Final answer:
The volume flow rate through the upper hemisphere of the unit sphere is π (pi) cubic meters per second.
Step-by-step explanation:
To compute the volume flow rate through the upper hemisphere of the unit sphere, we need to integrate the velocity field over the hemisphere's surface. Since the velocity field is given as v=(0,0,z), the only component of velocity that contributes to the flow rate is z. Therefore, the volume flow rate is the integral of z over the upper hemisphere:
Q = ∫∫ z dA
To perform this integration, we can switch to spherical coordinates. The limits of integration for θ and φ are 0 to 2π and 0 to π/2, respectively. The integral becomes:
Q = ∫∫ z r² sin(φ) dφ dθ
Since we are integrating over a unit sphere, we have r=1. Evaluating the integral, we get:
Q = ∫0^(2π) ∫0^(π/2) sin(φ) dφ dθ
Simplifying, we find that the volume flow rate through the upper hemisphere of the unit sphere is:
Q = π