Partly covering a hose outlet can make the emerging stream faster. That observation involves both flow rate and energy; it is not permission to say “fast fluid always has low pressure.” We will identify the conditions behind the useful rule.
We will derive both relations, combine them in concrete flows and examine where extra information is needed. The aim is to predict the flow from a model, rather than attach “faster means lower pressure” to every situation.
What you will learn to do
- Distinguish steady flow from constant parcel speed.
- Conserve mass through sections, branches and storage.
- Derive the pressure-work form of Bernoulli’s equation.
- Include elevation before comparing pressure and speed.
- Connect efflux to projectile motion and a changing tank level.
- Interpret stagnation pressure and account for pumps or losses.
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Continuity and energy in flowing fluids Learning goals: - Distinguish steady flow from constant parcel speed. - Conserve mass through sections, branches and storage. - Derive the pressure-work form of Bernoulli’s equation. - Include elevation before comparing pressure and speed. - Connect efflux to projectile motion and a changing tank level. - Interpret stagnation pressure and account for pumps or losses. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
BEFORE YOU READ
Bring back one idea
Try a sentence or a sketch before opening the comparison. If this is new, use the explanation to get started.
In steady incompressible flow through one unbranched tube, what must cross every section each second?
Compare your reasoning
The same volume of fluid must cross each section, assuming no leaks or accumulation. Thus Av is constant for the section-averaged speed. This is a conservation statement.
Revisit the explanation →01 / Continuity and energy in flowing fluids
Steady flow can still accelerate a moving fluid parcel
A fluid velocity field assigns a velocity to each position and time. In steady flow, the velocity at a fixed location does not change with time. A moving parcel can nevertheless accelerate because it travels into locations where the velocity is different. Water speeding up through a narrowing pipe can therefore be a steady flow.
A streamline is a curve tangent to the instantaneous velocity field. In steady flow it also traces a parcel’s path. The familiar Bernoulli relation will connect points along a streamline under additional assumptions. It should not be applied indiscriminately between points belonging to different flows or across a pump that adds energy.
Before calculating, specify whether density can be treated as constant, whether viscosity and losses are negligible and whether speed is approximately uniform across each section. The simple pipe equations below use cross-section average speed for continuity and a uniform-speed idealisation for the elementary energy balance.
STOP & THINK
In steady flow through a narrowing pipe, can a fluid parcel’s speed increase as it moves?
- No; steady means each parcel has constant speed
- Yes; the velocity can vary with position while remaining constant in time at each fixed point
- Only if mass disappears
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Continuity and energy in flowing fluids Focus: Steady flow can still accelerate a moving fluid parcel Learning goals: - Distinguish steady flow from constant parcel speed. - Conserve mass through sections, branches and storage. - Derive the pressure-work form of Bernoulli’s equation. - Include elevation before comparing pressure and speed. - Connect efflux to projectile motion and a changing tank level. - Interpret stagnation pressure and account for pumps or losses. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
02 / Continuity and energy in flowing fluids
Conservation of mass links the flow through successive sections
During a short interval Δt, fluid crossing a section of area A with uniform normal speed v occupies volume AvΔt. Its mass is ρAvΔt. Dividing by time gives volume flow rate Q = Av and mass flow rate ṁ = ρAv. For a nonuniform speed profile, integrate the normal velocity over the section or use the appropriate average.
In a steady unbranched pipe with no leaks or accumulation, the same mass passes every section per unit time. Thus ρ₁A₁v₁ = ρ₂A₂v₂. If density is constant, the relation reduces to A₁v₁ = A₂v₂. A smaller area then requires a larger average speed.
The diameter is not the area. For a circular pipe A = πd²/4, so halving diameter quarters area and multiplies speed by four at unchanged incompressible flow rate. Mixing diameter and area ratios is a frequent source of plausible-looking wrong answers.
Q has units m³/s; mass flow rate has units kg/s. Equal volume flow requires equal density as well as the no-accumulation condition.
WORKED EXAMPLE
Convert a diameter change into an area change
Water flows steadily through a pipe whose diameter narrows from 4 cm to 2 cm. The average speed in the wide section is 1.5 m/s. Assume incompressibility and no leakage. Find the narrow-section speed.
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STOP & THINK
A liquid has average normal speed 2 m/s through area 3 cm². What is the volume flow rate in litres per second?
