Water flows steadily through a planar nozzle whose walls converge towards the centreline

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

y = 0

. Over the region of interest the velocity field is modelled as

u = U_0\left(1 + \frac{x}{L}\right), \qquad v = -\frac{U_0 y}{L}, \qquad w = 0,

with

U_0 = 2.0\ \mathrm{m\,s^{-1}}

and

L = 0.40\ \mathrm{m}

, and

x

y

in metres. A dye is dissolved in the water with the steady concentration field

c = c_0 + \alpha x, \qquad c_0 = 2.0\ \mathrm{kg\,m^{-3}}, \qquad \alpha = 0.50\ \mathrm{kg\,m^{-4}}.

The field is held steady at every fixed point by dye diffusing in from sources along the walls, so a particle's concentration can change as it moves. Take the point

Q

x = 0.40\ \mathrm{m}

y = 0.10\ \mathrm{m}

. ((a)) Test the field for incompressibility by computing

\nabla\cdot\mathbf{u}

. ((b)) Find

Dc/Dt

for the particle at

Q

. ((c)) Find the components

a_x

and

a_y

of the acceleration of that particle.

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