# Is this expression for bulk modulus of a fluid incorrect?

Equation (6) of Lagrangian and Eulerian Representations of Fluid Flow:Kinematics and the Equations of Motion [PDF], by James F. Price is this:

A convenient measure of the stiffness or inverse compressibility of the material is $$B=\frac{S_{zz}}{\delta h/h}=-V_{0}\frac{\delta P}{\delta V}=\rho_{0}\frac{\delta P}{\delta\rho}$$ called the bulk modulus.

The $$z$$ component of stress is $$S_{zz},$$ and $$h$$ is measured in the $$z$$ direction.

I see no way that Equation (6) can be correct for a fluid, which is the primary type of material under discussion. In order for it to give a finite result, $$S_{zz}$$ would have to go to zero with $$\delta h$$. But that would require the fluid to have zero pressure when $$\delta h=0$$ As far as I am aware, that cannot happen with a fluid.

The left-most equivalence is plausible for a solid. For a fluid I propose the following modification of Equation (6):

$$B=\frac{\delta S_{zz}}{\delta h/h}=-V_{0}\frac{\delta P}{\delta V}=\rho_{0}\frac{\delta P}{\delta\rho}$$

which means $$B$$ is dependent on $$h$$ , and the expression is only valid near $$\delta h=0.$$ Is this reasonable?

• Your representation is much more appropriate, and the original version is definitely invalid. But it should also be mentioned that, in general, the bulk modulus refers to an isotropic state of loading, and the implication of using h and Szz is that the loading is not isotropic. Jun 21 '20 at 3:27
• If you care to turn that into an answer, I'll gladly accept it. I didn't see any way his equation could be correct, but I've missed subtleties in the past. Jun 24 '20 at 2:04

## 1 Answer

Your representation is much more appropriate, and the original version is definitely invalid. But it should also be mentioned that, in general, the bulk modulus refers to an isotropic state of loading, and the implication of using h and Szz is that the loading is not isotropic.