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

## Question:

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, uθ) for this flow field are ur = C/2πr and uθ = 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