1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

484 CHAPTER 11 • APPLICATIONS OF HARMONIC FUNCTIONS v Flow with circulation around a traditional airfoil. Figure 11.67 Flow with ...

11 .8 • T HE JOUKOWSKI AIRFOIL 485 ( c) Show that the line Lo is inclined at the angle ao = ~ -Arctan a. 4. Show that a line t h ...

486 CHAPTER ll • APPLICATIONS OF HARMONIC FUNCTIONS the angle of the trailing edge of the modified Joukowski airfoil A , forms a ...

11.9 • THE SCHWARZ- CHRISTOFFEL TRANSFORMATION 487 ...

488 CHAPTER, 11 • APPLICATIONS OF HARMONJC FUNCTIONS w =f(z} x Figure 11 .70 A Schwarz-Christoffel mapping with n = 5 and a 1 + ...

11 .9 • THE SCHWARZ-CHRlSTOFFEL TRANSFORMATION 489 J dz 1 = iarcsinz. (z^2 - l)' J dz i = log ( z + ( z2 - 1) t) -i;. (z^2 - l)" ...

490 CHAPTER. 11 • APPLICATIONS OF HARMONIC FUNCTIONS v Figure 11. 72 The region of interest. 'Tr 11 Using the image values f ( ...

11.9 • THE SCHWARZ- CHRJSTOFFEL TRANSFORMATION 491 v v (b) Figure 11. 7 4 The regions of interest. 7r 7r 0:1--+ 2,a2--+ -7r, and ...

492 CHAPTER 11 • APPLICATIONS OF HARMONIC FUNCTIONS will determine a mapping w = g(z ) of the upper half-plane onto the domain i ...

11.9 • THE SCHWARZ-CHRISTOFFEL TRANSFORMATION 493 Let a be a real constant. Use the Schwarz-Chru.'tOffel formula. to show that ...

494 CHAPTER 11 • APPLICATIONS OF HARMONIC FUNCTIONS as shown in Figure 1 1.79. Hint: Set xi= - l,x2 = O,xs = l,w 1 = i1r-d,W2 = ...

11 .9 • THE SCHWARZ-CHRISTOFF.EL TRANSFORMATION 495 Show that w = f (z) = (z - l)" [l + az/ (1 - a))^1 - " maps the upper half- ...

496 CHAPTER. 11 • APPLICATI ONS OF HARMONIC FUNCTIONS dz Show that w = f (z) = J ~ maps the upper half-plane onto a square. (z ...

10 • IMAGE OF A FLUID FLOW 497 (a) Flow over a step. (b) Flow around a blunt object. Figure 11.87 Furthermore, the image of ho ...

498 CHAPTER. 11 • APPLICATIONS OF HARMONIC FUNCTIONS Use Figure 11.89 to find the flow around an infinitely long rectangular ba ...

l 1.11 • SOURCES AND SINKS 499 Use Figure 11.91 to find the flow over a dam. Flow over a dam. Figure 11.91 For flow around an ...

500 CHAPTER 11 • APPLICATIONS OF HARMONIC FUNCTIONS (a) A source al lhe origin. (b) A sink at the origin. Figure 11.93 Sources a ...

11.11 • SOURCES AND SINKS 501 (0, 0, h) _qf2 l!.h L i"·. I • • • z \ .... ; ... ,.. .. .. .. ... ... ... (0, o. - h) • qt2 l!.h ...

502 CHAPTER 11 • APPLICATIONS OF HARMONIC FUNCTIONS Next, we use the change of variable h = r tan t and dh = r sec^2 t dt and th ...

11.11 • SOURCES ANO S rNKS 50 3 (a) Source and sink of equal strength. (b) Two sources of equal Strength. F igure 11.95 Fields d ...