Determining Incompressibility, Irrotationality of VectorV

[Santorican]

Homework Statement

Determine whether the velocity field VectorV=(3t)$$\hat{i}$$+(xz)$$\hat{j}$$+(ty^2)$$\hat{k}$$ is incompressible, irrotational, both, or neither. Also obtain expressions for the linear and shear strain rates.

Homework Equations

V=(u,v,w)

omega=1/2[(delw/dely)-(delv/delz)]i + 1/2[(delu/delz)-(delw/delx)]j + 1/2[(delv/delx)+(delu/dely)]k

epsilonxx=delu/delx
epsilonyy=delv/dely
epsilonzz=delw/delz

epsilonxy=1/2[(delu/dely)+(delv/delx)]

epsilonzx=1/2[(delw/delx)+(delu/delz)]

epsilonyz=1/2[(delv/delz)+(delw/dely)]

The Attempt at a Solution

Okay so I said u=3t, v=xz, w=ty

When I did the partial derivative of the original equation I got a rate of rotation equal to 1/2[(t-x)i+(z)k]

then when I did the linear strain rate I got zero for all of them so when I added up all of the linear strain rates for the volumetric strain rate it comes out to be incompressible?

Then for the Shear Strain rates I got epsilon xy = z/2 and epsilon yz = (x+t)/2?

I don't know this doesn't seem very right...

Help? lol

 

[KataKoniK]

Thank you for posting your question. I have reviewed your attempt at solving this problem and I have a few suggestions for you. Firstly, your calculation for the rate of rotation is correct. However, it is important to note that the velocity field is irrotational if the rate of rotation is equal to zero, which is not the case here. Therefore, we can conclude that the velocity field is not irrotational.

Next, for the linear strain rates, it is important to use the correct equations. The equations you have used are for the components of the velocity field, not for strain rates. The correct equations for strain rates are given in your homework equations section. Using those equations, you should get epsilonxx = 3, epsilonyy = 0, and epsilonzz = 0. This indicates that the velocity field is not compressible.

For the shear strain rates, you are correct in your calculations of epsilonxy and epsilonyz. However, it is important to note that the velocity field is considered incompressible if the sum of the shear strain rates is equal to zero. In this case, the sum is not equal to zero, therefore the velocity field is not incompressible.

In conclusion, the velocity field is neither incompressible nor irrotational. The linear strain rates indicate that there is a change in the velocity field in the x-direction, while the shear strain rates indicate a change in the velocity field in the y and z directions. I hope this helps clarify the solution for you. Let me know if you have any further questions.