(density x gravity acceleration, resulting in units such as lbf/ft3). Gerhart, et al. refer to Eqn. 1 as "the basic equation of fluid statics." Integrated and altered to express depth below a free liquid surface, such as water, it becomes
where p1 is the pressure at a deeper place, p2 is the pressure at a shallower depth, and h is the distance point 1 lies vertically below point 2 [z2 - z1 = h = (p1 - p2)/?]. Eqn. 2 is used to determine the forces exerted on submerged surfaces, as on gates below the water surface in a large reservoir. Vennard rearranges (2) usefully to reveal that everywhere in a static liquid:
p1/? + z1 = p2/? + z2 = Constant (3)
Fluid Kinematics and Conservation of Mass
Kinematics is the study of the changes of fluid quantities in space and time. The conservation of mass principle states that during any space-time changes in a fluid field, mass can be neither created nor destroyed. In differential form, for a fluid with density ? moving at velocities u, v, and w in the x, y, and z directions, respectively, over time, t:
This mass continuity equation, is true for unsteady, compressible or incompressible flows. For steady [(?/(t=0], incompressible [(?=0] liquid flows in a conduit from point 1 where velocity = V1 across a section with area A1 to point 2 with velocity = V2 and an area A2:
or, for compressible, varying-density fluids
Mass (and energy) are quantities that are conserved in fluid flow, whereas momentum and force are fluid entities that are not conserved. Momentum is the product of mass and its velocity. Since mass is conserved everywhere in a flowing fluid [Eqn. 5 or 6] while velocity can vary between A1 and A2, momentum does not have to remain constant. The change, or flux, of momentum is net force (because F = ma = m x dv/dt). The exact momentum exerted per second is
in which ¯ = a "momentum correction factor," and the other terms are as before. The factor ¯...