first paragraph counts as an attempt at solution, I would say. (Choosing C as parameter is unfortunate, though, if the problem statement already has a C.)
BvU said:
A pathline is the trajectory of a single fluid particle over time, while a streamline is a line that is tangent to the velocity vector at each point in the flow field. Essentially, a pathline shows the actual path of a particle, while a streamline represents the direction of flow at any given point.
Pathlines and streamlines are related because they both show the movement of fluid particles in a flow field. A pathline can be thought of as a collection of streamlines, with each streamline representing the direction of flow at a specific point in time.
Pathline and streamline analysis can be useful in many fields, such as fluid dynamics, aerodynamics, and oceanography. It can help in understanding and predicting the behavior of fluids in various systems, such as air flow over an aircraft wing or ocean currents.
To solve a pathline and streamline problem, one needs to first define the flow field and initial conditions, such as the velocity of the fluid particles at a certain time. Then, mathematical equations, such as the Navier-Stokes equations, can be used to calculate the pathlines and streamlines of the particles in the flow field.
One limitation of pathline and streamline analysis is that it assumes the fluid is incompressible and has constant viscosity, which may not always be the case in real-world situations. Additionally, it does not take into account factors such as turbulence and viscosity gradients, which can affect the accuracy of the results.