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[ F(z) = \fracm2\pi \ln\left( \fracz+az-a \right) ]
[ M_2^2 = \frac1 + \frac\gamma-12 M_1^2\gamma M_1^2 - \frac\gamma-12 ] [ \fracp_2p_1 = 1 + \frac2\gamma\gamma+1 (M_1^2 - 1) ] [ \fracT_2T_1 = \frac\left(1 + \frac\gamma-12 M_1^2\right) \left( \frac2\gamma\gamma+1 M_1^2 - \frac\gamma-1\gamma+1 \right)\left(1 + \frac\gamma-12 M_1^2\right) ] [ \fracp_02p_01 = \left[ \frac\frac\gamma+12 M_1^21 + \frac\gamma-12 M_1^2 \right]^\frac\gamma\gamma-1 \left[ \frac1\frac2\gamma\gamma+1 M_1^2 - \frac\gamma-1\gamma+1 \right]^\frac1\gamma-1 ] advanced fluid mechanics problems and solutions
The solutions provide exact analytical expressions for complex flow fields and forces. You can find further detailed problems in MIT OpenCourseWare's Advanced Fluid Mechanics or practice with resources like 2500 Solved Problems in Fluid Mechanics turbulent flow models Solution to Problem 6.04 - MIT OpenCourseWare [ F(z) = \fracm2\pi \ln\left( \fracz+az-a \right) ]
) plus the pressure forces must equal the net change in momentum flux: advanced fluid mechanics problems and solutions
The von Kármán integral equation for a flat plate (zero pressure gradient) is: $$ \fracd\thetadx = \frac\tau_w\rho U_\infty^2 $$ Where $\theta$ is the momentum thickness.
