Integrated potential

Complex potential is a function of two variables $x$ ${\ displaystyle x}$ $x$ and $y$ ${\ displaystyle y}$ $y$ , which is used in hydrodynamics to describe the plane stationary irrotational motion of an incompressible fluid of the form $f(z)=u(x,y)+iv(x,y)$ ${\ displaystyle f (z) = u (x, y) + iv (x, y)}$ ${\ displaystyle f (z) = u (x, y) + iv (x, y)}$ . The real part $u(x,y)$ ${\ displaystyle u (x, y)}$ $u (x, y)$ called a potential function, the imaginary part $v(x,y)$ ${\ displaystyle v (x, y)}$ ${\ displaystyle v (x, y)}$ called the current function. Lines $u(x,y)={\rm {const}}$ ${\ displaystyle u (x, y) = {\ rm {const}}}$ ${\ displaystyle u (x, y) = {\ rm {const}}}$ are called equipotential lines, or level lines. Lines $v(x,y)={\rm {const}}$ ${\ displaystyle v (x, y) = {\ rm {const}}}$ ${\ displaystyle v (x, y) = {\ rm {const}}}$ are called streamlines. Each fluid particle moves along a streamline. The value of the fluid flow rate is equal to the modulus of the derivative of the complex potential $V=|f'(z)|$ ${\ displaystyle V = | f '(z) |}$ ${\ displaystyle V = | f '(z) |}$ . The direction of fluid flow rate forms with a positive axis direction $O x$ ${\ displaystyle Ox}$ $Ox$ angle $\varphi =-\arg f'(z)$ ${\ displaystyle \ varphi = - \ arg f '(z)}$ ${\ displaystyle \ varphi = - \ arg f '(z)}$ . Using the Cauchy-Riemann conditions , it is possible to reconstruct the form of the complex potential from the equation of equipotential lines.

Literature

Krasnov M.L., Kiselev A.I., Makarenko G.I. Functions of a complex variable. Operational calculus. Theory of sustainability .. - 2nd. - M. , 1981.

Integrated potential

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Literature

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