1. What is “steady” or streamline flow?
When you open a tap gently, water glides out in smooth, well-behaved layers. At any fixed point the speed of each tiny water blob stays the same with time, although blobs at different points can move at different speeds. The smooth path traced by one blob is called a streamline. Because directions never clash, two streamlines can’t cross – if they did, a blob would have to wiggle in two directions at once! :contentReference[oaicite:0]{index=0}
- Visual tip : Where streamlines crowd together the fluid is racing; where they spread out, it saunters.
- Analogy : Imagine lanes on a highway – cars (fluid particles) stick to their lanes and never bump sideways into other lanes.
2. Equation of continuity – “what goes in must come out”
Conservation of mass insists that, during a tiny time slice \(\Delta t\), the mass flowing into one cross-section equals the mass leaving any other. Mathematically
\[ \rho_P A_P v_P \,\Delta t \;=\; \rho_R A_R v_R \,\Delta t \;=\; \rho_Q A_Q v_Q \,\Delta t \] :contentReference[oaicite:1]{index=1}
For an incompressible fluid the density \(\rho\) is the same everywhere, giving
\[ A_P v_P \;=\; A_R v_R \;=\; A_Q v_Q \]
or, in its most compact and memorable form,
\[ A\,v \;=\;\text{constant}\tag{9.11} \]
- The quantity \(A v\) is the volume flow rate (sometimes called flux). It stays fixed along the pipe. :contentReference[oaicite:2]{index=2}
- Narrow pipe (small \(A\)) → big speed \(v\). Wide pipe (large \(A\)) → gentle speed. That’s why rivers roar through gorges yet drift across wide plains.
3. Laminar vs turbulent flow
| Laminar (layered) flow | Directions are parallel; neighbouring layers slide smoothly past each other. The sketch in Fig. 9.8(a) shows evenly spaced streamlines. :contentReference[oaicite:3]{index=3} |
| Turbulent flow | Above a certain critical speed swirling eddies spring up, as in white-water rapids or behind a plate hit by a jet of air [Fig. 9.8(b)]. :contentReference[oaicite:4]{index=4} |
Key point : Whenever the exam mentions critical speed, think “onset of turbulence.”
4. Bernoulli’s principle – the energy balance along a streamline
Speed changes, height changes, and pressure changes are all tied together. If a fluid speeds up, some of its pressure energy converts into kinetic energy; if it climbs, pressure energy converts into gravitational potential energy. In steady flow the total energy per unit volume (pressure term + kinetic term + potential term) stays constant along any streamline. :contentReference[oaicite:5]{index=5}
Handy rule of thumb : Fast flow → lower pressure; slow flow → higher pressure.
5. Everyday examples you already know
- Hydraulic lift (Example 9.6) – A small force \(F_1\) on a tiny piston (radius 5 cm) raises a car sitting on a larger piston (radius 15 cm). Same pressure acts on both pistons, so the bigger area multiplies the force. :contentReference[oaicite:6]{index=6}
- Hydraulic brakes – Pressing the pedal raises the pressure of brake oil; that pressure reaches cylinders at all four wheels equally, giving balanced braking. :contentReference[oaicite:7]{index=7}
6. Quick checkpoints
- Streamlines never intersect.
- For incompressible flow, \(A v\) is constant along the pipe.
- If the pipe’s diameter halves, speed quadruples (area ∝ diameter²).
- Laminar turns turbulent once speed passes the critical value.
- Within a horizontal pipe, faster sections sit at lower pressure.
High-Yield Ideas for NEET
- Equation of continuity: \(A v = \text{constant}\) – expect numerical questions that mix area and speed.
- Critical speed & flow types: recognising when to treat a situation as laminar or turbulent.
- Streamline properties: tangent gives instantaneous velocity; streamlines never cross.
- Bernoulli’s energy picture: pressure drops in fast flow or at higher elevations.
Keep practising with small, varied problems – confidence grows as quickly as water speeds up in a narrow jet!
