As shown in Fig. P 3.90, a liquid column of height h is...

As shown in Fig. $\mathrm{P} 3.90$, a liquid column of height $h$ is confined in a vertical tube of cross-sectional area $A$ by a stopper. At $t=0$ the stopper is suddenly removed, exposing the bottom of the liquid to atmospheric pressure. Using a control volume analysis of mass and vertical momentum, derive the differential equation for the downward motion $V(t)$ of the liquid. Assume one-dimensional, incompressible, frictionless flow.

Key Concepts

Frictionless Flow

Assuming frictionless flow implies neglecting viscous forces and energy losses due to friction. This idealized condition simplifies the momentum conservation analysis by allowing the focus to be solely on inertial and pressure forces, which is especially useful when deriving the basic differential equations governing the fluid's motion.

Incompressible Flow

This assumption means that the density of the fluid remains constant throughout the motion. It significantly simplifies the formulation of both the mass conservation and momentum conservation equations, as changes in density do not need to be accounted for when analyzing the fluid dynamics.

Momentum Conservation

Momentum conservation, based on Newton’s second law, is used to relate the net forces acting on the fluid within the control volume to its rate of change of momentum. For fluid flow, this often involves analyzing pressure forces, gravitational forces, and any accelerations, which collectively drive the motion of the fluid.

Control Volume Analysis

This concept involves selecting a fixed region in space through which fluid flows and applying the conservation laws (mass and momentum) to that region. It simplifies the analysis of fluid motion by accounting for the net effects of fluxes crossing the boundaries rather than tracking individual fluid particles.

Mass Conservation

The mass conservation principle states that mass cannot be created or destroyed within a control volume. In the context of incompressible flow, it leads to a mathematical relationship that ensures the incoming mass flux equals the outgoing mass flux, which is fundamental for deriving relations between fluid velocity and changes in the system.

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