Answer in L/s.
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Continuity and energy in flowing fluids Focus: Conservation of mass links the flow through successive sections Learning goals: - Distinguish steady flow from constant parcel speed. - Conserve mass through sections, branches and storage. - Derive the pressure-work form of Bernoulli’s equation. - Include elevation before comparing pressure and speed. - Connect efflux to projectile motion and a changing tank level. - Interpret stagnation pressure and account for pumps or losses. Equations in this section (LaTeX): \dot m=\rho Av,\qquad Q=Av,\qquad\rho_1A_1v_1=\rho_2A_2v_2 Meaning: Q has units m³/s; mass flow rate has units kg/s. Equal volume flow requires equal density as well as the no-accumulation condition. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
03 / Continuity and energy in flowing fluids
Include branches and storage before equating two flow rates
At a steady junction, total incoming mass flow equals total outgoing mass flow. For a single incompressible liquid, the same statement applies to volume flow. The inlet need not match each individual outlet; it matches their sum.
If a tank is filling or draining, it stores or loses fluid. Then incoming and outgoing rates need not be equal. The difference gives the rate of change of stored mass or, for constant density, volume. For a vertical tank with constant cross-sectional area A, volume change is A dh, so A dh/dt = Qin − Qout.
For a compressible gas, even steady flow can have different volume flow rates at two sections because density changes. Mass flow remains the conserved quantity. The continuity relation follows from conservation, while the assumption of incompressibility is an extra simplification.
WORKED EXAMPLE
A mismatch of flow rates changes the water level
A vertical tank has constant cross-sectional area 0.50 m². Water enters at 3 L/s and leaves at 1 L/s. Find the rate at which the level rises. Assume uniform incompressible water and no other flow.
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Continuity and energy in flowing fluids Focus: Include branches and storage before equating two flow rates Learning goals: - Distinguish steady flow from constant parcel speed. - Conserve mass through sections, branches and storage. - Derive the pressure-work form of Bernoulli’s equation. - Include elevation before comparing pressure and speed. - Connect efflux to projectile motion and a changing tank level. - Interpret stagnation pressure and account for pumps or losses. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
04 / Continuity and energy in flowing fluids
Pressure work changes kinetic and gravitational energy
Consider an equal small volume of incompressible fluid entering one section and leaving another in steady flow. The upstream pressure does positive work p₁ΔV on the fluid; the downstream pressure contributes negative work −p₂ΔV. If no viscous dissipation, pump or turbine acts between the sections, the net pressure work changes the parcel’s kinetic and gravitational energy.
Divide the energy balance by ΔV. The result is p + ½ρv² + ρgy = constant along a streamline. Each term has units of energy per volume, which are the same as pressure. The equation can also be divided by ρg to express pressure head, velocity head and height in metres.
The three terms are not independent reservoirs that must each remain constant. Pressure can decrease while kinetic energy density increases, or height can change and alter both. The simple “faster means lower pressure” conclusion follows only after the other conditions, especially equal height and a common streamline with no added or lost energy, are established.
Use steady, incompressible, inviscid flow along a streamline with no shaft-work input or extraction between the points. Elementary section calculations assume a suitable uniform-speed model.
WORKED EXAMPLE
Combine continuity with energy in a horizontal constriction
Water of density 1000 kg/m³ flows steadily through an ideal horizontal pipe. Its speed changes from 2 m/s to 5 m/s. The upstream pressure is 150 kPa absolute. Neglect losses. Find the downstream pressure.
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Continuity and energy in flowing fluids Focus: Pressure work changes kinetic and gravitational energy Learning goals: - Distinguish steady flow from constant parcel speed. - Conserve mass through sections, branches and storage. - Derive the pressure-work form of Bernoulli’s equation. - Include elevation before comparing pressure and speed. - Connect efflux to projectile motion and a changing tank level. - Interpret stagnation pressure and account for pumps or losses. Equations in this section (LaTeX): p_1+\frac12\rho v_1^2+\rho gy_1=p_2+\frac12\rho v_2^2+\rho gy_2 Meaning: Use steady, incompressible, inviscid flow along a streamline with no shaft-work input or extraction between the points. Elementary section calculations assume a suitable uniform-speed model. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
05 / Continuity and energy in flowing fluids
A speed comparison alone cannot determine a pressure comparison
At different heights, gravitational energy matters. A flow descending to a lower level can gain pressure even if its speed also increases, provided the loss of gravitational energy per volume exceeds the kinetic-energy gain. Therefore the sign of a pressure difference must come from the full balance.
