(a) Water in a channel speeds up along its length, with velocity for , where and is…

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

Mathematical Methods for Physics, Ch Gradient, divergence and curl

Problem

(a) Water in a channel speeds up along its length, with velocity v=(γx, 0, 0)\mathbf v = (\gamma x,\ 0,\ 0) for x>0x>0, where γ=0.40 s−1\gamma = 0.40\ \mathrm{s^{-1}} and xx is measured in metres along the channel. Someone argues: "Every arrow points the same way, parallel to the banks, so the arrows never spread apart and nothing is being created. The divergence must be zero." Compute ∇⋅v\nabla\cdot\mathbf v, and say what a small box with faces across and along the channel sees. (b) A layer of fluid moves with velocity v=(0, βx2, 0)\mathbf v = (0,\ \beta x^2,\ 0), where β=2.0 m−1 s−1\beta = 2.0\ \mathrm{m^{-1}\,s^{-1}} and xx is in metres. Someone argues: "Every streamline of this flow is a straight line parallel to the yy-axis, so a paddle wheel dropped into the fluid cannot turn anywhere." Compute ∇×v\nabla\times\mathbf v, and find the rate at which a small paddle wheel with its axis along z^\hat{\mathbf z} turns at x=0.50 mx = 0.50\ \mathrm m and at x=0x = 0.

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