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p1120

# p1120 -

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Unformatted text preview: 11.20. CHAPTER 11, PROBLEM 20 23 11.20 Chapter 11, Problem 20 For ψ (x, y ) = x2 + 4y 2 the velocity components are u= ∂ψ = 8y, ∂y v=− ∂ψ = −2x ∂x At a stagnation point, we must have u = v = 0. For this condition to hold, necessarily x = y = 0. Thus, there is a single stagnation point at the origin. To determine the streamlines, we set ψ equal to a constant. Then, calling the constant C 2 , 2 2 x + 4y = C 2 x C =⇒ 2 y + C/4 2 =1 As shown in Figure 11.6, this is a family of ellipses with semimajor axis a = C and semiminor axis b = C/4. The vorticity for this flow is ω= ∂v ∂u − = −2 − 8 = −10 = 0 ∂x ∂y Therefore, the flow is rotational. y . . . .. .. ... . ... . . . .. . . ... .............................................. ............................................... . ......... .. .......... ......... . ......... . ........... ........... . ... ....... .... ....... .. . . .. .. ....... ..... ....... ..................................... .................................... ..... .. .. ....... . ........ .... . ....... ....... .... ..... . .. . ......... ......... ..... .. . ... ... . .. ..... .. ..... ... . ... . ..... ... ... ..... .......................... ........................... ... . .. ... . ... . .... .... . .... .. ... .. ... .. .. .. ..... . .. ... .. ..... . .. . .. . . . ........................................................................................................................................... .................................................. ....................................................................................... . . .. . . .. . ... . .. .... .. . . .. . ....... ... ... .. ......... . ............... .... ... ... . .. ... . .. . ......................... ..... ..... ........... . .... ... .... ..... ..... ... ....... ....... . ... .. . ... ..... . ..... ............. ............. .... ............................. ........ ..... .. . .. ...... ...... .............................. ........... ...... ...... ........ . ........ . ....... . ............. ......... ............. . ...................................................... ..................................................... . . . . . . . . . . . • Figure 11.6: Flow on elliptical streamlines. x ...
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