Use one vertical reference consistently. Only height differences matter, so setting one level to zero is convenient. A downward movement has y₂ − y₁ < 0 and contributes positively to p₂ when the equation is rearranged for downstream pressure.
As a limiting check, set both speeds to zero. Bernoulli reduces to p₂ − p₁ = ρg(y₁ − y₂), the hydrostatic relation already derived from static force balance. An energy equation that fails this limit has a height sign error.
WORKED EXAMPLE
A faster flow can still have higher pressure
An ideal steady water flow descends by 2 m while its speed increases from 1 m/s to 3 m/s. Take ρ = 1000 kg/m³ and g = 10 m/s². Find p₂ − p₁.
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Continuity and energy in flowing fluids Focus: A speed comparison alone cannot determine a pressure comparison Learning goals: - Distinguish steady flow from constant parcel speed. - Conserve mass through sections, branches and storage. - Derive the pressure-work form of Bernoulli’s equation. - Include elevation before comparing pressure and speed. - Connect efflux to projectile motion and a changing tank level. - Interpret stagnation pressure and account for pumps or losses. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
06 / Continuity and energy in flowing fluids
A large reservoir converts height into outlet speed
For a large tank open to the atmosphere with a small side opening, apply Bernoulli from the free surface to the emerging jet. Both points have approximately atmospheric pressure. If the free surface moves much more slowly than the jet and losses are negligible, its kinetic term can be neglected. A vertical depth h below the free surface then gives the ideal outlet speed √(2gh).
This is Torricelli’s result. The relevant height is the vertical difference between free surface and opening, not the opening’s height above the floor. Once the jet has left, its later trajectory is a projectile-motion problem under gravity and air-resistance assumptions.
A real sharp-edged opening can contract the jet and dissipate energy. Its volume flow rate need not equal the geometric hole area multiplied by the ideal speed without a discharge correction. Here we explicitly use the ideal opening and quasisteady large-reservoir model.
The free-surface area must greatly exceed outlet area a for its speed to be negligible. The approximation also neglects losses and jet contraction.
WORKED EXAMPLE
Separate outlet speed from the later flight time
A large open tank has a small horizontal outlet 1.8 m below the free surface and 0.80 m above a level floor. Use g = 10 m/s², ideal efflux and negligible air resistance. Find the ideal horizontal distance travelled by the jet before reaching the floor.
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Continuity and energy in flowing fluids
Focus: A large reservoir converts height into outlet speed
Learning goals:
- Distinguish steady flow from constant parcel speed.
- Conserve mass through sections, branches and storage.
- Derive the pressure-work form of Bernoulli’s equation.
- Include elevation before comparing pressure and speed.
- Connect efflux to projectile motion and a changing tank level.
- Interpret stagnation pressure and account for pumps or losses.
Equations in this section (LaTeX):
v_{\mathrm{out}}\approx\sqrt{2gh},\qquad Q\approx a\sqrt{2gh}
Meaning: The free-surface area must greatly exceed outlet area a for its speed to be negligible. The approximation also neglects losses and jet contraction.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.07 / Continuity and energy in flowing fluids
The fastest jet need not travel farthest horizontally
Suppose the free surface is a fixed height H above the floor and an outlet is at height y above the floor. Its pressure head is H − y, while its flight time depends on y. The horizontal range is therefore 2√[y(H − y)] in the ideal model.
A low outlet gives a high speed but a short time before the jet hits the floor. A high outlet gives more flight time but a smaller pressure head. The product y(H − y) is largest at y = H/2, found by completing the square or differentiating.
This optimisation assumes the water level is effectively fixed during the observation and the jet leaves horizontally. Obstructions, a draining level, nonhorizontal outlets or different downstream pressure require revisiting the model. A formula derived for one geometry is not a universal nozzle-design rule.
Compare outlet height with ideal landing distance
- Ideal range
- 0
WORKED EXAMPLE
Optimise the product rather than speed alone
A large open tank maintains its surface at H = 3.2 m above a floor. Under the ideal horizontal-efflux model, at what outlet height is range greatest, and what is that maximum range?
