Helmholtzian

Helmholtzian - in hydrodynamics a vector (converting a vector field into a vector field) linear differential operator defined for a moving liquid or gas. Helmholtzian is indicated by $\ \operatorname {helm}$ ${\ displaystyle \ \ operatorname {helm}}$ ${\ displaystyle \ \ operatorname {helm}}$ . Helmholtian over a vector field $\mathbf {A}$ ${\ displaystyle \ mathbf {A}}$ ${\ displaystyle \ mathbf {A}}$ defined by the formula

\ \operatorname {helm} \mathbf {A} \equiv {\frac {\partial \mathbf {A} }{\partial t}}+(v\cdot \nabla )\mathbf {A} -(\mathbf {A} \cdot \nabla )v+\mathbf {A} (\nabla \cdot v)

{\ displaystyle \ \ operatorname {helm} \ mathbf {A} \ equiv {\ frac {\ partial \ mathbf {A}} {\ partial t}} + (v \ cdot \ nabla) \ mathbf {A} - (\ mathbf {A} \ cdot \ nabla) v + \ mathbf {A} (\ nabla \ cdot v)}

{\ displaystyle \ \ operatorname {helm} \ mathbf {A} \ equiv {\ frac {\ partial \ mathbf {A}} {\ partial t}} + (v \ cdot \ nabla) \ mathbf {A} - (\ mathbf {A} \ cdot \ nabla) v + \ mathbf {A} (\ nabla \ cdot v)}

,

Where $\mathbf {v}$ ${\ displaystyle \ mathbf {v}}$ ${\ displaystyle \ mathbf {v}}$ Is the vector of the flow rate of a liquid or gas, $\nabla$ ${\ displaystyle \ nabla}$ $\ nabla$ - operator nabla .

The operator helmholtzian was introduced into everyday life by Alexander Alexandrovich Fridman , and named after Hermann Helmholtz .

The hydrodynamic meaning of helmholtzian is that equality $\ \operatorname {helm} \mathbf {A} =0$ ${\ displaystyle \ \ operatorname {helm} \ mathbf {A} = 0}$ ${\ displaystyle \ \ operatorname {helm} \ mathbf {A} = 0}$ means “frozen” vector field $\ \mathbf {A}$ ${\ displaystyle \ \ mathbf {A}}$ ${\ displaystyle \ \ mathbf {A}}$ into a moving fluid, understood in the sense that each vector line of this field (that is, a line tangent to which at any point in it has a vector direction $\ \mathbf {A}$ ${\ displaystyle \ \ mathbf {A}}$ ${\ displaystyle \ \ mathbf {A}}$ at this point) is stored, that is, all the time it consists of the same particles of liquid, and the intensity of the vector tubes (whose walls consist of vector lines), i.e. flows $\ \int \limits _{\sigma }(\mathbf {A} \cdot d\mathbf {\sigma } )$ ${\ displaystyle \ \ int \ limits _ {\ sigma} (\ mathbf {A} \ cdot d \ mathbf {\ sigma})}$ ${\ displaystyle \ \ int \ limits _ {\ sigma} (\ mathbf {A} \ cdot d \ mathbf {\ sigma})}$ of vector $\ \mathbf {A}$ ${\ displaystyle \ \ mathbf {A}}$ ${\ displaystyle \ \ mathbf {A}}$ through any sections $\ \mathbf {\sigma }$ ${\ displaystyle \ \ mathbf {\ sigma}}$ ${\ displaystyle \ \ mathbf {\ sigma}}$ of these tubes do not change with time ^[1] . Helmholtzian is used in geophysical hydrodynamics , magnetic hydrodynamics .

Notes

↑ Monin A.S. Theoretical Foundations of Geophysical Hydrodynamics. — L.: Gidrometeoizdat. — 1988. — P.17.

[1] Monin A.S. Theoretical Foundations of Geophysical Hydrodynamics. — L.: Gidrometeoizdat. — 1988. — P.17.

Helmholtzian

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