Pressure fields

Pressure is a scalar field. In a resting uniform fluid, pressure rises with depth because lower layers support the weight above them. Pressure is not a force by itself: it produces force only when combined with a specified surface area and normal direction.

Think like a programmer

Write pressure as a function of location, not a single number attached to a container. Define a depth coordinate and surface reference explicitly, then evaluate the same pure function at sample points. Keep pressure in pascals, density in kg/m³, gravity in m/s², and depth in metres; validate nonnegative density, gravity, and depth before calculation.

Model checklist

Inputs: Surface pressure, density, gravity magnitude, depth measured downward from a declared surface, and optional surface geometry.
State: Pressure samples, pressure differences, reference coordinate, and force-on-surface inputs.
Rule: Add weight per area of the fluid above each point: p = p₀ + ρgh for constant density at rest.
Output: Hydrostatic pressure, pressure difference, and derived force inputs.
Check: Depth zero returns p₀; doubling h doubles pressure increase; equal depth gives equal pressure in the same uniform resting fluid regardless of container shape; nonnegative inputs are required.

\[p(h)=p_0+\rho gh\]

Pressure and buoyancy sampler

Change depth and displaced volume. Pressure increases with depth; buoyancy is the weight of displaced fluid.

Depth: 2.0 mDisplaced volume: 0.020 m³

Depth 2.0 m; pressure 120620 Pa; displaced volume 0.020 m³; buoyant force 196.2 N.

The useful comparison is often Δp = ρgΔh; surface pressure cancels. For water with ρ = 1000 kg/m³, g = 9.81 m/s², and h = 2 m, the pressure increase is about 19.6 kPa. Two points at equal depth in a connected static uniform fluid have the same pressure even if their container walls curve differently. Wall forces, not a changed local equation, account for different container shapes.

To calculate net force, integrate pressure times the outward normal over the selected surface. Uniform pressure on opposite sides can cancel, while a pressure gradient produces buoyancy. The sampler reports pressure and displaced-fluid buoyancy from explicit input, but does not solve flow, density changes, or free-surface geometry.

Try this experiment

Prediction: Doubling depth doubles the pressure increase; equal depths in the same resting uniform fluid have equal pressure regardless of container shape.

Use p₀ = 101 kPa and compare water at depths 1 m and 2 m. Predict their pressure difference before sampling. Then describe two different container shapes with marked points at equal depth. State which geometry value is irrelevant to the ideal local calculation and which area/normal data is needed for a force calculation.

Where this model breaks

Density can vary with temperature, salinity, composition, and pressure. Accelerating or flowing fluids need momentum equations beyond hydrostatics, while surface tension and capillarity matter at small scales. Gravity can vary over large domains, and real pressure sensors have calibration, orientation, and finite-size limits.

Summary

Hydrostatic pressure is a location-dependent scalar field. Declare density, gravity, depth coordinate, and reference surface, then distinguish local pressure from the integrated force it can produce on a specified surface.

Glossary

Pressure: normal force per area.
Hydrostatic: fluid at rest.
Depth: distance below a chosen free surface.
Pressure gradient: spatial change in pressure that contributes to force density.
Pascal: newton per square metre, the SI unit of pressure.
Hydrostatic paradox: different container shapes can share pressure at equal depth.

Self-check

What field type is pressure?
Which variables set hydrostatic pressure?
Does container shape matter at equal depth?
What extra data turns pressure into net force?

Sources

Model contract

Treat the lesson as a small function before treating it as a fact to memorize. Give every value a unit, keep only the state needed for the next step, and make the output easy to inspect.

\[\text{observable output} = f(\text{inputs},\,\text{state})\]

Inputs: Quantities you set or measure, with units and useful bounds.
State: Values the program must retain to reproduce the next result.
Rule: The relationship or update that turns inputs and state into a result.
Check: A known limit, unit check, invariant, or measured result that can expose a bad model.

Implement the idea as a model

For Pressure Fields, write down the quantities you can control, the values your program must retain, and the result a reader could inspect. In Fluids, the useful program is not the drawing: it is the smallest explicit model that makes a prediction you can test.

Guided experiment

Prediction: changing one declared input while holding the others fixed should change only the outputs that the model connects to that input. Choose one input, predict the direction of change, then check a limiting case such as zero, a symmetric arrangement, or a familiar low-speed or small-change approximation.

Where this model breaks

This lesson is a teaching model, not a complete simulator. Before using it outside the stated question, check which interactions, scales, uncertainties, boundary conditions, and measurement limits it leaves out.

Summary

Treat Pressure Fields as a contract: named inputs and units enter a rule, the rule produces an observable result, and a known limit or invariant checks whether the implementation deserves trust.

Glossary

Input: a measured value or chosen parameter supplied to a model.
State: the smallest set of values needed to continue or reproduce a model.
Validation: comparing an output with a known result, limit, invariant, or measurement.

Self-check

Which values are inputs, and which values must remain state?
What observable result would tell you the model is behaving as expected?
Which assumption would you test first before applying the model to a real system?

Model review: turn Pressure Fields into a test

Compute hydrostatic pressure as a scalar field that increases with depth.

Name the inputs and units that the fluids model needs.
Separate the state you must keep from values you can calculate when needed.
Write one rule that maps the current state and inputs to an observable result.
Choose a limiting case, unit check, invariant, or known result before trusting an output.
State one assumption you would change before using this simplified model for a real decision.

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