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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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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