Page 371 - Introduction to Continuum Mechanics

Newtonian Viscous Fluid 355

Solution. Let the positive jc 3-axis be pointing vertically upward, then B = ~ge 3 so that

From Eqs. (i) and (ii), we see p is a function of x$ only, thus Eq. (iii) becomes

Assuming that p,p and © (absolute temperature) are related by the equation of state for ideal
gas, we have

where R is the gas constant for air. Thus, Eq. (iv) becomes

Integrating, we get

where p 0 is the pressure at the ground (x$ = 0), thus,

and from Eq. (v), if p 0 is the density at x$ - 0, we have

6.4 Newtonian Fluid
When a shear stress is applied to an elastic solid, it deforms from its initial configuration
and reaches an equilibrium state with a nonzero shear deformation, the deformation will
disappear when the shear stress is removed. When a shear stress is applied to a layer of fluid
(such as water, alcohol, mercury, air etc.) it will deform from its initial configuration and
eventually reaches a steady state where the fluid continuously deforms with a nonzero rate of
shear, as long as the stress is applied. When the shear stress is removed, the fluid will simply
remain at the deformed state, obtained prior to the removal of the force. Thus, the state of