Compressible Flow Homework: Show Particles in Cube Occupying Volume e at t=1

[geft]

Homework Statement

Consider the flow with velocity vector v = xi.
Show that the individual particles have the position vectors r(t) = c₁e^ti + c₂j + c₃k with constant c₁, c₂, c₃.
Show that the particles that at t = 0 are in the cube whose faces are portions of the planes x = 0, x = 1, y = 0, y = 1, z = 0, z = 1 occupy at t = 1 the volume e.

Homework Equations

Vector divergence formula regarding the condition of incompressibility, which is div v = 0.

The Attempt at a Solution

I don't understand what the question is asking. I only know that divergence explains the direction and magnitude of a vector field, as well as how to work it out. I also don't understand the notations describing the cube.

 

[mighty2000]

Can you please clarify?
Thank you for your post. It seems like you are having difficulty understanding the question and the notation used. Let me try to explain it in a simpler way.

The question is asking you to show that for a flow with velocity vector v = xi, the particles in the flow will have position vectors of the form r(t) = c1eti + c2j + c3k, where c1, c2, and c3 are constants. This means that the position of each particle in the flow can be described by this equation, where t is time.

To show this, you can use the vector divergence formula, which states that for an incompressible fluid (a fluid with no change in density), the divergence of the velocity vector is equal to zero. In this case, since the velocity vector is v = xi, the divergence of v is simply ∂x/∂x = 1. This means that the fluid is incompressible.

Next, the question asks you to show that the particles that start at the corners of a cube with faces at x = 0, x = 1, y = 0, y = 1, z = 0, z = 1 will occupy a volume of e at t = 1. This means that the particles will move along the flow and form a volume of e at t = 1.

I hope this helps clarify the question and how to approach it. Let me know if you need further assistance. Good luck with your work!