Bernoulli's Theorem & Continuity Bernoulli's Theorem & Continuity Bernoulli's Equation Terms Term Expression Energy Type Pressure Head Form Condition for Constancy Pressure, Velocity, and Height (PV-h) sum to a constant total head in perfect fluid paths. P + 1 2 v 2 + gh = constant Total mechanical energy per unit volume H = P g + v 2 2g + h (Total Head) Steady, incompressible, irrotational, and non-viscous flow Pressure energy per unit volume P g (Pressure Head) Consistent along a streamline in ideal fluids 1 2 v 2 Kinetic energy per unit volume v 2 2g (Velocity Head/Dynamic Head) Flow where velocity varies inversely with cross-section gh Potential energy per unit volume h (Static/Elevation Head) Fluid elements at different vertical heights in a gravity field P + 1 2 v 2 Static plus Dynamic pressure (Stagnation Pressure) P g + v 2 2g Applicable for horizontal flow where h 1 = h 2 bernoulli s equation terms Think of fluid motion as a careful budget of two things: mass and energy. The mass rule is the Equation of Continuity: in steady flow of an incompressible fluid, whatever volume enters a section of a pipe per second must leave the next section per second. If the pipe narrows, speed must rise so that the same volume fits through less area. The energy rule is Bernoulli’s principle: along a streamline of an ideal fluid, the sum of pressure energy per unit volume, kinetic energy per unit volume, and potential energy per unit volume stays constant. If a region of the flow speeds up, some of the available energy shifts into kinetic form, leaving less for pressure; so pressure drops where speed is higher. Together, these two rules explain why water jets speed up from a nozzle, how Venturi meters measure flow, why a wing produces lift, why a Pitot tube reads speed, and why a hole in a tank shoots out water with v = 2gh . In real life, viscosity and turbulence can steal mechanical energy and spoil the ideal picture, but even then, continuity remains a strong guide and Bernoulli still works qualitatively when losses are small. remember Garden hose = Continuity: covering the mouth makes the same water pass through a smaller area, so it speeds up. Crowded hallway = Bernoulli: when the corridor narrows, students run faster and push less on the walls, so wall pressure drops. Before formulas, settle the language. Steady (streamline) flow means each point in space sees the same velocity with time, even though different points can have different velocities. Laminar flow is a neat, layered version of steady flow. Turbulent flow is irregular, with eddies and mixing; it causes extra energy loss. For our formulas, we will mainly assume steady, incompressible, non-viscous (ideal) flow, and we will move along a single streamline. When these conditions are met well enough, the predictions match experiments beautifully. Streamline (Steady) Flow Flow in which the velocity at each fixed point does not change with time; fluid elements follow fixed streamlines. Orderly flow in layers (laminae) that slide over one another without mixing; a special case of steady flow. Laminar Flow Irregular, chaotic flow with eddies and mixing; occurs at high Reynolds number and leads to energy losses. Turbulent Flow Whether flow is laminar or turbulent is judged by the Reynolds number R e = v d . Smaller R e (narrow tube, low speed, low density, or high viscosity) tends to laminar; large R e tends to turbulent. In many biomedical flows (capillaries, small arteries), R e stays modest, so steady-flow ideas are quite useful, though pulsatility introduces additional details. Reynolds Number A dimensionless number R e = v d indicating the tendency to turbulence; small values imply laminar flow. Incompressible fluid means density is practically constant in the motion we consider. Most liquids meet this condition well; moderate-speed airflow over wings can also be treated as incompressible if the Mach number is small. This simplifies the mass balance to a simple and powerful area–speed relation. Incompressible Fluid A fluid whose density is effectively constant during flow (good approximation for liquids at moderate pressures). Mass crossing a section per second, m = A v . Mass Flow Rate Discharge (Volume Flow Rate) Q Volume crossing a section per second, Q = A v for an incompressible fluid. Equation of Continuity states A 1 v 1 = A 2 v 2 = Q . So if a tube’s area halves, the speed doubles to keep Q unchanged. Continuity applies to each connected streamtube in steady, incompressible flow. It is simply conservation of mass rewritten for a constant-density