Final answer:
To find the streamlines of the given flow field, we use the differential equation \(\frac{dx}{u} = \frac{dy}{v}\), substitute the velocity components to find the relationship \(|x|^2 = constant \times |y|\), and recognize that the streamlines are hyperbolas.
Step-by-step explanation:
To find and sketch the streamlines of the flow field given by u = kx, v = -2ky, w = 0 where k is a constant, we employ the concept of a streamline which is a path that is everywhere tangent to the velocity vector of the flow. For a two-dimensional flow in the xy-plane, the differential equations governing the streamlines can be written as:
\(\frac{dx}{u} = \frac{dy}{v}\)
Substituting the given velocity components into this equation:
\(\frac{dx}{kx} = \frac{dy}{-2ky}\)
Solving this differential equation by separating variables and integrating yields:
\(\frac{dx}{x} = -\frac{1}{2}\frac{dy}{y}\)
Integrating both sides:
ln|x| = -\frac{1}{2}ln|y| + C
Or in exponential form:
|x|^2 = e^{-2C}|y|
Sketching the streamlines involves plotting curves that satisfy this equation, indicating that each streamline is a hyperbola in the xy-plane with the property that the product of the x-coordinate squared and the y-coordinate is a constant.