Question 1.12: Given the steady two-dimensional velocity distribution

Since time does not appear explicitly in Eq. (1), the motion is steady, so that streamlines, pathlines, and streaklines will coincide. Since w = 0 everywhere, the motion is two-dimensional, in the xy plane. The streamlines can be computed by substituting the expressions for u and into Eq. (1.39):
\frac{dx}{Kx} =-\frac{dy}{Ky}
or
\int{\frac{dx}{x} } =-\int{\frac{dy}{y} }
Integrating, we obtain \ln x=-\ln y+\ln C , or xy=C
This is the general expression for the streamlines, which are hyperbolas. The complete pattern is plotted in Fig. E1.12 by assigning various values to the constant C. The arrowheads can be determined only by returning to Eq. (1) to ascertain the velocity component directions, assuming K is positive. For example, in the upper right quadrant (x\gt 0 , y\gt 0) ,u is positive and is negative; hence the flow moves down and to the right, establishing the arrowheads as shown.
Note that the streamline pattern is entirely independent of constant K. It could represent the impingement of two opposing streams, or the upper half could simulate the flow of a single downward stream against a flat wall. Taken in isolation, the upper right quadrant is similar to the flow in a 90° corner. This is definitely a realistic flow pattern and is discussed again in Chap. 8.
Finally note the peculiarity that the two streamlines (C = 0) have opposite directions
and intersect. This is possible only at a point where u = v = w = 0, which occurs at the origin in this case. Such a point of zero velocity is called a stagnation point.