Steady flow and streamlines
The study of fluids in motion is fluid dynamics. NCERT introduces it with a familiar observation: open a tap slowly and the water flows smoothly; open it wider and the smoothness breaks down. To make this precise we fix our attention not on a single particle but on a fixed point in space. The flow is said to be steady if, at any given point, the velocity of every fluid particle passing through that point stays constant in time.
This is a condition at fixed points, not a claim that the velocity is the same everywhere. A particle may speed up or slow down as it travels from one point to another — but every particle arriving at a given point behaves exactly like the one that passed there an instant before. The map of the flow does not change with time.
The path that a particle traces under steady flow is a streamline. NCERT defines it as a curve whose tangent at any point is in the direction of the fluid velocity at that point. A streamline is therefore a permanent map of the flow: it shows where the fluid goes, and at every point its tangent is the local velocity. Because the velocity at a point is unique, the streamline through that point is unique too.
This single-valuedness forces a key property: no two streamlines can ever cross. If they did, a fluid particle arriving at the crossing point would have two possible directions of travel, the velocity there would be ambiguous, and the flow could no longer be steady. A point cannot carry two tangents at once. The non-crossing rule is not a convention; it follows directly from the definition.
Streamline vs turbulent flow
Steady (streamline) flow is achieved only at low flow speeds. Beyond a limiting value — the critical speed — the flow loses its steadiness and becomes turbulent. NCERT's image is a fast stream meeting rocks: foamy, whirlpool-like "white-water rapids" appear where the orderly streamlines have broken into chaotic eddies. The two regimes contrast cleanly.
| Feature | Streamline (laminar) flow | Turbulent flow |
|---|---|---|
| Speed regime | Below the critical speed | Above the critical speed |
| Particle paths | Smooth streamlines that never cross | Irregular, mixing, eddying paths |
| Velocity at a point | Constant in time (steady) | Fluctuating in time |
| Streamline picture | Parallel layers; directions parallel even if magnitudes differ | Streamlines tangle and mix |
| Everyday cue | Slowly opened tap; smooth column | Stream over rocks; jet on a plate; fast tap |
| Predictability | Continuity and Bernoulli apply cleanly | Energy dissipates in eddies; idealised laws break |
The tube of flow
To turn streamlines into a quantitative law, NCERT bundles a set of them. Consider planes drawn perpendicular to the flow at three points P, R and Q. Choose the plane patches so that their boundaries are determined by the same set of streamlines. Because streamlines never cross, no fluid can leak in or out through the sides of this bundle — fluid enters only through one end-face and leaves only through the other. This bundle is a tube of flow, and it behaves like a pipe with invisible walls made of streamlines.
The crucial consequence: the number of fluid particles crossing the surfaces at P, R and Q in a given time is the same. Mass cannot accumulate inside the tube in steady flow, so whatever mass enters one face per second must leave the other face per second. This is the seed of the continuity equation.
The equation of continuity
Let the cross-sectional areas at P, R, Q be $A_P, A_R, A_Q$ and the corresponding fluid speeds be $v_P, v_R, v_Q$, with local densities $\rho_P, \rho_R, \rho_Q$. In a small interval $\Delta t$, the mass crossing the face at P is the mass of a slab of length $v_P\,\Delta t$ and area $A_P$:
$$\Delta m_P = \rho_P A_P v_P\,\Delta t.$$
The same construction holds at R and Q. Since mass flowing out of the tube equals mass flowing in — no leaks through the streamline walls, no accumulation in steady flow — these are all equal:
$$\rho_P A_P v_P\,\Delta t = \rho_R A_R v_R\,\Delta t = \rho_Q A_Q v_Q\,\Delta t.$$
Cancelling $\Delta t$ gives the general mass-conservation form $\rho A v = \text{constant}$. For an incompressible fluid the density is the same everywhere, $\rho_P=\rho_R=\rho_Q$, so $\rho$ cancels and we are left with
$$A_P v_P = A_R v_R = A_Q v_Q, \qquad \text{or in general} \qquad Av = \text{constant}.$$
This is the equation of continuity. For two cross-sections it is written
$$\boxed{A_1 v_1 = A_2 v_2}.$$
NCERT states its meaning plainly: it is a statement of conservation of mass in the flow of incompressible fluids. The quantity $Av$ is the volume flux — the volume of fluid crossing any section per unit time — and it stays constant all along the tube.
Why a narrowing pipe speeds the fluid up
Rearranging $Av=\text{constant}$ gives $v = \text{constant}/A$, so the speed is inversely proportional to the cross-sectional area. Where the tube narrows, the streamlines crowd together and the fluid must speed up; where it widens, the streamlines spread out and the fluid slows down. Halving the area doubles the speed.
