# 4.4.3: Fluid Statics in Geological System

Acknowledgement

This author would like to express his gratitude to Ralph Menikoff for suggesting this topic.

In geological systems such as the Earth provide cases to be used for fluid static for estimating pressure. It is common in geology to assume that the Earth is made of several layers. If this assumption is accepted, these layers assumption will be used to do some estimates. The assumption states that the Earth is made from the following layers: solid inner core, outer core, and two layers in the liquid phase with a thin crust. For the purpose of this book, the interest is the calculate the pressure at bottom of the liquid phase.

Fig. 4.18 Earth layers not to scale.

This explanation is provided to understand how to use the bulk modulus and the effect of rotation. In reality, there might be an additional effects which affecting the situation but these effects are not the concern of this discussion. Two different extremes can recognized in fluids between the outer core to the crust. In one extreme, the equator rotation plays the most significant role. In the other extreme, at the north–south poles, the rotation effect is demished since the radius of rotation is relatively very small (see Figure 4.19). In that case, the pressure at the bottom of the liquid layer can be estimated using the equation (66) or in approximation of equation (77). In this case it also can be noticed that $$g$$ is a function of $$r$$. If the bulk modulus is assumed constant (for simplicity), the governing equation can be constructed starting with equation (??). The approximate definition of the bulk modulus is
$\label{static:eq:iniGov} B_T = \dfrac{\rho \, \Delta P }{ \Delta \rho} \Longrightarrow \Delta \rho = \dfrac{\rho \, \Delta P }{ B_T} \tag{137}$
Using equation to express the pressure difference (see Example 4.6 for details explanation) as
$\label{static:eq:govBT1} \rho(r) = \dfrac{\rho_0}{1 - \displaystyle \int_{R_0}^r \dfrac{g(r)\rho(r)}{B_T(r)} dr } \tag{138}$
In equation (138) it is assumed that $$B_T$$ is a function of pressure and the pressure is a function of the location. Thus, the bulk modulus can be written as a function of the location radius, $$r$$. Again, for simplicity the bulk modulus is assumed to be constant.