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 ...

, Lamb vector is the

cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...

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 ...

vector and

velocity Velocity is a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector (geometry), vector Physical q ...

vector of the flow field, named after the physicist

Horace Lamb Sir Horace Lamb (27 November 1849 – 4 December 1934R. B. Potts,, '' Australian Dictionary of Biography'', Volume 5, MUP, 1974, pp 54–55. Retrieved 5 Sep 2009) was a British applied mathematician and author of several influential texts on ...

. The Lamb vector is defined as :

\mathbf l = \mathbf \times \boldsymbol

where

\mathbf

is the velocity field and

\boldsymbol=\nabla\times\mathbf

is the vorticity field of the flow. It appears in the

Navier–Stokes equations The Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances. They were named after French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician Georg ...

through the

material derivative In continuum mechanics, the material derivative describes the time rate of change of some physical quantity (like heat or momentum) of a material element that is subjected to a space-and-time-dependent macroscopic velocity field. The material de ...

term, specifically via convective acceleration term, :

\mathbf u\cdot \nabla\mathbf u =  \frac\nabla\mathbf u^2 - \mathbf\times\boldsymbol =  \frac\nabla\mathbf u^2 - \mathbf l

In irrotational flows, the Lamb vector is zero, so does in

Beltrami flow In fluid dynamics, Beltrami flows are flows in which the vorticity vector \boldsymbol and the velocity vector \mathbf are parallel to each other. In other words, Beltrami flow is a flow in which the Lamb vector is zero. It is named after the Italia ...

s. The concept of Lamb vector is widely used in turbulent flows. The Lamb vector is analogous to

electric field An electric field (sometimes called E-field) is a field (physics), physical field that surrounds electrically charged particles such as electrons. In classical electromagnetism, the electric field of a single charge (or group of charges) descri ...

, when the Navier–Stokes equation is compared with

Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, Electrical network, electr ...

Gromeka–Lamb equation

The

Euler equations In mathematics and physics, many topics are eponym, named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include their own unique function, e ...

written in terms of the Lamb vector is referred to as the Gromeka–Lamb equation, named after Ippolit S. Gromeka and Horace Lamb.Majdalani, J. (2022). On the generalized Beltramian motion of the bidirectional vortex in a conical cyclone. Physics of Fluids, 34(3). This is given by :

\nabla H = \mathbf l.

Properties

The divergence of the lamb vector can be derived from vector identities, :

\nabla\cdot\mathbf l = \mathbf u\cdot\nabla\times\boldsymbol\omega -\boldsymbol\omega\cdot H.

At the same time, the divergence can also be obtained from Navier–Stokes equation by taking its divergence. In particular, for incompressible flow, where

\nabla\cdot\mathbf u=0

, with body forces given by

-\nabla U

, the Lamb vector divergence reduces to :

\nabla\cdot\mathbf l=-\nabla^2 H,

where :

H= \frac + \frac\mathbf u^2+ U.

In regions where

\nabla\cdot\mathbf l\geq 0

, there is tendency for

\Phi

to accumulate there and vice versa.

References

{{reflist, 30em Fluid dynamics Vector calculus