Bioengineers interested in birds flight often place birds in a wind tunnel. By measuring velocities in some parts of the flow, the researchers can infer characteristics of other parts of the flow. For this experiment, the wind tunnel is oriented vertically, with airflow (known density ) directed downwards onto the bird, who hovers untethered steadily in place. The flow into the tunnel (of diameter D) is approximately uniform with known velocity V_in. Downstream of the bird, the flow is broken up into two regions. In the circular wake below the bird (known diameter d), the flapping of the wings create a higher velocity (V_w). In the annular region around this wake, there is a slightly diminished velocity, V_a, that is as-of-yet unknown. The wind tunnel is designed such that the walls of the tunnel impart negligible shear stress onto the passing air.

Determine an expression for V_a in terms of D, d, V_in, and V_w

Determine an expression for m in terms of g, , D, d, V_w, V_in, and V_a.