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1. The theory of complex variable comes in naturally in the study of fluid phenomena.Let us define, w = 6 + ių (1) with o velocity potential and v stream function of a two dimensional fluid flow. w is called the complex potential function.Then w turns out to be analytic because the Cauchy-Riemann conditions, dy' dy 1 are exactly the natural flow conditions that have to be satisfied. Suppose, velocity potential for a two dimensional fluid flow is given by the function Ó(x, y) = - 2.ry.Find out the stream function.Also find the complex potential 1² + y? function for the fluid flow. ||

1. The theory of complex variable comes in naturally in the study of fluid phenomena.Let us define, w = 6 + ių (1) with o velocity potential and v stream function of a two dimensional fluid flow. w is called the complex potential function.Then w turns out to be analytic because the Cauchy-Riemann conditions, dy' dy 1 are exactly the natural flow conditions that have to be satisfied. Suppose, velocity potential for a two dimensional fluid flow is given by the function Ó(x, y) = - 2.ry.Find out the stream function.Also find the complex potential 1² + y? function for the fluid flow. ||

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1. The theory of complex variable comes in naturally in the study of fluid phenomena.Let us define, w = 6 + ip (1) with ø velocity potential and v stream function of a two dimensional fluid flow. w is called the complex potential function.Then w turns out to be analytic because the Cauchy-Riemann conditions, (2) %3D dy' dy 1 are exactly the natural flow conditions that have to be satisfied. Suppose, velocity potential for a two dimensional fluid flow is given by the function $(x, y) = - 2ry.Find out the stream function.Also find the complex potential 1² + y? function for the fluid flow. 2. Evatuate the following Complex Integration: (a) Evaluate I (3ry + iy?) dz along the straight line joining z = i and z = 2 – i. (b) Evaluate cz? – 32 + 2 dz, where C is the circle |2 – 2| =!. 1 3. Expand the function f(2) : given by (a) 0 < |z – 2| < 2 (b) 0 < |z – 3| <1 (c) |z – 2| > 2. in a Laurent series valid for the annular region z(z – 2) a +x² 4. Use residues to evaluate the improper integral / dr; where a is a real constant. ||
Transcribed Image Text:1. The theory of complex variable comes in naturally in the study of fluid phenomena.Let us define, w = 6 + ip (1) with ø velocity potential and v stream function of a two dimensional fluid flow. w is called the complex potential function.Then w turns out to be analytic because the Cauchy-Riemann conditions, (2) %3D dy' dy 1 are exactly the natural flow conditions that have to be satisfied. Suppose, velocity potential for a two dimensional fluid flow is given by the function $(x, y) = - 2ry.Find out the stream function.Also find the complex potential 1² + y? function for the fluid flow. 2. Evatuate the following Complex Integration: (a) Evaluate I (3ry + iy?) dz along the straight line joining z = i and z = 2 – i. (b) Evaluate cz? – 32 + 2 dz, where C is the circle |2 – 2| =!. 1 3. Expand the function f(2) : given by (a) 0 < |z – 2| < 2 (b) 0 < |z – 3| <1 (c) |z – 2| > 2. in a Laurent series valid for the annular region z(z – 2) a +x² 4. Use residues to evaluate the improper integral / dr; where a is a real constant. ||
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