COMPLEX ANALYSIS: Mathematica 4.1 Notebooks
(c) John H. Mathews, and
Russell W. Howell, 2002
Complimentary software to accompany our textbook:

COMPLEX ANALYSIS: for Mathematics and Engineering, 
4th Edition,  2001,  ISBN: 0-7637-1425-9
Jones & Bartlett Publishers, Inc.  
40 Tall Pine Drive, Sudbury, MA 01776  
Tele.  (800) 832-0034, FAX:  (508)  443-8000  
E-mail:  mkt@jbpub.com  
Internet:  http://www.jbpub.com/  
This free software is compliments of the authors.
John H. Mathews,   mathews@fullerton.edu 
Russell W. Howell,  howell@westmont.edu

CHAPTER 10  APPLICATIONS of HARMONIC FUNCTIONS

a = A = b = B = c = k = R = T = u = U = v = V = w = W = x = X = y = Y = z = Z = 
                                        0; 
Clear[a, a0, a1, a2, a3, A, b, B, 
        CR, Δx, Δy, eq1, eq2, eqns, f, f1, F, F1, 
        F2, F3, F4, g, G, lines, n, P, Ψ, ϕ, Φ, 
        r, r0, solset, θ, t, T, T1, T2, Q, u, u0, U, U1, v, v0, V, V1, 
        w, w0, W, W1, x, x1, x2, x3, y, z, z1, z2, z3, 
        z4]; 
Needs["Graphics`ComplexMap`"]; 
Needs["Graphics`ParametricPlot3D`"]
Needs["Graphics`Colors`"];

Section 10.1	Preliminaries

Section 10.2	Invariance of Laplace's Equation and the Dirichlet Problem

Section 10.3	Poisson's Integral Formula for the Upper Half Plane

Section 10.4	Two Dimensional Mathematical Models

Section 10.5	Steady State Temperatures

Section 10.6	Two-Dimensional Electrostatics

Section 10.7	Two-Dimensional Fluid Flow

Section 10.8  The Joukowski Airfoil

The function  J (z) = z  + (1/z)  was studied by the Russian scientist Nikolai Egorovich Joukowsky.  It will be shown that the image of a circle passing through  (Subscript[z, 1]) = 1  and containing the point  (Subscript[z, 2]) = -1  is mapped onto a curve that is shaped like the cross section of an airplane wing.  We call this curve the Joukowski airfoil.  If the streamlines for a flow around the circle are known, then their images under the mapping  w = J (z)  will be streamlines for a flow around the Joukowski airfoil.

The Joukowski Airfoil, Page 445.  Find the image of a fluid flow past the Joukowski airfoil.

Solution for Joukowski Airfoil, Page 445.

Construct the four distinct inverse functions used in graphing a Joukowski airfoil.

Remove[w, z, Z]; 
Clear[F, F1, F2, F3, F4, dx, dy, r0]; 
Δx  =  -(1/10); 
Δy  =   (1/5); 
r0  =  (Sqrt[(((1 - Δx))^2) + (((0 - Δy))^2)]); 
(Subscript[Z, 1])  =   ((z  +  (Sqrt[(z^2)  -  4]))/2); 
(Subscript[Z, 2])  =   ((z  -  (Sqrt[(z^2)  -  4]))/2); 
(Subscript[Z, 3]) =  ((-z  -  (Sqrt[(z^2)  -  4]))/2); 
(Subscript[Z, 4])  =  ((-z  + (Sqrt[ (z^2)  -  4]))/2); 
w[
          z_]  =   r0 z  +  Δx  +  I Δy; 
F[z_]  =   z  +  (1/z); 
F1[z_]  =   F[w[(Subscript[Z, 1])]]; 
F2[z_]  =   F[w[(Subscript[Z, 2])]]; 
F3[z_]  =   F[w[(Subscript[Z, 3])]]; 
F4[z_]  =   F[w[(Subscript[Z, 4])]]; 
\(\(Print[\*""\!\(F\_1\)[z] = "", F1[z]];\)\ \)
\(\(Print[\*""\!\(F\_2\)[z] = "", F2[z]];\)\ \)
\(\(Print[\*""\!\(F\_3\)[z] = "", F3[z]];\)\ \)
\(\(Print[\*""\!\(F\_4\)[z] = "", 
        F4[z]];\)\ \)

