Consider the same duct and velocity field

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

Fluid Dynamics, Ch Kinematics in three dimensions and Cauchy's equation of motion

Problem

Consider the same duct and velocity field u(x)=U0+kxu(x)=U_0+kx, with U0=2.0 m s−1U_0=2.0\,\mathrm{m\,s^{-1}}, k=5.0 s−1k=5.0\,\mathrm{s^{-1}}, and cross-section A=0.010 m2A=0.010\,\mathrm{m^2}, but now follow the material (fluid) volume that at this instant occupies the same stretch 0≤x≤1.0 m0\le x\le1.0\,\mathrm m of the duct; its boundaries move with the local fluid velocity, not fixed in space. ((a)) Compute dV/dt\mathrm dV/\mathrm dt for this material volume from the rate at which its two end faces, each moving with the local fluid velocity, separate: dV/dt=A [u(1.0 m)−u(0)]\mathrm dV/\mathrm dt=A\,[u(1.0\,\mathrm m)-u(0)]. ((b)) Compute the same rate independently from the Reynolds transport theorem with F=1F=1: dV/dt=∫V∇⋅u dV\mathrm dV/\mathrm dt=\int_V\nabla\cdot\mathbf u\,\mathrm dV. ((c)) Confirm the two results agree.

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