Radial inflow kinematics: In a steady radial flow into an intake where the radial velocity varies as u(r) = K / r^2, the magnitude of radial acceleration is proportional to which power of r?

Introduction / Context:
Steady 2-D radial flows arise near intakes or sinks. Even with no tangential velocity, there is convective acceleration due to spatial variation of speed. Evaluating acceleration helps assess pressure gradients and potential cavitation near intakes.

Given Data / Assumptions:

• Purely radial flow: vθ = 0.
• Steady, incompressible conditions.
• Radial velocity u(r) = K r^(-2), with K > 0 constant.

Concept / Approach:

Radial acceleration in polar coordinates for purely radial flow is a_r = u du/dr (since vθ = 0). With u = K r^(−2), compute du/dr and then u du/dr to obtain the r-dependence.

Step-by-Step Solution:

Verification / Alternative check:

Dimensional analysis: u ~ r^(−2), gradient ~ r^(−3), product ~ r^(−5), consistent with the computed result. The negative sign indicates acceleration directed inward (toward decreasing r), as expected for inflow.

Why Other Options Are Wrong:

• 1/r, 1/r3, 1/r4: Do not match the product of r^(−2) and r^(−3).
• Constant: Incorrect; acceleration varies rapidly near the intake.

Common Pitfalls:

• Including centripetal term −vθ^2/r when vθ = 0.
• Missing the convective term u du/dr in steady flows.

Final Answer:

1/r5