Fluids and Conservation Laws

Learning Goals

After this lesson, you should be able to:

• calculate volume flow rate from volume, time, area, and average speed;
• derive and apply the continuity equation for an incompressible fluid;
• use area ratios and radius ratios to compare speeds;
• interpret pressure as mechanical energy transferred per unit volume;
• derive and apply Bernoulli's equation along an ideal-fluid flow path;
• compare pressure, speed, and height using energy-density bar charts;
• derive and apply Torricelli's theorem for a large open tank;
• design experiments that test continuity or Torricelli's prediction; and
• recognize when viscosity, compressibility, unsteady flow, or an unfilled pipe invalidates the simple model.

1. Model Assumptions Come First

Unless stated otherwise, AP Physics 1 models the fluid as ideal and the pipe as completely filled. For this lesson:

• Incompressible: density remains constant;
• Nonviscous: no mechanical energy is dissipated by internal friction;
• Steady flow: conditions at a fixed location do not change with time; and
• Filled pipe: the relevant cross section is occupied by flowing fluid.

Continuity follows from mass conservation. Bernoulli additionally requires negligible dissipative transfer along the compared flow path.

3. A Pressure Difference Can Drive Flow

Different pressures on opposite faces of a fluid region produce a net pressure force. If that force is not balanced by gravity or other forces, the fluid accelerates and begins to flow.

Once a steady flow is established, pressure, speed, and height are linked by conservation laws. “Fluid always flows from high pressure to low pressure” is incomplete when gravity, changing area, or external devices also transfer energy.

4. Volume Flow Rate

Volume flow rate is the volume crossing a section per time:

Its SI unit is \(\mathrm{m^3/s}\). A useful conversion is

Flow rate is not fluid speed. \(Q\) measures volume per time; \(v\) measures distance per time.

5. Deriving \(Q=Av\)

In time \(\Delta t\), fluid with average speed \(v\) travels distance \(\Delta x=v\Delta t\). Through a cross-sectional area \(A\), the swept volume is

Divide by time:

The velocity component perpendicular to the cross section is relevant. In AP problems, \(v\) usually denotes the average flow speed across that section.

6. Mass Flow Rate

Since \(\Delta m=\rho\Delta V\),

The SI unit of mass flow rate is \(\mathrm{kg/s}\). For steady flow through a tube without sources, sinks, or leaks, the mass entering per time equals the mass leaving per time:

7. Continuity for an Incompressible Fluid

For an incompressible fluid, \(\rho_1=\rho_2\), so density cancels:

A smaller cross-sectional area requires a larger average speed. Continuity alone does not calculate pressure; it relates flow rate, area, and speed.

8. Radius and Diameter Ratios Are Squared

For a circular pipe, \(A=\pi r^2=\pi d^2/4\). Therefore,

If a pipe radius is halved, its area becomes one fourth and its speed becomes four times as large for the same incompressible steady flow—not twice as large.

9. Worked Example: A Narrowing Pipe

Water flows at \(1.50\,\mathrm{m/s}\) through a pipe of radius \(3.00\,\mathrm{cm}\), then enters a section of radius \(1.50\,\mathrm{cm}\).

Answer: The narrow-section speed is \(6.00\,\mathrm{m/s}\), and the flow rate is \(4.24\times10^{-3}\,\mathrm{m^3/s}\), or \(4.24\,\mathrm{L/s}\).

10. Branches and Conservation of Flow

At a junction in steady incompressible flow, total incoming volume flow rate equals total outgoing volume flow rate:

If one incoming pipe carries \(5.0\,\mathrm{L/s}\) and one branch carries away \(1.8\,\mathrm{L/s}\), the remaining branch carries \(3.2\,\mathrm{L/s}\). Apply \(Q=Av\) separately to find a branch speed.

11. Pressure Work Transfers Energy

Pressure force can do work on a moving fluid volume. When a volume \(\Delta V=A\Delta x\) enters at pressure \(P\),

Thus pressure has units of energy per volume:

This makes pressure compatible with the other terms in Bernoulli's equation.

12. Deriving Bernoulli's Equation

Follow an equal volume \(\Delta V\) of incompressible fluid from point 1 to point 2. Net pressure work is

For a nonviscous flow, this work changes kinetic and gravitational potential energy:

Divide by \(\Delta V\) and rearrange:

13. The Three Bernoulli Terms

If one term increases along the ideal flow path, one or both other terms must decrease by the same total amount.

14. Level Flow: Speed and Pressure

At equal heights, \(y_1=y_2\), so Bernoulli reduces to

Along the same ideal level flow, greater speed corresponds to lower pressure. This conclusion requires the Bernoulli assumptions and comparison along the relevant flow path; it is not a universal rule for every pair of points in every moving fluid.

15. Worked Example: Pressure in a Constriction

Water flows through a level pipe. At point 1, \(v_1=2.0\,\mathrm{m/s}\) and \(P_1=1.80\times10^5\,\mathrm{Pa}\). At a narrower point 2, \(v_2=6.0\,\mathrm{m/s}\).

Answer: The narrow-section pressure is \(1.64\times10^5\,\mathrm{Pa}\), a decrease of \(1.6\times10^4\,\mathrm{Pa}\).

16. Equal Area but Different Height

If a constant-area pipe carries steady incompressible flow, continuity gives \(v_1=v_2\). Bernoulli then gives

For water rising \(4.0\,\mathrm m\),

The pressure must decrease to supply the increase in gravitational potential energy per volume when speed is unchanged.

