Water in a cylindrical tank has been turning long enough to rotate as a rigid body…

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

Fluids and Continuum Physics, Ch Kinematics of flow: fields, streamlines and the material derivative

Problem

Water in a cylindrical tank has been turning long enough to rotate as a rigid body about the vertical axis

x = y = 0

. In Cartesian coordinates the velocity field is

\mathbf{u} = (-\Omega y,\; \Omega x,\; 0), \qquad \Omega = 3.0\ \mathrm{s^{-1}},

with

x

y

in metres. One side of the tank is warmed, and a steady temperature field is maintained in the water,

T = T_0 + Ax, \qquad T_0 = 293\ \mathrm{K}, \qquad A = 8.0\ \mathrm{K\,m^{-1}},

which reads the same at every fixed point at all times. Heating and heat conduction together hold this field steady; how they do so does not matter here. Consider the point

R

x = 0.20\ \mathrm{m}

y = 0.10\ \mathrm{m}

. ((a)) Compute

\nabla\cdot\mathbf{u}

and state what it says about the volume of a fluid parcel. ((b)) Find

DT/Dt

for the particle at

R

. ((c)) Find

a_x

and

a_y

for that particle, and say in one sentence in which direction

\mathbf{a}

points.

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