Final answer:
The streamline equation is derived from integrating the ratio of differential components of the velocity field, and the streamlines can be sketched by plotting this equation to visualize the path taken by fluid particles.
Step-by-step explanation:
The velocity field provided suggests that this is a flow field problem typically studied in fluid mechanics, a branch of Engineering. To find the equation of the streamline, we need to recognize the relationship between flow rate and velocity.
For a two-dimensional flow, the flow rate (Q) is the product of the cross-sectional area (A) and the velocity (U), as described by the equation Q = AU. For incompressible fluids, the flow rate is constant throughout the flow field. Additionally, streamlines can be visualized by solving the differential equation that represents the path followed by a fluid particle, which equates the velocity components in both the x and y directions.
The equations relating the magnitude and direction of velocity (v), and its components (vx and vy), along the x- and y-axes are crucial. The given velocity field has components U = Cx/(x² + y²) and V = Cy/(x² + y²). To find the equation for streamlines, we set dY/V = dX/U and integrate, which yields the streamline equation.
To sketch the streamlines, a plot of this equation is needed. Streamlines will show the path taken by fluid particles and do not intersect because at any given point, the flow has only one velocity vector.