Question 10.4: Find the velocity and acceleration in a uniform flow in the X...

From Eqs. 10.10a–10.10c, the velocity is defined by U = dX/dt, V = dY/dt, and W = dZ/dt. Taking derivatives of the particle path functions, we obtain:

U = \frac{dX}{dt}=\frac{d}{dt} (X₀ + U_∞(t − t₀)) = U_∞,       V =\frac{dY}{dt}=\frac{d}{dt} (Y₀) = 0

and

W =\frac{dZ}{dt}=\frac{d}{dt} (Z₀) = 0

Thus, the velocity vector is V = U_∞i + 0j + 0k, and, since this velocity is independent of time, acceleration is zero. The results are consistent with the description of these particle paths as a uniform flow.