Deriving poroelastic equation in differential form

In summary, The conversation is about a poroelastic model of tissue, specifically a simple spherical model with a central sphere for liquid production, followed by two brain tissue spheres with different permeabilities, and a final impermeable shell representing the skull. The general poroelastic equations are simplified for the spherically symmetric geometry to give a differential-form representation. The speaker is looking for insights on how to get from the general equations to the differential form for both the poroelastic equations and Hooke's law.
  • #1
enc08
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Hi,

I'm trying to fill in the gaps in my notes - looking at a poroelastic model of tissue.

We have the simple spherical model below. The centre sphere is where liquid is produced, then the two following spheres are brain tissue with different permeabilities, and the final sphere is an impermeable shell representing the skull.

p1_zps3cd2a08e.png


The general poroelastic equations are

p2_zps4c762c3e.jpg


My notes indicate that these equations are inverted and simplified for the spherically symmetric geometry we have assumed, to give this differential-form representation:

p3_zps81ec2f40.jpg


Any specific thoughts on how I can get from the general equations to the differential form above?

Thanks a lot.
 
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  • #2
Any thoughts on how one can go from the general Hooke's law to differential form?
 

1. What is the poroelastic equation in differential form?

The poroelastic equation in differential form is a mathematical equation that describes the behavior of fluid flow and deformation in porous materials. It is derived from the principles of fluid mechanics and solid mechanics and takes into account the interactions between the fluid and solid phases within the porous material.

2. Why is it important to derive the poroelastic equation in differential form?

Deriving the poroelastic equation in differential form allows for a more detailed and accurate understanding of fluid flow and deformation in porous materials. This is important in various fields such as geology, engineering, and biomedicine, where the behavior of porous materials plays a crucial role.

3. What are the assumptions made in deriving the poroelastic equation in differential form?

Some of the assumptions made in deriving the poroelastic equation in differential form include: the fluid and solid phases are incompressible, the fluid is Newtonian and the solid is elastic, and the porous material is homogeneous and isotropic.

4. How is the poroelastic equation in differential form solved?

The poroelastic equation in differential form can be solved using various numerical methods, such as finite element or finite difference methods. These methods involve discretizing the equation into smaller parts and solving it iteratively to obtain a numerical solution.

5. What are the applications of the poroelastic equation in differential form?

The poroelastic equation in differential form has various applications, including modeling groundwater flow, simulating fluid flow in oil reservoirs, predicting the behavior of biological tissues, and understanding the effects of fluid injection on the surrounding rock formations in geothermal energy production.

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