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Continuity and energy in flowing fluids Focus: The fastest jet need not travel farthest horizontally Learning goals: - Distinguish steady flow from constant parcel speed. - Conserve mass through sections, branches and storage. - Derive the pressure-work form of Bernoulli’s equation. - Include elevation before comparing pressure and speed. - Connect efflux to projectile motion and a changing tank level. - Interpret stagnation pressure and account for pumps or losses. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
08 / Continuity and energy in flowing fluids
Continuity can turn an instantaneous flow law into a time history
In a tank draining through a small bottom outlet, the driving head h decreases with time. The outlet speed and flow rate therefore decrease. Using the initial flow rate for the entire process would underestimate the drain time.
For a vertical tank of constant area A and a small ideal outlet a, continuity gives A dh/dt = −a√(2gh). Separate variables: dh/√h = −(a/A)√(2g)dt. Thus √h decreases linearly with time even though h itself does not.
The resulting expression applies until h first reaches zero, and only while the quasisteady ideal-efflux assumptions remain suitable. Near the end, real flow geometry and losses can matter. Extending the squared formula beyond the calculated emptying time would falsely predict the tank filling itself.
Use the interval with nonnegative square-root height. The fixed tank area and small outlet are part of the derivation.
WORKED EXAMPLE
Find a changing-flow drain time
An ideal vertical tank of area 0.20 m² initially contains liquid to height 0.80 m above a small outlet of effective area 0.0010 m². Use g = 10 m/s², incompressible quasisteady ideal flow and negligible jet contraction. Estimate the emptying time.
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Continuity and energy in flowing fluids
Focus: Continuity can turn an instantaneous flow law into a time history
Learning goals:
- Distinguish steady flow from constant parcel speed.
- Conserve mass through sections, branches and storage.
- Derive the pressure-work form of Bernoulli’s equation.
- Include elevation before comparing pressure and speed.
- Connect efflux to projectile motion and a changing tank level.
- Interpret stagnation pressure and account for pumps or losses.
Equations in this section (LaTeX):
\sqrt{h(t)}=\sqrt{h_0}-\frac{a}{2A}\sqrt{2g}\,t,\qquad t_{\mathrm{empty}}=\frac Aa\sqrt{\frac{2h_0}{g}}
Meaning: Use the interval with nonnegative square-root height. The fixed tank area and small outlet are part of the derivation.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.09 / Continuity and energy in flowing fluids
A stagnation measurement converts speed into a pressure rise
A fluid brought smoothly to rest along an ideal streamline at unchanged height has its kinetic-energy density converted into a pressure increase. Its stagnation pressure is p₀ = p + ½ρv². The difference between stagnation and local static pressure can therefore estimate speed under the model’s conditions.
A probe facing the flow samples a stagnation condition; a properly arranged side pressure tap samples static pressure. Confusing the two readings can lead to treating p + ½ρv² as the static pressure everywhere. They are different measurements.
For compressible or strongly lossy flow, the elementary incompressible relation needs correction. A pressure difference also has a sign: the ideal stagnation pressure is greater than or equal to static pressure, with equality at zero speed. A negative inferred v² signals inconsistent data, reference pressures or assumptions.
WORKED EXAMPLE
Infer speed from two pressure readings
An ideal incompressible water flow has stagnation pressure 2.0 kPa above its static pressure at the same elevation. Take density 1000 kg/m³. Find the speed.
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Continuity and energy in flowing fluids Focus: A stagnation measurement converts speed into a pressure rise Learning goals: - Distinguish steady flow from constant parcel speed. - Conserve mass through sections, branches and storage. - Derive the pressure-work form of Bernoulli’s equation. - Include elevation before comparing pressure and speed. - Connect efflux to projectile motion and a changing tank level. - Interpret stagnation pressure and account for pumps or losses. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
10 / Continuity and energy in flowing fluids
Pumps and losses belong in the energy balance
A pump adds mechanical energy to a flow, while a turbine extracts it and viscous effects dissipate it. The simple constant Bernoulli sum cannot be carried unchanged across those elements. In practical head accounting, add pump head and subtract loss or extracted head with consistent signs.
For an incompressible flow rate Q lifted through a net required head H, the useful hydraulic power is ρgQH. If the drive efficiency is η, input power is useful power divided by η. This uses the energy transferred to each cubic metre multiplied by the cubic metres transferred per second.
