Mathematical Tools for Physics

9—Vector Calculus 1 250

each of these areas the fluid has a velocity~vk. This may not be a constant, but as usual with integrals, you pick
a point somewhere in the little area and pick the~vthere; in the limit as all the pieces of area shrink to zero it
won’t matter exactly where you picked it. The flow rate through one of these pieces is Eq. ( 1 ),~vk.ˆnk∆Ak, and

the corresponding estimate of the total flow through the surface is, using the notation∆A~k=ˆnk∆Ak,

∑N

k=1

~vk.∆A~k

This limit as the size of each piece is shrunk to zero and correspondingly the number of pieces goes to infinity is
the definition of the integral
∫
~v.dA~= lim
∆Ak→ 0

∑N

k=1

~vk.∆A~k (2)

Example of Flow Calculation
In the rectangular pipe above, suppose that the flow exhibits shear, rising from zero at the bottom tov 0 at the
top. The velocity field is

~v(x,y,z) =vx(y)ˆx=v 0

y

b

xˆ (3)

The origin is at the bottom of the pipe and they-coordinate is measured upward from the origin. What is the
flow rate through the area indicated, tilted at an angleφfrom the vertical? The distance in and out of the plane
of the picture (thez-axis) is the lengtha. Can such a fluid flow really happen? Yes, real fluids such as water
have viscosity, and if you construct a very wide pipe but not too high, and make the top surface movable you can
slide the top part of the pipe to the right. That will drag the fluid with it so that the fluid just next to the top
is moving at the same speed that the top surface is while the fluid at the bottom is kept at rest by the friction
with the bottom surface. In between you get a gradual transition in the flow that is represented by Eq. ( 3 ).

y

x

φ

ˆnk

b