fluid. Equation of Continuity For steady, incompressible flow along a streamtube, the product of area and speed is constant. Interpretation: A v is the pipeline’s carrying capacity for volume per second. If several parallel streamlines merge or split, continuity still holds for each streamtube, and overall for the whole cross-section by summing flows. The rule does not tell us the actual value of Q ; that depends on what drives the flow (pressure difference, gravity) and on losses (viscosity, turbulence). tip Continuity limits: incompressible, steady flow, full pipe, and no sources/sinks between the sections. For gases at low speeds with small pressure change, the approximation is still often good. Bernoulli’s principle is energy conservation written per unit volume of an ideal fluid. The three terms are: static pressure P (the squeezing force the fluid exerts on walls), dynamic pressure 1 2 v 2 (kinetic energy density), and hydrostatic term g h (gravitational potential energy density). Their sum is constant along a streamline when no energy is added or removed by pumps, friction, or heat. Along a streamline for steady, incompressible, non-viscous flow: P + 1 2 v 2 + g h = constant . Bernoulli’s Equation The isotropic pressure P a fluid exerts on walls or a probe at rest with the local flow. Static Pressure The kinetic energy per unit volume: 1 2 v 2 ; equals the rise in static pressure needed to bring the flow to rest isentropically. Dynamic Pressure After dividing by g : pressure head P g , velocity head v 2 2g , and elevation head h ; their sum is constant in ideal flow. Head (Pressure/Velocity/Potential) Units check: P is in Pa , 1 2 v 2 is also Pa , and g h is Pa . Head form expresses each as a length (metres of fluid column) by dividing by g . This is convenient when using manometers. Bernoulli (pressure form) Energy conservation per unit volume along a streamline for an ideal fluid. Each term is a height (head). Manometer readings naturally give head differences. Bernoulli (head form) Bernoulli assumptions: steady flow, incompressible, inviscid, along a single streamline, no pump/turbine between points, and no significant heat exchange. If viscosity is important, a head-loss term must be added. tip P + 1 2 v 2 + g h = constant (along a streamline) Steady, incompressible, inviscid flow Analysis along a streamline No pump/turbine and negligible heat exchange Bernoulli’s Equation from the work–energy theorem Using Bernoulli between two points 1 and 2 on the same streamline gives P 1 + 1 2 v 1 2 + g h 1 = P 2 + 1 2 v 2 2 + g h 2 . Combine with continuity to eliminate one speed. In a horizontal pipe ( h 1 = h 2 ), a narrower section has higher speed and therefore lower pressure; the difference is what a Venturi meter converts into a measurable head difference. Pressure drop arises from kinetic energy gain and/or elevation change. Two-point Bernoulli Horizontal case: h 1 = h 2 . Then P 1 - P 2 = 1 2 (v 2 2 - v 1 2) . Vertical rise: even at equal speeds, pressure falls with height as g (h 2 - h 1) like hydrostatics. When both effects occur together, treat them with care and keep signs consistent. Venturi effect: as a pipe narrows to a throat, continuity forces v to rise. Bernoulli then predicts that static pressure dips in the throat. Tapping the pipe at three locations and connecting to a manometer shows the middle column standing lower than the side columns. The measured head difference maps directly to the flow speed. Venturi tube with inlet, narrow throat, and outlet. Blue streamlines crowd in the constriction; manometer taps P1, P2, P3 show the throat level (P2) lower than P1 and P3, indicating higher speed and lower pressure at the throat. Venturi effect visualization with three pressure columns and accelerated flow in the constriction. A Venturi meter has a known inlet area A 1 and throat area A 2 . Using continuity v 2 = (A 1/A 2) v 1 , and Bernoulli between the sections, the pressure difference P = P 1 - P 2 equals 1 2 (v 2 2 - v 1 2) . Solving for v 1 (or Q = A 1 v 1 ) yields the discharge in terms of measured P from a manometer. Real instruments include a discharge coefficient to account for losses; in ideal theory it is 1. Venturi discharge (ideal) If P is read as g h for a simple water manometer, replace P accordingly. For a different manometer fluid of density m , use P = ( m - ) g h . This formula applies to the steady, incompressible, and ideal flow of a fluid through a Venturi meter, derived from Bernoulli's principle. A Pitot tube faces the flow and brings it to rest at a