Steady Flow, Continuity and Bernoulli’s Principle

Steady does not mean the same speed everywhere

Turn a tap on gently and the jet is initially smooth. Increase the flow and the jet becomes irregular. Fluid dynamics studies such motion. Instead of following every molecule, we describe the velocity at each position. Flow is steady when the velocity at any fixed point does not change with time: successive fluid elements passing that point have the same velocity. That velocity can still differ from the velocity at another point. A particle can therefore accelerate as it moves through a steady, narrowing pipe.

A streamline is a curve whose tangent at each point has the direction of the local fluid velocity. In steady flow, the streamline pattern stays fixed and a moving fluid particle follows a streamline. At an ordinary point with nonzero velocity two distinct streamlines cannot cross: a crossing would assign two velocity directions to the same point at the same instant. The drawing is a map of a velocity field, not a set of physical strings inside the liquid.

Laminar motion has orderly layers that slide past one another. Adjacent layers can move at different speeds; equal speed everywhere is not required. At sufficiently large speeds for a given geometry and fluid, irregular eddies and fluctuations can develop: this is turbulent motion. River rapids around rocks are a familiar example. The critical speed depends on the fluid and the size and geometry of the flow; it is not one universal speed for all liquids and pipes.

Deriving continuity from conservation of mass

Imagine a tube whose side boundary is made from streamlines. Fluid does not cross that side boundary because velocity is tangent to it. In a short time Δt, fluid at a section of area A travels distance vΔt. The volume that crosses is AvΔt when velocity is approximately uniform over that section; for a nonuniform profile use the cross-sectional mean speed. Multiplying by density gives the transported mass ρAvΔt.

In steady flow no mass accumulates inside the tube segment. The mass entering during Δt must equal the mass leaving. Cancel the same time interval to obtain ρ₁A₁v₁ = ρ₂A₂v₂. If density is constant, it cancels too. The resulting volume flow rate Q = Av is then the same at every section of a single unbranched tube.

If a circular pipe’s diameter halves, its area becomes one quarter and its mean speed becomes four times as large for the same volume flow. Do not use a simple inverse-diameter rule. Closely spaced streamlines can show a small stream-tube area and higher speed when the drawn lines represent equal flow portions. Arbitrarily drawing extra lines does not physically increase velocity.

Why a pressure difference is needed to change speed

Continuity determines how velocities at different sections are related. It does not by itself explain the force that produces the acceleration. In a horizontal ideal-fluid constriction, the surrounding fluid exerts a net pressure force that accelerates the flowing material. To relate that pressure change to speed and elevation, apply conservation of mechanical energy. This is Bernoulli’s principle.

The pressure-work derivation of Bernoulli’s equation

Choose two sections of a pipe along a streamline. Let their pressures be P₁ and P₂, their heights above a common datum be y₁ and y₂, and their speeds be v₁ and v₂. Assume steady incompressible flow with negligible viscous dissipation. During a short interval Δt, the same volume ΔV enters section 1 and leaves section 2. Most of the fluid between the sections is replaced by identical fluid in the same steady state. The net energy change can therefore be found by comparing the entering volume with the leaving volume.

At the inlet, the pressure force is P₁A₁ in the direction of motion. Its point of application moves v₁Δt, so work done on the fluid is P₁A₁v₁Δt = P₁ΔV. At the outlet, the downstream fluid pushes back against the flow. Work done on the chosen fluid there is −P₂A₂v₂Δt = −P₂ΔV. Thus the net pressure work is (P₁ − P₂)ΔV. The minus sign at the outlet is essential.

The transported mass is Δm = ρΔV. Its kinetic energy changes by one half this mass times v₂² − v₁². Its gravitational potential energy changes by the same mass times g(y₂ − y₁). Pressure work supplies both changes. Equivalently, if gravity is counted as work, W_pressure + W_gravity = ΔK; moving the gravity term to the other side gives W_pressure = ΔK + ΔU.

Divide by the common nonzero ΔV and rearrange so that the section-1 terms stand on one side and section-2 terms on the other. The result states that a specific sum stays constant along the streamline. It is conservation of energy per unit volume, not an additional conservation law unrelated to mechanics.

Using the equation without turning it into a slogan

At the same elevation along a suitable streamline, higher speed corresponds to lower static pressure. If the heights differ, the ρgy change must also be included: “faster always means lower pressure” is not a universal rule. In static fluid v = 0, Bernoulli reduces to P₁ + ρgy₁ = P₂ + ρgy₂, which reproduces the hydrostatic pressure difference. This is a useful consistency check on signs.

Gauge pressures may replace absolute pressures if both are measured relative to the same atmospheric pressure: subtracting a common constant from both sides leaves the equation unchanged. If the reference atmospheric pressures are different, those different subtractions do not cancel. Keep absolute pressures or include the difference explicitly.

Viscous drag converts organised mechanical energy into internal energy. In a real pipe an extra pressure drop is needed to maintain flow; the downstream pressure is lower than a loss-free prediction for otherwise comparable conditions. Turbulent rapids have substantial dissipation and fluctuating velocities, so the elementary steady Bernoulli formula cannot be applied directly as a complete description. Low viscosity does not justify ignoring every loss in every geometry.

A vertical open tube attached to a horizontal pipe reads the local pressure through its stationary liquid height. With the same liquid in both tubes, a taller column means higher static pressure. At the constriction continuity predicts greater speed; Bernoulli then predicts lower pressure and a shorter column. A picture with a taller constriction column is incompatible with those assumptions. Decide the speed relation first, then the pressure relation; do not infer everything from Bernoulli alone.

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