# Lamb vector

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{{Short description|Mathematical object used in fluid dynamics}}
In [fluid dynamics](/source/fluid_dynamics), '''Lamb vector''' is the [cross product](/source/cross_product) of [vorticity](/source/vorticity) vector and [velocity](/source/velocity) vector of the flow field, named after the physicist [Horace Lamb](/source/Horace_Lamb).<ref>Lamb, H. (1932). Hydrodynamics, Cambridge Univ. Press,, 134–139.</ref><ref>Truesdell, C. (1954). The kinematics of vorticity (Vol. 954). Bloomington: Indiana University Press.</ref> The Lamb vector is defined as

:<math>\mathbf l = \mathbf{u} \times \boldsymbol{\omega}</math>

where <math>\mathbf{u}</math> is the velocity field and <math>\boldsymbol{\omega}=\nabla\times\mathbf{u}</math> is the vorticity field of the flow. It appears in the [Navier–Stokes equations](/source/Navier%E2%80%93Stokes_equations) through the [material derivative](/source/material_derivative) term, specifically via convective acceleration term,

:<math> \mathbf u\cdot \nabla\mathbf u =  \frac{1}{2}\nabla\mathbf u^2 - \mathbf{u}\times\boldsymbol{\omega} =  \frac{1}{2}\nabla\mathbf u^2 - \mathbf l</math>

In irrotational flows, the Lamb vector is zero, so does in [Beltrami flow](/source/Beltrami_flow)s. The concept of Lamb vector is widely used in turbulent flows. The Lamb vector is analogous to [electric field](/source/electric_field), when the Navier–Stokes equation is compared with [Maxwell's equations](/source/Maxwell's_equations).

==Gromeka–Lamb equation==
The [Euler equations](/source/Euler_equations_(fluid_dynamics)) written in terms of the Lamb vector is referred to as the Gromeka–Lamb equation, named after [Ippolit S. Gromeka](/source/Ippolit_S._Gromeka) and Horace Lamb.<ref>Majdalani, J. (2022). On the generalized Beltramian motion of the bidirectional vortex in a conical cyclone. Physics of Fluids, 34(3).</ref> This is given by

:<math>\nabla H = \mathbf l.</math>

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

:<math>\nabla\cdot\mathbf l = \mathbf u\cdot\nabla\times\boldsymbol\omega -\boldsymbol\omega\cdot H.</math>

At the same time, the divergence can also be obtained  from Navier–Stokes equation by taking its divergence. In particular, for incompressible flow, where <math>\nabla\cdot\mathbf u=0</math>, with body forces given by <math>-\nabla U</math>, the Lamb vector divergence reduces to

:<math>\nabla\cdot\mathbf l=-\nabla^2 H,</math>

where

:<math>H= \frac{p}{\rho} + \frac{1}{2}\mathbf u^2+ U.</math>

In regions where <math>\nabla\cdot\mathbf l\geq 0</math>, there is tendency for <math>\Phi</math>{{what|date=March 2025}} to accumulate there and vice versa.

==References==
{{reflist|30em}}

Category:Fluid dynamics
Category:Vector calculus

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