17. Bernoulli Energy-Density Bar Charts

At each selected point, draw bars for \(P\), \(\frac12\rho v^2\), and \(\rho gy\) using one common height zero and one consistent pressure reference. Choose convenient references that keep the displayed bars nonnegative when possible; otherwise represent a negative referenced value explicitly. The algebraic sum of the three terms must match between points.

• Use continuity first to compare speeds.
• Use vertical position to compare gravitational terms.
• Assign the remaining difference to pressure.

Changing the height or pressure reference shifts corresponding terms consistently but does not change a physically meaningful pressure or energy difference.

18. Combining Continuity and Bernoulli

Many problems require both conservation laws:

1. Use geometry to calculate \(A_1\) and \(A_2\).
2. Use \(A_1v_1=A_2v_2\) to relate speeds.
3. Insert the speed relationship into Bernoulli's equation.
4. Solve for the requested pressure, speed, or height.

Continuity answers how speed changes when area changes. Bernoulli answers how pressure and height participate in the energy exchange.

19. Deriving Torricelli's Theorem

Compare the free surface of a large open tank with a small opening a vertical distance \(\Delta y\) below it. Both points are exposed to the same atmospheric pressure, so \(P_1=P_2\). The tank surface area is much larger than the hole area, so continuity gives \(v_1\approx0\).

Cancel pressure and density, with \(\Delta y=y_1-y_2\):

The result matches the speed acquired by an object falling through the same vertical distance in the ideal model.

20. Worked Example: Tank Exit Speed

A small opening is \(1.80\,\mathrm m\) below the water surface of a large open tank. Neglect viscosity and motion of the upper surface.

Answer: The ideal exit speed is \(5.94\,\mathrm{m/s}\). The hole area determines the volume flow rate \(Q=A_{\mathrm{hole}}v_{\mathrm{exit}}\), even though it does not appear in the ideal speed formula.

21. Torricelli Graph Linearization

Squaring Torricelli's theorem gives

A graph of \(v^2\) vertically versus \(\Delta y\) horizontally should be a straight line through the origin with slope \(2g\). Therefore,

A nonzero intercept can indicate a height-origin or speed-calibration offset; curvature can indicate draining during measurement or nonideal losses.

22. Experiment: Test Torricelli's Prediction

1. Use a tall container with a small horizontal opening and mark the hole's vertical position.
2. Measure the free-surface height above the hole immediately before each trial.
3. Measure exit speed directly with video, or use horizontal range and projectile motion to infer it.
4. Repeat for at least six height differences while keeping the hole unchanged.
5. Graph \(v^2\) versus \(\Delta y\) with uncertainty bars.
6. Fit a line, calculate \(g=\text{slope}/2\), and compare with the accepted value.
7. Evaluate whether the intercept is consistent with zero.
8. Discuss surface-level change during a trial, jet breakup, range measurement, hole geometry, viscosity, and video scale.

23. Experiment: Test Continuity

Send steady water through two transparent pipe sections with known diameters. Measure collected volume over time to determine \(Q\), and measure or infer speed in each section.

1. Calculate each cross-sectional area from its measured diameter.
2. Calculate \(v=Q/A\) for both sections.
3. Compare \(A_1v_1\) and \(A_2v_2\) with propagated measurement uncertainty.
4. Repeat at several pump settings.
5. Graph \(v\) versus \(1/A\) at fixed \(Q\); the predicted slope is \(Q\).

A leak, trapped air, fluctuating pump, or partially filled section breaks the intended model.

24. Model Limits and Common Nonideal Effects

• Viscosity transfers mechanical energy into thermal energy, so the ideal Bernoulli sum decreases without pump input.
• Compressibility prevents density from canceling in the simple continuity equation.
• Turbulence makes local velocities fluctuate and increases dissipation.
• Unsteady flow means conditions at a point change during measurement.
• A pipe that is not full has a flowing cross section different from the pipe's geometric area.
• A pump or turbine transfers energy into or out of the fluid and requires an extended energy equation.
• A small tank surface may have non-negligible \(v_1\), so the simple Torricelli approximation must be revised.

25. A Reliable Fluid-Flow Workflow

1. Mark points 1 and 2 and record \(A\), \(v\), \(P\), and \(y\) at each.
2. State the fluid and pipe assumptions.
3. Use \(Q=Av\) for a single section.
4. Use continuity to relate speeds or flow rates.
5. Choose one height reference and one pressure type.
6. Apply Bernoulli only along an appropriate ideal-fluid flow path.
7. Cancel equal terms before substituting numbers.
8. Use Torricelli only when the surface and exit pressures match and the large-tank approximation is justified.
9. Check units: \(Q\) in \(\mathrm{m^3/s}\), speed in \(\mathrm{m/s}\), and every Bernoulli term in pascals.
10. Test whether the result respects mass conservation and energy exchange.

26. Common Traps and Key Takeaways

• Flow rate and speed are different quantities.
• \(Q=Av\) uses cross-sectional area perpendicular to flow.
• For incompressible steady flow, \(A_1v_1=A_2v_2\).
• Radius ratios must be squared.
• Continuity conserves mass; Bernoulli conserves mechanical energy.
• Every Bernoulli term is energy per volume and has pressure units.
• At equal height in ideal flow, greater speed corresponds to lower pressure.
• Use both continuity and Bernoulli when area and pressure change together.
• Torricelli requires matched surface and exit pressures plus a nearly stationary tank surface.
• Real viscosity makes ideal predictions overestimate available mechanical energy.