stagnation point. Bernoulli between a moving point (speed v ) and the stagnation point (speed 0) at the same height gives P stag - P = 1 2 v 2 . Connecting a manometer between these two ports yields a head difference that directly gives v = 2(P stag -P)/ . If the manometer uses mercury with density m , then P stag -P = ( m - ) g h . Torricelli’s law is Bernoulli applied between the free surface of a large tank and a small orifice at depth h below it. The free surface moves slowly so we take v surface 0 . Then 1 2 v 2 = g h and the efflux speed is v = 2 g h . Real jets contract slightly just outside the hole (vena contracta) and viscosity causes loss, so actual speed is C v 2gh with C v slightly less than 1. Large tank so free-surface speed is negligible Same fluid at both points; heights measured from the same reference Orifice is small and horizontal Torricelli’s speed of efflux v = 2 g h Ideal efflux speed from a small orifice at depth h below the free surface. Torricelli’s law This law determines the ideal speed of efflux from an orifice based on the height difference between the free surface and the exit point. Lift on an aerofoil emerges when the flow over the top becomes faster than below. Bernoulli then gives a smaller static pressure on top and a larger one below, creating an upward resultant. In reality, wing curvature and angle of attack guide the flow to turn downward; conservation of momentum and Bernoulli are consistent stories of the same physics. For NEET, use the pressure difference P = 1 2 (v top 2 - v bottom 2) and Lift = P wing area. Wing with red fast lines on top, blue slow lines below, and an upward lift arrow. Airflow over a wing: red streamlines over the top are denser (higher speed, lower pressure) than blue streamlines below (lower speed, higher pressure). The pressure difference produces lift. Medicine link: A narrowed artery (stenosis) accelerates blood due to continuity; Bernoulli then predicts a local pressure drop. Downstream turbulence (if R e grows) dissipates energy as heat, raising the overall pressure drop needed to maintain the same flow. Measuring pressure differences and speeds by Doppler and catheterization relies on these same ideas. Dimensional hygiene saves marks. In Bernoulli, each term has dimensions of pressure. In head form, each term is a length. Discharge has units m 3 , s -1 , while speed is m , s -1 —do not mix them. For manometers, be clear about which fluid is in the gauge: P = ( m - ) g h if the manometer fluid is different from the flowing fluid. Velocity profile Smooth, often parabolic in pipes Irregular, flattened by mixing Parabolic profile underlies Poiseuille’s law Energy loss Low High (eddies, vortices) Bernoulli must include head loss if turbulent Reynolds number Typically R e 1000 R e 2000 Transition in between; depends on geometry Aspect Laminar Turbulent NEET Note Higher velocity always means lower pressure everywhere in the flow. Bernoulli links pressure and speed along a streamline when no external work or losses occur. Pumps, fans, viscous heating, or comparing different streamlines can break the simple inverse trend. Water must slow down in a narrow section due to friction. Continuity for incompressible steady flow demands higher speed in a narrower section. Friction reduces the overall flow rate Q for a given pressure drop but does not reverse the area–speed trade-off locally. neet-alert Trap: In Venturi/Pitot problems, compute pressure difference correctly. If a different manometer fluid is used, use P = ( m - ) g h , not g h . Trap: Do not apply Bernoulli between two points at the same location but on different streamlines in rotational or highly sheared flow. Stay on a single streamline unless the flow is irrotational and the constant is global. neet-alert Continuity: “A-V stays Alive.” Read it as Area × Velocity stays constant for steady incompressible pipes: A v = Q . Conservation of mass for incompressible steady flow: A 1 v 1 = A 2 v 2 = Q . Along a streamline: P + 1 2 v 2 + g h = constant . In head form, sums of heads remain constant. Static surface-tension balance: h = 2 g r . Though a static result, it often pairs with Bernoulli in fluid topics. Capillary rise height Static equilibrium Cylindrical tube of radius r Contact angle is constant Liquid density uniform h = 2 g r Capillary rise is a static balance between surface tension pull and weight; Bernoulli does not enter. It appears in the same syllabus block and shares the “head” language with Bernoulli: both finally weigh energies or forces per unit area against a height. Keep