Application of Analytic Function

N. B. Vyas

Department of Mathematics,Atmiya Institute of Tech. and Science,

Rajkot (Guj.)

N.B.V yas − Department of Mathematics, AITS − Rajkot

Fluid Flow

For a given flow of an incompressible fluid there exists ananalytic function

F (z) = φ(x, y) + iψ(x, y)

F(z) is called Complex Potential of the flow.

ψ is called the Stream Function.

The function φ is called the Velociy Potential.

The velocity of the fluid is given by

V = V1 + iV2 = F ′(z)

φ(x, y) = Const is called Equipotential Lines.

Points were V is zero are called Stagnation Points of flow.

N.B.V yas − Department of Mathematics, AITS − Rajkot

Electrostatic Fields

The force of attraction or repulsion between charged particleis governed by Coloumb’s law.

This force can be expressed as the gradient of a function φ,called the Electrostatic Potential

The electrostatic potential satisfies Laplace’s equation

O2φ =∂2φ

∂x2+∂2φ

∂y2= 0

The surfaces φ = Const. are called EquipotentialSurfaces.

This φ will be the real part of some analytic functionF (z) = φ(x, y) + iψ(x, y)

N.B.V yas − Department of Mathematics, AITS − Rajkot

Heat Flow Problems

Laplace’s equation governs heat flow problems that aresteady, i.e. time - independent.Heat conduction in a body of Homogeneous material is givenby the heat equation

∂T

∂t= c2O2T

Where function T is temperature, t is time and c2 is apositive constant.Here the problem is steady,

∂T

∂t= 0

Heat equation reduces to

∂2T

∂x2+∂2T

∂y2= 0

N.B.V yas − Department of Mathematics, AITS − Rajkot

Heat Flow Problems

T (x, y) is called the Heat Potential.

It is the real part of Complex Heat Potential i.e.

F (z) = T (x, y) + iψ(x, y)

T (x, y) = Const. are called Isotherms

ψ(x, y) = Const. is heat flow lines.

N.B.V yas − Department of Mathematics, AITS − Rajkot