Application of analytic function
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This document discusses the application of analytic functions to fluid flow, electrostatic fields, and heat flow problems. It explains that for incompressible fluid flow, the complex potential F(z) describes the flow, with its real part giving the velocity potential and imaginary part the stream function. It also describes how the electrostatic potential satisfies Laplace's equation and can be written as the real part of a complex potential. Finally, it explains that steady heat conduction problems are governed by Laplace's equation and the heat potential is the real part of a complex heat potential, with constant values representing isotherms and heat flow lines.
Application of analytic function
- 1. Application of Analytic Function N. B. Vyas Department of Mathematics, Atmiya Institute of Tech. and Science, Rajkot (Guj.) N.B.V yas − Department of M athematics, AIT S − Rajkot
- 2. Fluid Flow For a given flow of an incompressible fluid there exists an analytic 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 M athematics, AIT S − Rajkot
- 3. Electrostatic Fields The force of attraction or repulsion between charged particle is 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 2 ∂ 2φ ∂ 2φ φ= + =0 ∂x2 ∂y 2 The surfaces φ = Const. are called Equipotential Surfaces. This φ will be the real part of some analytic function F (z) = φ(x, y) + iψ(x, y) N.B.V yas − Department of M athematics, AIT S − Rajkot
- 4. Heat Flow Problems Laplace’s equation governs heat flow problems that are steady, i.e. time - independent. Heat conduction in a body of Homogeneous material is given by the heat equation ∂T = c2 2 T ∂t Where function T is temperature, t is time and c2 is a positive constant. Here the problem is steady, ∂T =0 ∂t Heat equation reduces to ∂ 2T ∂ 2T + =0 ∂x2 ∂y 2 N.B.V yas − Department of M athematics, AIT S − Rajkot
- 5. 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 M athematics, AIT S − Rajkot