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Application of Analyltic function ( in short )
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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