Question

A two-dimensional, unsteady velocity field is given by

u = 5x(1 + t) and v = 5y(- 1 + t) , where u is the x-velocity component and v the y-velocity component. Find x(t) and y(t) if x = x and y = yo at t = 0 Do the velocity components represent an Eulerian description or a Lagrangian description?

Answer 1

To find x(t) and y(t), we need to integrate the velocity components u and v with respect to time:

dx/dt = u = 5x(1 + t)
dy/dt = v = 5y(-1 + t)

Separating variables and integrating, we get:

∫ dx/x = ∫ 5(1 + t) dt
ln|x| = 5t + (5/2)t^2 + C1

∫ dy/y = ∫ 5(-1 + t) dt
ln|y| = -5t + (5/2)t^2 + C2

where C1 and C2 are constants of integration.

Solving for x and y, we have:

x = ± e^(5t + (5/2)t^2 + C1)
y = ± e^(-5t + (5/2)t^2 + C2)

where we take the positive or negative sign depending on the initial values x = x0 and y = y0 at t = 0.

The velocity components u and v represent a Lagrangian description, which describes the motion of individual particles in the fluid as they move with the flow. This is because the velocity components are given in terms of the spatial coordinates x and y and the time t, which allows us to track the motion of individual particles over time. In contrast, an Eulerian description would describe the flow at fixed points in space as a function of time.