Use Mathematica to graph the Joukowski airfoil.

gr1  =  
        CartesianMap[
          F1, {0.00001, 3.20001, 0.2}, {0.00001, 1.20001, 
            0.2}, 
AxesLabel -> {"u", "v"}, 
          PlotStyle -> {Magenta, Pink}, 
          DisplayFunction -> Identity];  
gr2  =  
        CartesianMap[
          F2, {-3.20001, -0.00001, 0.2}, { 0.00001, 1.20001, 0.2}, 
          AxesLabel -> {"u", "v"}, 
          PlotStyle -> {Magenta, Pink}, 
          DisplayFunction -> Identity];  
gr3  =  
        CartesianMap[F3, {0.00001, 3.20001, 0.2}, {0.00001, 1.20001, 0.2}, 
          AxesLabel -> {"u", "v"}, 
          PlotStyle -> {Magenta, Pink}, 
          DisplayFunction -> Identity];  
gr4  =  
        CartesianMap[
          F4, {-3.20001, -0.00001, 0.2}, { 0.00001, 1.20001, 0.2}, 
          AxesLabel -> {"u", "v"}, 
          PlotStyle -> {Magenta, Pink}, 
          DisplayFunction -> Identity];  
\(\(Show[gr1, gr2, gr3, 
        gr4, 
\ \ \ \ \ Ticks -> {Range[\(-2\), 2, 0.5], 
            Range[\(-2\), 2, 0.5]}, 
        DisplayFunction -> $DisplayFunction];\)\ \)
Print["A Joukowski airfoil formed with the mapping"]; 
Print["f[z] = ", F1[z]];

Library Research Experience for Undergraduates

Project I. Write a report on Joukowski transformation and airfoils.

1. Bera, Rajendra K., (1989), ''A New Look at Theodorsen's Method in Aerofoil Theory,'' Int. J. for Numerical Methods in Fluids, Vol. 9, pp. 251-262.

2. Bergman, Stefan, ''Two-Dimensional Subsonic Flows of a Compressible Fluid and Their Singularities,'' Trans. of the Am. Math. Soc., Vol. 62, No. 3. (Nov., 1947), Jstor.   

3. Bergman, Stefan, ''Two-Dimensional Transonic Flow Patterns,'' Stefan Bergman American J. of Math., Vol. 70, No. 4. (Oct., 1948), pp. 856-891, Jstor.   

4. Burington, R. S., ''On the Use of Conformal Mapping in Shaping Wing Profiles,'' Am. Math. M., Vol. 47, No. 6. (Jun. - Jul., 1940), pp. 362-373, Jstor.   

5. Cunningham, John, (1965), Complex Variable Methods in Science and Technology, London, N.Y., Van Nostrand.

6. D'Appen, Heinz, (1987), ''Wind-tunnel wall corrections on a two-dimensional plate by conformal mapping,'' AIAA J., V. 25, pp. 1527-3015.

7. Jeffrey, Alan, (1992), Complex Analysis and Applications, Boca Raton, FL., CRC Press.

8. Mancill, J. D. and Betty Thomas, ''On the Equation of Joukowski's Aerofoils (in Discussions and Notes),'' Am. Math. M., Vol. 53, No. 3. (Mar., 1946), pp. 147-149, Jstor.   

9. Oskam, B., (1985), ''Transonic panel method for the full potential equation applied to multicomponent airfoils,'' AIAA J., V. 23, pp. 1327-1334.

10. Paliouras, John D., (1990), Complex Variables for Scientists and Engineers, 2nd ed., N.Y., Macmillan.

11. Pennisi, Louis Legendre, (1976), Elements of Complex Variables, 2nd Ed., N.Y., Holt, Rinehart and Winston.

12. Sacksteder, Richard C., (1978), ''On Oscillatory Flows,'' Math. Intell., V. 1, No. 1, pp. 45- 51.

13. Stewart, Ian and David Tall, (1983), Complex Analysis (The Hitchhiker's Guide to the Plane), N.Y.: Cambridge University Press.

Section 10.8  Exercises for The Joukowski Airfoil
See textbook page 451.

Section 10.9	The Schwarz-Christoffel Transformation

Section 10.10	Image of a Fluid Flow

Section 10.11	Sources and Sinks

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