The relation between incompressibility and divergence-free?

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In summary, for both incompressible solids and fluids, a divergence-free vector field (displacement for solids and velocity for fluids) can be used to characterize them. However, for large deformations in solids, the determinant of the deformation gradient must be one, which is not equivalent to divergence-free. In the case of fluids, since they use the Eularian frame, the divergence-free condition is equivalent to incompressibility in any case.
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phdggg
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Hi all,

One thing I am really confused is the relation between incompressiblity and divergence-free. Since I am coding a finite element code that use mixed displacement and pressure formulation. From what I got from your book. Both incompressible solid and fluid can be characterized by divergence-free vector field (displacement for solid and velocity for fluid). But is this only for small deformation in the case of solid? Since for large deformation, the determinant of deformation gradient must be one, which seems not equivalent to divergence-free.

What about for fluid, because it uses Eularian frame, the divergence-free should be equivalent to incompressibility for any case. Is this correct? Thanks in advance for your clarification!


Mengda
 
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phdggg said:
What about for fluid, because it uses Eularian frame, the divergence-free should be equivalent to incompressibility for any case. Is this correct? Thanks in advance for your clarification!

You are correct.
 

1. What is the concept of incompressibility in fluid mechanics?

Incompressibility refers to the property of a fluid where its density remains constant regardless of the pressure applied to it. This means that the volume of the fluid does not change even when subjected to external forces.

2. How is incompressibility related to divergence-free?

Incompressible fluids are also divergence-free, meaning that the flow of the fluid is such that the flow rate into a certain region is equal to the flow rate out of that region. This is because incompressible fluids cannot accumulate in a certain area, resulting in a net flow of zero.

3. What is the significance of incompressibility in fluid dynamics?

Incompressible fluids are commonly used in fluid dynamics because they simplify the equations and make it easier to analyze the flow of fluids. It also allows for the assumption that the density of the fluid is constant, making it easier to predict the behavior of the fluid.

4. Can a fluid be both compressible and divergence-free?

No, a fluid cannot be both compressible and divergence-free. Compressibility implies that the fluid's density can change, while divergence-free means that the fluid's density is constant. These two properties are contradictory and cannot exist simultaneously in a fluid.

5. How does incompressibility affect the speed of fluid flow?

Incompressible fluids have a constant density, resulting in a constant speed of flow. This means that the flow of an incompressible fluid will not change unless external forces are applied to it. This property is important in many applications, such as in pumps and turbines.

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