A steady two-dimensional water flow near a stagnation point is described

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

A steady two-dimensional water flow near a stagnation point is described, in the continuum picture, by the velocity field

\mathbf{u} = (Sx,\; -Sy,\; 0), \qquad S = 2.0\ \mathrm{s^{-1}},

with

x

y

in metres and

u

v

in metres per second. The water carries the temperature field

T = T_0 + Ax, \qquad T_0 = 288\ \mathrm{K}, \qquad A = 5.0\ \mathrm{K\,m^{-1}},

which does not change with time at any fixed point, held so by heating and heat conduction that we need not describe. Consider the point

P

x = 0.40\ \mathrm{m}

y = 0.30\ \mathrm{m}

. ((a)) Evaluate

\nabla\cdot\mathbf{u}

and say whether the flow is incompressible. ((b)) Find

DT/Dt

for the fluid particle that is at

P

. ((c)) Find the two components

a_x

a_y

of the acceleration of that particle, and the magnitude of the acceleration.

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