potential density; or, more concretely, the quantity directly related to the

kinetic energy In physics, the kinetic energy of an object is the form of energy that it possesses due to its motion. In classical mechanics, the kinetic energy of a non-rotating object of mass ''m'' traveling at a speed ''v'' is \fracmv^2.Resnick, Rober ...

in the flow model that corresponds to

dissipation In thermodynamics, dissipation is the result of an irreversible process that affects a thermodynamic system. In a dissipative process, energy ( internal, bulk flow kinetic, or system potential) transforms from an initial form to a final form, wh ...

effects in the fluid. It is particularly useful in the study of turbulent flows, and is often identified in the study of thrusters as well as in

combustion Combustion, or burning, is a high-temperature exothermic redox chemical reaction between a fuel (the reductant) and an oxidant, usually atmospheric oxygen, that produces oxidized, often gaseous products, in a mixture termed as smoke. Combustion ...

theory and

meteorology Meteorology is the scientific study of the Earth's atmosphere and short-term atmospheric phenomena (i.e. weather), with a focus on weather forecasting. It has applications in the military, aviation, energy production, transport, agricultur ...

. Given a domain

\Omega \subseteq \R^n

and a once-weakly differentiable vector field

u \in H^1(\R^n)^n

which represents a fluid flow, such as a solution to the Navier-Stokes equations, its enstrophy is given by:where

, \nabla \mathbf, ^2 = \sum_^n \left,  \partial_i u^j \^2

. This quantity is the same as the squared

seminorm In mathematics, particularly in functional analysis, a seminorm is like a Norm (mathematics), norm but need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some Absorbing ...

, \mathbf, _^2

of the solution in the

Sobolev space In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...

H^1(\Omega)^n

Incompressible flow

In the case that the flow is

incompressible Incompressible may refer to: * Incompressible flow, in fluid mechanics * incompressible vector field, in mathematics * Incompressible surface, in mathematics * Incompressible string, in computing {{Disambig ...

, or equivalently that

\nabla \cdot \mathbf = 0

, the enstrophy can be described as the integral of the square of the

vorticity In continuum mechanics, vorticity is a pseudovector (or axial vector) field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point an ...

\mathbf

:Doering, C. R. and Gibbon, J. D. (1995). ''Applied Analysis of the Navier-Stokes Equations'', p. 11, Cambridge University Press, Cambridge. . :

\mathcal(\boldsymbol \omega) \equiv \int_\Omega , \boldsymbol \omega, ^2 \,d \mathbf x

or, in terms of the

flow velocity In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...

: :

\mathcal(\mathbf) \equiv \int_\Omega , \nabla \times \mathbf u, ^2 \,d \mathbf x

In the context of the incompressible Navier-Stokes equations, enstrophy appears in the following useful result: :

\frac \left( \frac \int_\Omega , \mathbf, ^2 \right) = - \nu \mathcal(\mathbf)

The quantity in parentheses on the left is the kinetic energy in the flow, so the result says that energy declines proportional to the

kinematic viscosity Viscosity is a measure of a fluid's rate-dependent drag (physics), resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for e ...

\nu

times the enstrophy.

Incompressible flow

See also

References

Further reading