4:1 Fluid Pressures - Accelerating Fluids - Linear and Rotational

Overview

Derek Elsworth derives how Newton’s second law explains pressure distributions in fluids that are static relative to an accelerating container. He applies the equations to horizontally accelerating tanks, elevator motion, and rotating containers, where the free surface tilts linearly or forms a paraboloid; earthquake-driven seiches and railway tank cars illustrate the effects.

Key takeaways

For a fluid at rest relative to a container accelerating horizontally, the pressure gradient is ∂P/∂y = −ρaᵧ and the open free-surface slope has magnitude |aᵧ|/g when vertical acceleration is zero.
Vertical container acceleration changes effective gravity: upward acceleration increases the pressure gradient to ρ(g + a_z), while downward acceleration reduces it.
A rotating liquid’s pressure rises radially according to ∂P/∂r = ρω²r, producing a free surface described by z = ω²r²/(2g) + C.
Resonance can amplify water motion: Elsworth cites Bay of Fundy tides of about 30 feet and a seiche triggered in Hebgen Lake by the 1959 earthquake.
Railway tank cars use raised end geometry to accommodate fluid surges during acceleration and braking, which can otherwise generate damaging pressure loads.

Chapters

0:00 Earthquake Shaking, Resonance, and Seiches

Elsworth connects water sloshing during earthquakes to resonance between shaking periods and a pool or lake’s natural response.
The 1959 Hebgen Lake earthquake produced a seiche in the lake; the 2011 Mineral, Virginia, earthquake also illustrates regional shaking.
Bay of Fundy tides can reach roughly 30 feet when the bay’s response aligns with tidal forcing; tsunami travel time likewise depends on water depth.

7:00 Railway Tank Cars and the Exam Review

Railway tank cars have raised ends because fluid surges under acceleration and braking can create pressures capable of damaging a flat-topped tank.
Elsworth clarifies that the upcoming exam covers viscosity, pressure on planar structures, and accelerating fluids, including the material through section 4.1.
He recommends practicing prior exam problems and notes that an equation sheet will be provided.

13:00 Revisiting Fluid Statics from Newton’s Second Law

A differential fluid cube with dimensions dx, dy, and dz is analyzed by balancing pressure forces, weight, and acceleration using F = ma.
Setting acceleration to zero recovers the familiar hydrostatic relations: pressure is constant horizontally and changes vertically at a rate of −ρg.
Integrating the vertical pressure gradient gives P = P₀ + ρgh, with h measured downward from the reference pressure.

22:30 Pressure Gradients in an Accelerating Fluid

Allowing nonzero acceleration adds inertial terms to the pressure equations derived for the differential fluid element.
For horizontal acceleration in the y direction, the pressure gradient is ∂P/∂y = −ρaᵧ.
In the vertical direction, the gradient becomes ∂P/∂z = −ρ(g + a_z), using upward-positive z coordinates.

24:00 How Horizontal Acceleration Changes Tank Pressure

In a closed aquarium accelerating across a room, pressure varies horizontally as well as with depth.
The horizontal pressure difference follows ∂P/∂y = −ρaᵧ, so the fluid’s pressure distribution cannot be described by depth alone.
The example separates pressure changes caused by container acceleration from ordinary hydrostatic pressure.

27:00 Deriving the Free-Surface Slope in an Accelerating Tank

At an open tank’s free surface, pressure remains atmospheric even when the surface tilts under acceleration.
Combining the horizontal and vertical pressure gradients gives a free-surface slope with magnitude |dz/dy| = |aᵧ/(g + a_z)|.
For a bus accelerating horizontally with no vertical acceleration, the slope magnitude is |aᵧ|/g.

36:00 Elevator Acceleration and Apparent Fluid Weight

An aquarium accelerating upward has a vertical pressure gradient of −ρ(g + a_z), making the fluid behave as though effective gravity has increased.
Downward acceleration reduces the effective pressure gradient; at 9.81 m/s² downward, the idealized fluid is weightless.
Elsworth relates the weightless condition to parabolic aircraft flights such as NASA’s “Vomit Comet.”

41:30 Rotating Fluids and Angular Velocity

A rotating can of liquid experiences radial acceleration even at constant rotation rate because the velocity direction continually changes.
Angular velocity ω is related to tangential velocity by vθ = ωr, with ω measured in radians per second.
One revolution per second corresponds to 2π radians per second.

44:30 Radial Pressure Gradients in a Spinning Container

For a rotating fluid, pressure increases with radius according to the centripetal-acceleration balance ∂P/∂r = ρω²r.
Pressure does not change with angular position at a fixed radius and elevation, while the vertical pressure gradient remains hydrostatic.
The radial pressure difference drives the free surface upward toward the container wall.

49:20 Integrating the Rotating Free Surface into a Paraboloid

Combining radial and vertical pressure gradients yields dz/dr = ω²r/g for the free surface.
Integration gives z = ω²r²/(2g) + C, the equation of a paraboloid whose curvature increases with rotation rate squared.
A known surface height at a reference radius determines C, allowing the height at another radius to be calculated.

53:10 Mass Conservation and the Transition to Fluid Dynamics

Conservation of mass constrains how the rotating liquid’s central depression and raised outer surface balance within the tank.
Pressure on a submerged wall can still be calculated from P = ρgh using the local fluid height above the point.
Elsworth closes the fluid-statics unit and previews Bernoulli’s equation for moving fluids.

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.