Physics question about air flow?

Question

Consider a steady flow of air through the diffuser portion of an axisymmetric wind tunnel. The air speed along the center line decreases from a value of 10 m/s at x=0 to 5 m/s at x=2 meters. The wall of the diffuser is a conical surface whose radius at the entrance is 1.0 meter.

What is an expression for the centerline speed as a function of x?

What is the acceleration along the centerline at x=1 meter?

Answer 1

Assuming no compression, the halving of velocity at x=2 means a doubling of the area of the expanding conical section; thus its radius has increased to sqrt(2) * the inlet value, and if sampled every 2 m, the radius is an arithmetical series: 1+0*(sqrt(2)-1), 1+1*(sqrt(2)-1), 1+2*(sqrt(2)-1), etc.
radius(x) = 1+x/2*(sqrt(2)-1)
area(x)/area(0) = (1+x/2*(sqrt(2)-1))^2 / 1^2
centerline speed
v(x) = 10*area(0)/area(x) = 10 / (1+x/2*(sqrt(2)-1))^2 m/s
spatial speed gradient
dv/dx = -10*(sqrt(2)-1)/(1+x/2*(sqrt(2)-1))^3 (m/s)/m
acceleration dv/dt = dx/dt*dv/dx = v(x)*dv/dx
At x=0,
v(x) = 10 m/s
dv/dx = -4.142 (m/s)/m
acceleration = -41.42 m/s^2