them conceptually separate: capillarity is interfacial physics; Bernoulli is flow energy. v2 and Q easy v2 in m/s, Q in m 3 /s d1 = 6.0 cm d2 = 3.0 cm v1 = 2.0 m/s Fluid: water (incompressible) A horizontal pipe narrows from diameter 6.0 cm to 3.0 cm carrying water steadily. If the speed in the wide section is 2.0 m/s, find the speed in the narrow section and the discharge Q . Use continuity: A 1 v 1 = A 2 v 2 = Q , with A = d 2/4 . Interpretation: halving the diameter quarters the area, so the speed must rise fourfold to keep the same Q . The discharge is a few litres per second ( 5.6 10 -3 , m 3/s 5.6 , L/s ). medium m 3 /s d1 = 5.0 cm d2 = 2.5 cm h = 0.10 m (mercury-water manometer) rho Hg = 13600 kg/m 3 rho = 1000 kg/ m 3 g = 9.8 m/ s 2 Venturi-type reading with different manometer fluid A Venturi meter has inlet diameter 5.0 cm and throat diameter 2.5 cm. Water flows through it. The differential mercury manometer across the two sections reads h = 10.0 cm. Find the discharge Q . Take Hg = 13 , 600 , kg/m 3 , water = 1000 , kg/m 3 , g=9.8 , m/s 2 . Use P = ( Hg - ) g h . Combine with P = 1 2 (v 2 2 - v 1 2) and continuity A 1 v 1 = A 2 v 2 . Check: a few millilitres per second would be too small; a few litres per second would be too large for this small meter. The computed 2.5 10 -3 , m 3/s equals about 2.5 , L/s , which is reasonable. A = 20 m 2 v top = 70 m/s v bottom = 60 m/s rho air = 1.2 kg/m 3 hard Lift L Use P = 1 2 (v bottom 2 - v top 2) (bottom has lower speed → higher pressure). Then L = P A . An aircraft wing of area A = 20 , m 2 experiences speeds v top = 70 , m/s and v bottom = 60 , m/s in level flight. Air density is 1.2 , kg/m 3 . Estimate the lift force due to pressure difference using Bernoulli. Wing-lift by Bernoulli Compressible-flow caution: at high speeds (large Mach number), density changes matter and the simple incompressible Bernoulli fails. In liquids, another extreme appears if pressure dips close to vapor pressure—bubbles form (cavitation) and collapse, damaging pump blades. Such edge cases are outside ideal Bernoulli and demand added physics. Split image comparing laminar straight streamlines with turbulent swirls. Side-by-side water-tank demo: left shows laminar, straight blue lines; right shows turbulent flow with swirling eddies when speed/Reynolds number is higher. As the area shrinks toward the throat, velocity head rises while pressure head dips; the sum stays nearly flat if losses are negligible. Position along Venturi (inlet → throat → outlet) 0 to L custom Energy trade-off along a Venturi in ideal flow. Relative head (pressure head and velocity head) 0 to max throat Lowest P, highest v pressure head: minimum dependent Pressure head dependent Velocity head How to attack Bernoulli problems Sketch the geometry, choose two points on the same streamline, and mark heights. Write Continuity to relate speeds if areas differ. Write Bernoulli between the two points; cancel common pressures (e.g., both at atmosphere). Insert manometer relation for pressure difference when present. Solve for the unknown (speed, discharge, or pressure). Check units and reasonableness. Useful constants and conversions Standard gravity: g 9.8 , m/s 2 . Water density near room temperature: 1000 , kg/m 3 . Mercury density: about 13 , 600 , kg/m 3 . 1 m 3/s = 1000 , L/s ; 1 L/s = 10 -3 , m 3/s . Pressure head h = P g for the same fluid; with a different manometer fluid use P = ( m - ) g h . Subtle but important: a Pitot tube measures the difference between stagnation and static pressure; a Venturi measures the difference between two static pressures caused by different local speeds. Both use Bernoulli but with different port arrangements. If you remember which port senses what, sign errors vanish. remember Big picture: Continuity sets how speeds must adjust when geometry changes. Bernoulli tells how pressure adjusts when speeds and heights change. Measurements (Venturi, Pitot) are just clever ways to convert one into the other. Volume per second through a section, Q = A v . Discharge Q Pressure felt by a probe at rest with the local flow. Static Pressure 1 2 v 2 1 2 v 2 , the kinetic energy density. Dynamic Pressure Bernoulli Constant Sum P + 1 2 v 2 + g h along a streamline. P total Flow meter using area change to convert speed into a measurable pressure difference. Venturi Meter Pitot Tube Speed probe using stagnation pressure rise to infer velocity. Efflux speed from a small hole: v = 2gh . Torricelli’s Law R e = v d , indicator of laminar vs turbulent flow. Reynolds Number Key terms recap