Determining Incompressibility, Irrotationality of VectorV

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In summary, the velocity field VectorV=(3t)\hat{i}+(xz)\hat{j}+(ty^2)\hat{k} is neither incompressible nor irrotational. The linear strain rates are epsilonxx = 3, epsilonyy = 0, and epsilonzz = 0, while the shear strain rates are epsilonxy = z/2 and epsilonyz = (x+t)/2.
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Santorican
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Homework Statement


Determine whether the velocity field VectorV=(3t)[tex]\hat{i}[/tex]+(xz)[tex]\hat{j}[/tex]+(ty^2)[tex]\hat{k}[/tex] 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
 
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  • #2




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.
 

Related to Determining Incompressibility, Irrotationality of VectorV

1. What is the definition of incompressibility of a vector field?

Incompressibility refers to the property of a vector field where the magnitude of the vector remains constant at every point in the field. This means that the divergence of the vector field is equal to zero, indicating that there is no net flow of the vector field into or out of a closed surface.

2. How is incompressibility of a vector field determined?

Incompressibility is determined by calculating the divergence of the vector field using the vector calculus operator del (∇). If the divergence is equal to zero, then the vector field is considered incompressible.

3. What does it mean for a vector field to be irrotational?

An irrotational vector field is one in which the curl of the vector field is equal to zero, indicating that there is no rotation or circulation of the vectors in the field. This means that the vectors are aligned in a parallel manner and do not form any closed loops.

4. How can the irrotationality of a vector field be determined?

The irrotationality of a vector field can be determined using the vector calculus operator del (∇). If the curl of the vector field is equal to zero, then the field is considered irrotational.

5. What are the practical applications of determining incompressibility and irrotationality of a vector field?

Determining incompressibility and irrotationality of a vector field is important in various fields of science and engineering, such as fluid dynamics, electromagnetism, and aerodynamics. It allows for the analysis and prediction of the behavior of vector fields, which can help in designing efficient and stable systems.

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