14-6 The Equation of Continuity#

Prompts

What four properties define an ideal fluid? Give a brief meaning of each (steady, incompressible, nonviscous, irrotational).
What is a streamline? Why can two streamlines never intersect?
State the equation of continuity. If a pipe narrows, does the flow speed increase or decrease? Why?
Define volume flow rate \(R_V\) and mass flow rate \(R_m\). How are they related?
You partially close a garden hose with your thumb. The water shoots out faster. Explain using the equation of continuity.

Lecture Notes#

Overview#

Ideal fluid: Steady, incompressible, nonviscous, irrotational flow.
Streamlines and tube of flow: geometric tools to describe flow.
Equation of continuity: \(Av = \text{constant}\)—narrower cross section → faster flow.
Volume flow rate \(R_V = Av\); mass flow rate \(R_m = \rho Av\).

Ideal fluid#

We model fluid flow with an ideal fluid—a simplification that yields useful results:

Property	Meaning
Steady flow	Velocity at any fixed point does not change with time (laminar, not turbulent)
Incompressible	Density \(\rho\) is constant
Nonviscous	No internal friction; no viscous drag
Irrotational	Fluid elements do not rotate about their own center of mass

Real fluids (water, air) approximate these under certain conditions. Turbulent flow, compressibility, and viscosity complicate the analysis.

Streamlines and tube of flow#

Streamline: The path followed by a tiny fluid element. The velocity \(\vec{v}\) is always tangent to the streamline.
Tube of flow: A bundle of streamlines forming an imaginary tube. No fluid crosses the boundary; all flow stays inside.

Streamlines cannot intersect—otherwise a fluid element would have two velocities at the intersection.

Equation of continuity#

For steady, incompressible flow through a tube (or tube of flow): the volume of fluid entering one cross section in time \(\Delta t\) equals the volume leaving another. So

(29)#\[ A_1 v_1 = A_2 v_2 = \text{constant} \]

\(A\): cross-sectional area
\(v\): flow speed (magnitude of velocity)

Physical meaning: Same volume per second passes every cross section. Narrower → faster; wider → slower.

Garden hose

Partially closing the hose opening reduces \(A\) at the exit. Since \(Av\) is constant, \(v\) increases—water shoots out faster.

Poll: Where does the extra kinetic energy come from?

Water flows through a horizontal pipe that narrows. The flow speed increases. The kinetic energy density \(\frac{1}{2}\rho v^2\) increases. Where did this energy come from?

(A) The pump added it
(B) It came from pressure energy—pressure is lower in the narrow section
(C) Gravity
(D) It is free—no source needed

Poll: Mississippi River

The Mississippi River flows at 2 m/s in one section and 4 m/s in another. If the width is the same in both sections, how does the depth in the faster section compare to the slower section?

(A) 1/4 as deep
(B) 1/2 as deep
(C) 2× as deep
(D) 4× as deep

Volume and mass flow rates#

Volume flow rate (volume per second):

(30)#\[ R_V = A v = \text{constant} \]

Mass flow rate (mass per second), for incompressible flow:

(31)#\[ R_m = \rho R_V = \rho A v = \text{constant} \]

Both are conserved along a tube of flow for steady, incompressible flow.

Summary#

Ideal fluid: Steady, incompressible, nonviscous, irrotational
Streamline: Path of fluid element; \(\vec{v}\) tangent to it
Equation of continuity: \(A_1 v_1 = A_2 v_2\); smaller \(A\) → larger \(v\)
Volume flow rate \(R_V = Av\); mass flow rate \(R_m = \rho Av\)