Fluids, Pressure and Hydrostatic Balance

What makes a material a fluid?

Water pours, air escapes from a balloon, and oil takes the shape of its bottle. Liquids and gases are fluids because they can flow. A solid can maintain its shape under a small steady shear force; an ordinary fluid cannot sustain a static shear stress and remain at rest. Continued shear makes neighbouring layers move relative to one another. A fluid can nevertheless resist that relative motion while it flows: that resistance is viscosity, studied later. “Able to flow” does not mean “offering no resistance to motion”.

A liquid has an approximately definite volume under ordinary conditions but takes the shape of its container and can have a free surface. A gas spreads through the available volume. Both can be compressed; the distinction is quantitative. Liquids and solids change volume relatively little under ordinary pressure changes, whereas gases often change volume substantially. Calling water incompressible is a useful approximation, not a claim that its volume can never change. Fluid ideas describe rivers and atmosphere as well as blood, plant sap and many other biological processes.

Pressure describes how a normal force is distributed

A sharp needle produces a much larger local effect than a broad spoon pushed with the same force because the needle shares its load over a smaller contact area. Similarly, a strong board can distribute a load across a larger area. Pressure describes normal force per unit area, rather than the total force alone. For a flat area with uniform pressure, doubling the area doubles the force without changing the pressure.

The local definition matters when the pressure varies across a large surface. Divide that surface into sufficiently small patches, find each normal force, and combine the forces with their directions. Do not use one pressure value multiplied by a large area unless the pressure is uniform or the chosen value is its appropriate average.

In a fluid at rest the force on a boundary or imaginary internal surface is perpendicular to that surface. If a tangential component remained, an ordinary fluid would shear and move, contradicting the assumed rest. On an immersed curved body, the normal direction changes from point to point; the pressure can be the same while the force directions differ. Pressure also exists between neighbouring parts of the fluid, even where there is no solid wall.

Pressure is a scalar. In F⊥/A the numerator is a normal force magnitude, not an unrestricted force vector. Its unit is the pascal: 1 Pa = 1 N/m², and its dimensions are M L⁻¹ T⁻². A pressure needs no arrow attached to it. A force calculation still needs the orientation of the patch. Do not conclude that equal scalar pressures on two differently oriented surfaces give equal force vectors.

Density and relative density

Mass density tells us the mass contained in a given volume. For example, relative density 2.7 corresponds to 2700 kg/m³ when the reference water density is 1000 kg/m³. Density and pressure are different quantities: a dense fluid is not automatically at high pressure. Both are positive scalars in the ordinary fluid states considered here. Liquid density is approximately constant for this chapter’s pressure calculations; gas density generally depends strongly on pressure and temperature.

These are the chapter’s representative density values. Its STP convention is 0°C and 1 atm; the extremely dilute interstellar entry is an illustrative comparison, not an STP gas sample. Notice the enormous density contrast between ordinary liquids and gases and between mercury and water. Always use the specified density in a numerical problem rather than treating every liquid as water.

Why pressure at a point is the same in every direction

Imagine an extremely small triangular wedge of fluid. Give its inclined face area Aₙ and let that face make angle θ with the horizontal. The vertical face has area Aₓ = Aₙ sin θ, and the horizontal face has area Aᵧ = Aₙ cos θ. The surrounding fluid pushes normally inward on each face. Take the wedge small enough that the weight, which scales with volume, is negligible compared with the face forces, which scale with area.

The angle factors cancel because the same geometry controls force components and projected areas. Rotating the imaginary wedge gives equal static pressure for any surface orientation through that point. Now isolate a horizontal fluid bar. Its end forces must balance, so equal-height points in the same connected resting liquid have equal pressure. Points separated vertically require a different balance because the intervening fluid has weight.

Deriving the pressure increase with depth

Choose a vertical cylinder of resting fluid with area A, height h and uniform density ρ. Label the upper pressure P₁ and lower pressure P₂. The pressure force on the top pushes downward, the force on the bottom pushes upward, and the weight ρAhg acts downward. The fluid is at rest, so the net vertical force must vanish.

Area cancels because doubling the cylinder area doubles both its fluid weight and its pressure forces. If the upper point is a free surface at pressure P₀ and the lower point is depth h below it, the result becomes P = P₀ + ρgh. A sealed vessel need not have P₀ equal to atmospheric pressure. With height z measured upward, the same result says pressure decreases as height increases.

The hydrostatic paradox is that containers with different shapes and water masses can have the same bottom pressure when the liquid, surface pressure and vertical depth are the same. It is pressure, not necessarily total bottom force, that is equal. For equal bottom areas their bottom forces are equal too; forces from sloping walls help balance the different fluid weights. Connected open vessels therefore settle to a common surface level. This equal-level argument requires connected fluid at rest and the same pressure over the free surfaces.

In a standing person, the feet are below the brain. The hydrostatic contribution to blood pressure is therefore larger at the feet. Real circulation includes a beating heart and flowing, viscous blood; the depth law identifies the gravitational contribution rather than a complete circulatory model.

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