Incompressible Fluids

For incompressible fluids density is a constant. In addition as stated before, for most applications of practical interest acceleration due to gravity is also a constant. As a consequence the pressure equation is greatly simplified and Eq. 2.15 is readily integrated. Thus for incompressible fluids,

$$\int_{p_1} ^{p_2} dp ~=~-\rho~g~\int_{z_1}^{z_2}dz~=~-\gamma~\int_{z_1}^{z_2}~dz$$ (2.16)

which on integration yields

$$p_2-p_1~=~-\gamma~(z_2-z_1)$$ (2.17)

A convenient form of the above equation is

$$p_1~=~\gamma h + p_2$$ (2.18)

where $h$ is the difference in elevation, $z_2-z_1$.

By rewriting the above equation, we have,

$$h~=~{{p_1~-~p_2} \over {\gamma}}$$ (2.19)

These equations demonstrate that pressure difference between two points in an incompressible fluid is proportional to the difference in elevation or height between the two points. The term $h$ is sometimes defined as the pressure head and is the height of the fluid column of density $\rho$ (and specific weight, $\gamma$) that supports a pressure difference of $p_1-p_2$.

The above is used to determine pressures in atmosphere and ocean depths. For this, it is advisable to choose a convenient datum or reference. Depending upon the application as shown in the figure, sealevel(for atmospheric pressure) or free surface (for measurements in oceans and lakes) seems to be ideally suited.

For an ocean or a lake, if $p_a$ is the pressure acting on the free surface pressure at any depth $h_2$ is given by

$$p~=~p_a~+~\gamma_{water}~h_2$$ (2.20)

On the other hand for the atmosphere with $p_a$ being the sealevel pressure, we have,

$$p~=~p_a~-~\gamma_{air}~h_1$$ (2.21)

Figure 2.4 : Measurement of pressure in atmosphere, oceans and lakes.

 

(c) Aerospace, Mechanical & Mechatronic Engg. 2005
University of Sydney