having a depth h as illustrated. Construct a set of x, y axes with y= 0 represent-
ing the bottom of the channel and y=h representing the top of the channel. If
the velocity of the fluid in the channel is given by the one-dimensional vector field
v =α y ˆe 1 , for 0 ≤y≤hand αis some proportionality constant, then the vector field
associated with this flow can be graphically illustrated by sketching the vectors v at
various depths in the channel. The resulting images represent one way of illustrating
a vector field. The resulting sketch is called a vector field plot.
Example 6-24.
Consider the two-dimensional vector field v =v (x, y ) = yˆe 1 +xˆe 2. There are
computer programs that can graphically illustrate this vector field by plotting vectors
at selected points within a specified region. The resulting images of all the vectors
illustrated at a finite set of points is called a vector field plot. The figure 6-16
illustrates a vector field plot for the above vector v sketched at selected points over
the region R={(x, y ) |− 5 ≤x≤ 5 , − 5 ≤y≤ 5 }.
An alternative method of illustrating a vector field is to define a set of curves,
called field lines, where each curve has the property that at each point (x, y )on any
curve, the tangent to the curve at (x, y )has the same direction as the vector field at
that point. If r =xˆe 1 +yˆe 2 is a position vector to a point (x, y )on a field line, then
dr gives the direction of the tangent line and if this direction is to have the same
direction as v , then the two directions must be proportional and requires that
dr =dx ˆe 1 +dy ˆe 2 =kv (x, y ) = k[yˆe 1 +xeˆ 2 ] = ky eˆ 1 +kx ˆe 2 (6 .78)
where kis some proportionality constant.
Figure 6-16. Vector field plot for v =v (x, y ) = yeˆ 1 +xeˆ 2
