Page 370 - Introduction to Continuum Mechanics

354 Equations Of Hydrostatics

From Eq. (ii), p is independent of x 2, from Eq. (i)

and from Eqs. (iii) and (iv)

Thus,

i.e.,

The integration constant c can be determined from the fact that on the free surface, the
pressure is equal to the ambient pressure p 0. Let the origin of the coordinate axes (fixed with
respect to the earth) be located at a point on the free surface at the instant of interest, then

Thus, the pressure inside the fluid at any point is given by

To find the equation for the free surface, we substitute/? = p 0 in Eq. (vi) and obtain

Thus, the free surface is a plane with the angle of inclination given by

(b) Referring to Fig. 6.3, we have fa—h) /x\ = tan 6, thus, XT, = h + xi(a /g), therefore

i.e., the pressure at any point inside the fluid depends only on the depth h of that point from
the free surface directly above it and the pressure at the free surface.

Example 6.3.4

For minor altitude differences, the atmosphere can be assumed to have constant tempera-
ture. Find the pressure and density distributions for this case.