6:3 Conservation of Mass - Static, Moving and Deforming Control Volumes

Overview

Derek Elsworth develops conservation of mass for static, translating rigid, and moving deforming control volumes, emphasizing that flow across a moving boundary must be measured relative to that boundary. He applies all three formulations to filling a beaker, shows they give the same fill time, and connects the mass-balance framework to continuity constraints and the upcoming conservation-of-momentum analysis.

Key takeaways

Mass flow across a moving control surface depends on fluid velocity relative to the surface; using the stationary-frame velocity alone can give the wrong balance.
A beaker with cross-sectional area A2 filled by an inlet of area A1 and speed V1 takes A2H/(V1A1) to reach height H, regardless of whether the control volume is fixed, translating, or deforming.
For a constant-density free jet, increasing fall speed requires decreasing area so that A·V remains constant along the jet.
The continuity constraint ρ1V1A1 = ρ2V2A2 relates upstream and downstream velocities and can eliminate an unknown when paired with Bernoulli’s equation.
The Reynolds transport framework carries over from mass to momentum: choosing momentum as the extensive property makes velocity the intensive property and requires accounting for vector components and forces.

Chapters

0:00 Three Control-Volume Types and the Beaker-Filling Problem

Elsworth introduces filling a 16-ounce glass at 1 ounce per second: the intuitive fill time is 16 seconds.
The lecture compares a static non-deforming control volume, a moving rigid one, and a moving deforming one.
Examples include flow through a fixed pipe, a translating jet engine, and an expanding balloon.

3:00 Mass Accumulation and the Sign Convention for Flux

For mass conservation, the extensive property is mass and its intensive property is 1.
Accumulation in a control volume can result from increasing volume, changing density, or both.
The balance combines density and volume changes with mass flux across the control surface; inflow is negative and outflow is positive.

8:00 Moving Boundaries Require Relative Fluid Velocity

Elsworth uses a moving boat and fire hose to distinguish the water velocity seen from shore from its velocity relative to the boat.
The stationary-observer velocity is the vector sum of control-surface velocity and fluid velocity relative to the surface.
For a translating rigid control volume, opposite sides sweep equal areas with opposite signs, so the net contribution from uniform surface motion cancels.

13:00 A Falling Jet Narrows as Its Speed Increases

Gravity accelerates the free jet as it falls, increasing its speed with vertical distance.
For constant-density water, continuity requires the product of cross-sectional area and velocity to remain constant, so the jet narrows.
The beaker problem is recast as water entering through an area A1 at velocity V1 and raising a free surface of area A2.

17:30 Fixed Control Volume: Deriving the Beaker Fill Time

With a stationary control volume at the beaker, control-surface velocity and control-volume change are both zero.
The inlet mass rate is −ρV1A1 and the rising-water outlet term is +ρV2A2; balancing them relates the inlet flow to the rise rate.
Using beaker volume A2H and inlet volumetric flow V1A1 gives the fill time as A2H/(V1A1), matching volume divided by flow rate.

21:00 Moving Rigid Control Volume: Tracking the Rising Surface

The second construction moves a non-deforming control volume upward with the beaker’s rising water surface.
Mass flow must be evaluated relative to the moving boundary, rather than using only the fluid’s stationary-frame velocity.
The moving-boundary formulation again leads to the same fill-time result when the inlet flow and swept volume are accounted for.

25:00 Deforming Control Volume: Letting the Volume Expand

In the third construction, the control volume’s lower edge stays at the beaker bottom while its upper boundary rises and the enclosed volume expands.
When the upper boundary follows the water surface, fluid has no relative flow across that boundary; the inlet supplies the growing control-volume contents.
Setting the volume growth rate equal to V1A1 and using volume A2H reproduces the same fill time as the fixed and moving rigid cases.

30:00 Continuity in Bernoulli Problems and the Momentum Preview

The familiar continuity relation ρ1V1A1 = ρ2V2A2 is the practical mass-balance constraint used alongside Bernoulli’s equation.
For a fixed control surface, fluid velocity across the boundary is its ordinary velocity; for a moving surface, it is relative velocity.
Elsworth previews momentum conservation using a water jet striking a person or a Pelton wheel, where jet momentum transfer produces force and turbine rotation.
Changing the conserved extensive property from mass to momentum gives B = mv and intensive property b = v, making the resulting balance vector-valued.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Derek Elsworth.