A steady flow has physical (unit-vector) components ,

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

Conditions of Validity, Ch Gradient, divergence and curl without coordinates, and with any coordinates

Problem

A steady flow has physical (unit-vector) components Fr^=krF_{\hat r}=kr, Fθ^=0F_{\hat\theta}=0, Fz^=0F_{\hat z}=0 in cylindrical coordinates (r,θ,z)(r,\theta,z), with k=0.50 s−1k=0.50\ \mathrm{s^{-1}}. Someone computes the divergence at r=1.5 mr=1.5\ \mathrm{m} by writing ∇⋅F=∂Fr^∂r+∂Fz^∂z=k,\nabla\cdot\mathbf F=\frac{\partial F_{\hat r}}{\partial r}+\frac{\partial F_{\hat z}}{\partial z}=k, treating the cylindrical formula as if it were the Cartesian one. ((a)) Find the error in this working and identify the term it drops. ((b)) Give the correct value of ∇⋅F\nabla\cdot\mathbf F at this point, in s−1\mathrm{s^{-1}}.

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