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v ξ - i v η = d w d ς = d w d z d z d ς = v x - i v y d z d ς

This shows that the magnitude of the velocity is changed, in the transformation from the z -plane to the ζ -plane, by the reciprocal of the factor by which linear dimensions of small figures are changed. Thus the kinetic energy of the fluid contained within a closed curve in the z -plane is equal to the kinetic energy of the corresponding flow in the region enclosed by the corresponding in the ζ -plane.

Flow around elliptic cylinder (Batchelor, 1967). The transformation of the region outside of an ellipse in the z -plane into the region outside a circle in the ζ -plane is given by

z = ς + λ 2 ς ς = 1 2 z + 1 2 z 2 - 4 λ 2 1 / 2

where λ is a real constant so that

x = ξ 1 + λ 2 ς 2 , y = η 1 - λ 2 ς 2

This converts a circle of radius c with center at the origin in the - plane into the ellipse

x 2 a 2 + y 2 b 2 = 1

in the z -plane, where

λ = 1 2 a 2 - b 2 1 / 2

If the ellipse is mapped into a circle in the ζ -plane, it is convenient to use polar coordinates ( r , θ ) , especially since the boundary corresponds to a constant radius. The radius that maps to the elliptical boundary is ( ellipse.m in the complex directory)

r o = 1 2 log a + b a - b

The transformation from the polar coordinates to the z - plane is defined by

z = 2 λ cosh ω where ω = r + i θ
Transformation of cylindrical polar coordinates into orthogonal, elliptical coordinates

The polar coordinates ( r , θ ) , transform to an orthogonal set of curves which are confocal ellipses and conjugate hyperbolae.

This transformation can be substituted into the complex potential expression for the flow of a fluid past a circular cylinder.

w = - 1 2 a + b U - i V e ω - r o + U + i V e r o - ω

It is convenient to write - for the angle which the direction of motion of the flow at infinity makes with the x - axis so that

U + i V = U 2 + V 2 1 / 2 e - i α

The complex potential now becomes

w = - U 2 + V 2 1 / 2 a + b cosh ω - r o + i α

The corresponding velocity potential and stream function are

ϕ = - U 2 + V 2 1 / 2 a + b cosh r - r o cos θ + α ψ = - U 2 + V 2 1 / 2 a + b sinh r - r o sin θ + α

The velocity potentials and streamlines are illustrated below for flow past an elliptical cylinder ( fellipse.m in the complex directory). Note the stagnation streamlines on either side of the body. These two stagnation points are regions of maximum pressure and result in a torque on the body. Which way will it rotate?

Flow past an ellipse of an inviscid fluid that is in steady translation at infinity.

Pressure distribution. When an object is in a flow field, one may wish to determine the force exerted by the fluid on the object, or the 'drag' on the object. Since the flow field discussed here has assumed an inviscid fluid, it is not possible to determine the viscous drag or skin friction directly from the flow field. It is possible to determine the 'form drag' from the normal stress or pressure distribution around the object. However, one must be critical to determine if the calculated flow field is physically realistic or if some important phenomena such as boundary layer separation may occur but is not allowed in the complex potential solution.

The Bernoulli theorems give the relation between the magnitude of velocity and pressure. We have assumed irrotational, incompressible flow. If in addition we assume the body force can be neglected, then the quantity, H , must be constant along a streamline.

Questions & Answers

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The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31
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Q2 x+(x+2)+(x+4)=60 3x+6=60 3x+6-6=60-6 3x=54 3x/3=54/3 x=18 :. The numbers are 18,20 and 22
Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
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Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
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x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
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Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
Need help solving this problem (2/7)^-2
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A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
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Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
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Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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Source:  OpenStax, Transport phenomena. OpenStax CNX. May 24, 2010 Download for free at http://cnx.org/content/col11205/1.1
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