Water flows steadily along the axis of a tapering nozzle

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Kinematics, conservation laws and ideal flow · Kinematics, conservation laws and ideal flow

Fluids and Continuum Physics, Ch Kinematics of flow: fields, streamlines and the material derivative

Problem

Water flows steadily along the axis of a tapering nozzle. On the axis the flow is purely axial and the measured speed is u(x)=U0(1+xL)2,0≤x≤L,u(x) = U_0\left(1 + \frac{x}{L}\right)^{2}, \qquad 0 \le x \le L, with U0=1.2 m s−1U_0 = 1.2\ \mathrm{m\,s^{-1}} and L=0.40 mL = 0.40\ \mathrm{m}, xx in metres measured downstream from the inlet. The transverse components are negligible on the axis. Student's solution. "The flow is steady, so at every point ∂u/∂t=0\partial u/\partial t = 0. The acceleration of a fluid particle is the rate of change of the velocity field with time, a=∂u/∂ta = \partial u/\partial t. Hence the acceleration on the axis is zero everywhere, including at x=0.20 mx = 0.20\ \mathrm{m}." ((a)) State in one sentence what is wrong with the reasoning. ((b)) Calculate the correct axial acceleration of a fluid particle at x=0.20 mx = 0.20\ \mathrm{m}.

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