The velocity field for a plane vortex sink is given by , where q = 2 m3/s/m and K = 1 m3/s/m. The fluid density is 1000 kg/m3. Find the acceleration at (1, 0), (1, π/2), and (2, 0). Evaluate ∇p under the same conditions.
Consider the mass conservation equation in cylindrical coordinates with only gravity force from the equation 6.3a in the text book.
Reduce the equation along the r- components.
Here, is the component of acceleration along the r-axis, is the component of velocity along the r-axis, is the component of velocity along the -axis.
Substitute for and for .
Calculate the r-component of acceleration at point (1,0).
Here, r is the radius of the plane vortex.
Substitute for q, for K, and 1 for r.
Therefore, the r-component of acceleration at point (1,0) is .