Verify that the spiraling line vortex/sink flow in the r-plane of Prob. 917 satisfies the two-dimensional incompressible

Verify that the spiraling line vortex/sink flow in the rθ-plane of Prob. 9–17 satisfies the two-dimensional incompressible continuity equation. What happens to conservation of mass at the origin? Discuss.

Data from Problem 17

Consider a spiraling line vortex/sink flow in the xy-or rθ-plane as sketched in Fig. P9–17. The two-dimensional cylindrical velocity components (ur, u_θ) for this flow field are u_r = C/2πr and u_θ = Γ/2pr, where C and G are constants (m is negative and Γ is positive). Transform these two-dimensional cylindrical velocity components into two dimensional Cartesian velocity components (u, v). Your final answer should contain no r or θ—only x and y. As a check of your algebra, calculate V² using Cartesian coordinates, and compare to V² obtained from the given velocity components in cylindrical components.

Figure 9-17

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