An incompressible flow has velocity field , , (lengths in metres)

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

An incompressible flow has velocity field u=3x s−1u=3x\,\mathrm{s^{-1}}, v=y s−1v=y\,\mathrm{s^{-1}}, w=−4z s−1w=-4z\,\mathrm{s^{-1}} (lengths in metres), so that the rate-of-strain tensor at every point is eij=diag(3,1,−4) s−1e_{ij}=\mathrm{diag}(3,1,-4)\,\mathrm{s^{-1}}. ((a)) Confirm that this flow is incompressible. ((b)) A short straight material line element is oriented along the unit vector n^=(1/2, 0, 1/2)\hat{\mathbf n}=(1/\sqrt2,\,0,\,1/\sqrt2). Compute its instantaneous fractional stretching rate n^ieijn^j\hat n_ie_{ij}\hat n_j. ((c)) State whether the line is instantaneously lengthening or shortening.

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