Use the ideal equation to understand the transfers and limiting cases, then state where reality needs more information: velocity profiles, losses, compressibility, unsteadiness or vapour-pressure limits. A calculated pressure that the assumed liquid state cannot sustain is a warning that the flow model must be reconsidered, not an instruction to report an impossible pressure as a physical prediction.
WORKED EXAMPLE
Include the head lost as well as the height gained
A pump moves water at 0.002 m³/s between open reservoirs with a 5 m surface-height difference. The required loss head is 1 m, and surface speeds are negligible. Take ρ = 1000 kg/m³, g = 10 m/s² and overall efficiency 60%. Find useful hydraulic power and input power.
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STOP & THINK
Can the simple lossless Bernoulli constant be used unchanged across a working pump?
- Yes, because the liquid is incompressible
- No; the pump adds energy that must appear in the balance
- Only if pressure is measured in kPa
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Continuity and energy in flowing fluids Focus: Pumps and losses belong in the energy balance Learning goals: - Distinguish steady flow from constant parcel speed. - Conserve mass through sections, branches and storage. - Derive the pressure-work form of Bernoulli’s equation. - Include elevation before comparing pressure and speed. - Connect efflux to projectile motion and a changing tank level. - Interpret stagnation pressure and account for pumps or losses. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
BRING THE IDEAS TOGETHER
What should stay with you
- Continuity conserves mass; equal volume flow also needs constant density.
- Area ratios use squared diameter ratios for circular sections.
- Bernoulli’s simple constant applies along a streamline under its stated ideal assumptions.
- Pressure comparisons require speed and height terms together.
- A draining head changes the flow rate with time.
- Pumps, losses and compressibility require a broader energy model.
RETRIEVE, THEN EXPLAIN
Try it without the examples
Work these out before checking. If an answer surprises you, identify which assumption or step needs another look. A correct choice is most useful when you can explain why the alternatives fail.
SELF CHECK 1
An incompressible steady flow passes from area 6 cm² at 2 m/s to area 3 cm². What is the second average speed?
Answer in m/s.
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SELF CHECK 2
For an ideal small opening 0.45 m below a large open reservoir’s surface, what is the outlet speed if g = 10 m/s²?
Answer in m/s.
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SELF CHECK 3
A steady incompressible junction has one inlet and two outlets. What must be equal?
- Inlet volume flow and each outlet separately
- Inlet volume flow and the sum of outlet flows
- All three speeds regardless of area
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SELF CHECK 4
Why can a descending flow become faster and also gain static pressure?
- Gravity can supply more energy per volume than the kinetic-energy increase
- Bernoulli never includes height
- Pressure is a vector that rotates
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SELF CHECK 5
Why is initial tank volume divided by initial outlet rate generally too short a drain-time estimate for a gravity-fed tank?
- The outlet rate decreases as the head falls
- The liquid mass is not conserved
- The initial rate is always zero
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APPLY THE CHAPTER
Take the ideas into a problem
Try a complete problem next. Commit to a method before revealing a hint, then compare your reasoning with the worked solution.
Pressure pays for a speed increaseIntermediate · Free sample · Hints and guided reasoningWhat happens when the pipe narrows?Beginner · Hints and guided reasoningSpeed from a pressure headIntermediate · Hints and guided reasoningSize the narrow sectionIntermediate · Hints and guided reasoningA draining jet also fallsAdvanced · Hints and guided reasoningCOME BACK TO THE IDEA
What can you explain later?
Try these from memory at your next study session, then again a few days later. Say your reasoning aloud or sketch it before looking. Adjust the gap to what you can explain.
Next study session
Along a horizontal streamline in ideal steady flow, why can increased speed accompany decreased pressure?
Compare your reasoning
The increase in kinetic energy per volume is balanced by a decrease in pressure in Bernoulli’s relation. The conclusion uses the same streamline and excludes added pumps and dissipative losses.
Revisit the explanation →A few days later: change the situation
Would you apply the same loss-free Bernoulli constant across a pump or a long resistive pipe? Explain.
Compare your reasoning
A pump adds energy and viscosity dissipates mechanical energy, so the simple unchanged constant is insufficient. Add the relevant energy transfer or loss terms rather than ignore the apparatus.
Revisit the explanation →Compare the physical reason, the assumptions and the conclusion. Your wording can differ. If one part is missing, revisit that section and try explaining it again after a break.
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