NCERT notes the dynamical consequence directly from its figure: with $A_R > A_Q$, continuity forces $v_R < v_Q$, so the fluid is accelerated as it passes from the wider region R to the narrower region Q. An acceleration requires a net force, supplied by a pressure difference along the pipe — which is precisely the bridge to Bernoulli's principle. This is why a thumb over a hose nozzle, or a tap stream that thins as it falls, speeds the water up: the cross-section shrinks and continuity does the rest.
Volume flux and flow rate
The product $Av$ deserves a name of its own. It is the volume flux or volume flow rate $Q$ — the volume of fluid crossing a cross-section per unit time:
$$Q = Av \qquad (\text{SI unit: } \text{m}^3\,\text{s}^{-1}).$$
For an incompressible fluid in a tube of flow, $Q$ is the same at every section. Multiplying by density returns the mass flow rate $\rho A v$ (unit $\text{kg s}^{-1}$), which is the quantity that is fundamentally conserved. A NEET problem may quote a flow rate in litres per second, an area in $\text{cm}^2$, and a speed in $\text{m s}^{-1}$; the only discipline is to convert everything to a single consistent unit system before applying $A_1 v_1 = A_2 v_2$.
Worked examples
Water flows steadily through a horizontal pipe whose cross-sectional area falls from $8~\text{cm}^2$ to $2~\text{cm}^2$. If the speed in the wide section is $1.5~\text{m s}^{-1}$, find the speed in the narrow section and the volume flow rate.
Apply continuity. $A_1 v_1 = A_2 v_2$, so $v_2 = \dfrac{A_1 v_1}{A_2} = \dfrac{8 \times 1.5}{2} = 6~\text{m s}^{-1}$. The areas appear as a ratio, so the $\text{cm}^2$ units cancel and no conversion is needed for the speed.
Flow rate. $Q = A_1 v_1 = (8\times10^{-4}~\text{m}^2)(1.5~\text{m s}^{-1}) = 1.2\times10^{-3}~\text{m}^3\,\text{s}^{-1}$, i.e. $1.2~\text{L s}^{-1}$. The same $Q$ holds at the narrow end: $(2\times10^{-4})(6) = 1.2\times10^{-3}~\text{m}^3\,\text{s}^{-1}$. Quartering the area quadrupled the speed.
A pipe of radius $R$ branches into $n$ identical small pipes, each of radius $r$. If the speed in the main pipe is $v$, find the speed in each small pipe (incompressible, steady flow).
Mass in equals mass out. The volume flux of the main pipe is shared equally among the $n$ branches: $\pi R^2 v = n\,(\pi r^2)\,v'$.
Solve. $v' = \dfrac{R^2 v}{n r^2}$. Continuity here is applied to a junction, not a single tube — the total $Av$ entering equals the total $Av$ leaving. This branching form is a frequent NEET variant of $A_1 v_1 = A_2 v_2$.
Continuity tells you the speed change; the pressure change that accompanies it comes from Bernoulli's principle.
Where continuity leads next
The equation of continuity is the first half of a two-step story. It tells you that a narrowing pipe accelerates the fluid, but acceleration demands a net force, and that force is a pressure difference. Supplying the energy bookkeeping for that pressure change is the work of Bernoulli's principle — which is conservation of energy applied along a streamline, and which uses continuity as an input.
The other forward link concerns the boundary of validity. Everything here assumed steady, streamline flow below the critical speed. The quantitative threshold at which streamline flow gives way to turbulence is set by the dimensionless Reynolds number: low values mean laminar flow, high values mean turbulent flow. Continuity itself, being mass conservation, survives the transition; the smooth-streamline picture does not.
Streamline flow and continuity in one breath
- Flow is steady if the velocity at every fixed point is constant in time; the path a particle then traces is a streamline (tangent = local velocity).
- Streamlines never cross — a crossing point would carry two velocities and break steadiness.
- Below the critical speed flow is laminar; above it, turbulent (eddies, mixing).
- A bundle of streamlines is a tube of flow: no leaks through the sides, so mass in = mass out.
- Equation of continuity: $\rho A v=\text{constant}$ in general; $A_1 v_1 = A_2 v_2$ for an incompressible fluid. It is mass conservation.
- $v \propto 1/A$: narrower section → higher speed. $Av = Q$ is the volume flow rate, constant along the tube.
- Continuity holds even where Bernoulli's energy balance does not.