11.7 • TWO-DIMENSIONAL FLUID FLOW 467
y
Figure 11.47 A two-dimensional vector field.
Both p and q are continuously differentiable, so we can use the rnean value
theorem to show that
p (x + C!i.x, t) - p (x, t) = Px (x1, t) C!i.x and
q(t, y+C!i.y)- q(t,y) = Qy(t, Y2)C!i.y,
(11-32)
where x < x 1 < x + C!i.x and y < Y2 < y + C!i.y. Substitution of the expressions in
Equation (11-32) into Equation (11-31) and subsequently dividing through by
D.x C!i.y result in
1 1!1+~Y 1 rx+~x
C!i.y Y Px (x1, t) dt + C!i.x f , q 11 (t, y2) dt = O.
We can use the rnean value theorem for integrals with this equation to show that
Px (x1, Y1) + Qy (x2, Y2) = 0,
where y < Y1 < y + C!i.y and x < x2 < x + C!i.x. Letting C!i.x --+ 0 and C!i.y --+ 0 in
this equation yields
p,, (x, y) + q 11 (x, y) = 0, (11-33)
which is called the equation of continuity.
The curl of the vector field in Equation (11-30) has magnitude
!curl V (x, y)I = qx (x, y) - Py (x, y)
and is an indication of how the field swirls in the vicinity of a point. Imagine that
a "fluid element" at the point (x,y) is suddenly frozen and then moves freely in
the fluid. The fluid element will rotate with an angular velocity given by
1 1 1
2qY (x, y) - 2Px (x, y) = 2 lcurl V (x, y)I.
We consider only fluid flows for which the curl is zero. Such fluid flows are
called irrotational. This condition is more precisely characterized by requiring
