In a horizontal plane

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

In a horizontal plane, the steady wind field near the upwind end of a long fence is modelled as

\mathbf{u} = (U_0,\; k x,\; 0), \qquad U_0 = 5.0\ \mathrm{m\,s^{-1}},\quad k = 0.40\ \mathrm{s^{-1}},

valid for

0 \le x \le 4.0\ \mathrm{m}

, with

x

and

y

horizontal coordinates in metres. Treat the air as a continuum and the field as exactly the one given. ((a)) By integrating the tangency condition

\mathrm{d}x/u = \mathrm{d}y/v

, find the equation of the streamline that passes through the origin, and give the value of

y

on it at

x = 3.0\ \mathrm{m}

. ((b)) What is the slope

\mathrm{d}y/\mathrm{d}x

of that streamline at

x = 3.0\ \mathrm{m}

? (Sketch the family of streamlines for yourself: they are parabolas that all open the same way.) ((c)) Two records are made in this flow: (1) a sonic anemometer bolted to a post at the fixed point

(3.0\ \mathrm{m},\,0.36\ \mathrm{m})

, logging the velocity there against time; (2) a small neutrally buoyant soap bubble released into the wind and filmed as it drifts. Which record is the Eulerian one? Answer with its number, 1 or 2.

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