# Question 11.9: The streamlines for the 2D, inviscid, constant density ﬂow o...

The streamlines for the 2D, inviscid, constant density ﬂow over a cylinder are shown in Figure 11.8. The streamfunction for this ﬂow is given in cylindrical coordinates by ψ(r, θ) = U_{∞}r(1− R²/r²) sin θ, where U_{∞} is the freestream velocity and R is the cylinder radius. If the body force is neglected, the pressure distribution is given by p(r, θ) = p_{∞}+ 1 2ρU^2 _∞ [1−(1− R^{2}/r2)^{2} −4(R^{2}/r^{2}) sin^{2} θ]. Show that the velocity ﬁeld in this case is described by

v_r=U _∞\left(1-\frac{R^2}{r^2}\right)\cos \theta , v_θ=U _∞\left(1-\frac{R^2}{r^2}\right)\sin \theta , and v_{z } = 0

and that the continuity and Euler equations are satisﬁed. Comment on the boundary conditions.

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