Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Question

1.

1. Consider two dimensional irrotational motion in a plane parallel to xy - plane. The velocity v of fluid can be expressed as, Vại + vyj V = Since the motion is irrotational, a scalar function ø(x, y) gives the velocity components. V= φ(7, ) - +j = 1 Əx ду dy On comparing (1) and (2), we get Vr Vy dy The scalar function (x, y) which gives the velocity component is called the velocity potential function. As the fluid is incompressible, Vv = 0 → dy Vgi + Vyj dy dx Putting the values of and Vy from (3) and (4), we get Vx dy? This is Laplace equation. The function o is harmonic and it is a real part of analytic function f(2) = ¢(x, y) + inp (x, y) By using (3) it can be shown that, dy Vy ie Vx dx dy Here the resultant velocity v? + v? of the fluid is along the tangent to the curve p(x, y) = C' Such curves are known as stream lines and (x, y) is known as stream func- tion. The curves represented by ø(x, y) As ø(x, y) and Þ(x, y) are conjugates of analytic function f(z), the equipoten- tial lines and and stream lines intersects each other orthogonally. Therefore, = c are called eqipotential lines. f'(2) = vn – ivy The magnitude of the resultant velocity is /v? + v3. The function f(2) which V represents the flow pattern is called complex potential. (а) If w %3 p = x² – y? + O + ip represents the complex potential of a flow and determine the function o. x2 + y2' (b) Find the equation for stream lines and equipotential lines represented by f(z) = az², where a is a constant and z = x + iy. (c) For the function in (b) show that the speed is everywhere proportional to the distance from the origin. ||
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Transcribed Image Text:1. Consider two dimensional irrotational motion in a plane parallel to xy - plane. The velocity v of fluid can be expressed as, Vại + vyj V = Since the motion is irrotational, a scalar function ø(x, y) gives the velocity components. V= φ(7, ) - +j = 1 Əx ду dy On comparing (1) and (2), we get Vr Vy dy The scalar function (x, y) which gives the velocity component is called the velocity potential function. As the fluid is incompressible, Vv = 0 → dy Vgi + Vyj dy dx Putting the values of and Vy from (3) and (4), we get Vx dy? This is Laplace equation. The function o is harmonic and it is a real part of analytic function f(2) = ¢(x, y) + inp (x, y) By using (3) it can be shown that, dy Vy ie Vx dx dy Here the resultant velocity v? + v? of the fluid is along the tangent to the curve p(x, y) = C' Such curves are known as stream lines and (x, y) is known as stream func- tion. The curves represented by ø(x, y) As ø(x, y) and Þ(x, y) are conjugates of analytic function f(z), the equipoten- tial lines and and stream lines intersects each other orthogonally. Therefore, = c are called eqipotential lines. f'(2) = vn – ivy The magnitude of the resultant velocity is /v? + v3. The function f(2) which V represents the flow pattern is called complex potential. (а) If w %3 p = x² – y? + O + ip represents the complex potential of a flow and determine the function o. x2 + y2' (b) Find the equation for stream lines and equipotential lines represented by f(z) = az², where a is a constant and z = x + iy. (c) For the function in (b) show that the speed is everywhere proportional to the distance from the origin. ||
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