6.3 Equation of Continuity | Fluid Mechanics | Physics 11

The equation of continuity is a fundamental principle in fluid dynamics derived from the law of conservation of mass. It states that for an ideal fluid flowing through a pipe or channel, the mass flow rate is constant at all points. This means the product of the fluid's cross-sectional area and its velocity remains constant along a streamline.

The equation for an ideal, incompressible fluid is:

$A_{1} v_{1} = A_{2} v_{2} or A \cdot v = constant$

$A_{1}, A_{2}$ : Cross-sectional areas at points 1 and 2.
$v_{1}, v_{2}$ : Fluid velocities at points 1 and 2.

Essentially, where the pipe is narrower, the fluid must speed up, and where it is wider, the fluid slows down to ensure that the same amount of mass passes through every section in the same amount of time. This principle is essential in understanding fluid flow and is directly applied in Bernoulli's Equation→.

Conservation of Mass

The core principle is that mass is neither created nor destroyed within the fluid flow. The mass of fluid entering a section of a pipe in a given time interval must equal the mass of fluid exiting it in the same interval.

Ideal Fluid

The simplified version of the equation ( $A v = constant$ ) assumes the fluid is "ideal." An ideal fluid is a theoretical concept with the following properties:

Property	Description
Incompressible	The density ( $ρ$ ) of the fluid remains constant.
Non-viscous	There is no internal friction (viscosity) within the fluid.
Steady Flow	The velocity of the fluid at any given point does not change over time.
Irrotational Flow	The fluid flows without rotation or turbulence.

Real vs. Ideal Fluids

For real fluids (like gases), density can change. The equation of continuity is therefore generalized to include density:

$ρ_{1} A_{1} v_{1} = ρ_{2} A_{2} v_{2} or ρ A v = constant$

This is the most complete form of the equation, as it accounts for fluids that can be compressed.

Figure: Fluid flow through a pipe of varying cross-section showing velocity changes — Fluid flow through a pipe of varying cross-section showing velocity changes

Mathematical Derivation

The equation is derived directly from the principle of mass conservation.

Mass Conservation: Consider a fluid flowing through a pipe. The mass entering at point 1 ( $Δ m_{1}$ ) in a time interval $Δ t$ must equal the mass exiting at point 2 ( $Δ m_{2}$ ) in the same time. $Δ m_{1} = Δ m_{2}$
Relating Mass, Density, and Volume: Mass is the product of density ( $ρ$ ) and volume ( $V_{vol}$ ). $Δ m = ρ \cdot Δ V_{vol}$
Expressing Volume in Terms of Flow: The volume of a fluid segment passing a point is its cross-sectional area ( $A$ ) times the distance it travels ( $Δ x$ ). $Δ V_{vol} = A \cdot Δ x$ Since distance is velocity times time ( $Δ x = v \cdot Δ t$ ), the volume is: $Δ V_{vol} = A \cdot v \cdot Δ t$
Formulating the Mass Flow Rate: Substituting this into the mass equation gives the mass passing a point in time $Δ t$ : $Δ m = ρ \cdot A \cdot v \cdot Δ t$
Equating Mass Flow at Two Points: Applying the conservation of mass ( $Δ m_{1} = Δ m_{2}$ ): $ρ_{1} A_{1} v_{1} Δ t = ρ_{2} A_{2} v_{2} Δ t$
Final Equation: After canceling the time interval $Δ t$ , we get the general form of the equation of continuity: $ρ_{1} A_{1} v_{1} = ρ_{2} A_{2} v_{2}$ For an ideal, incompressible fluid, the density is constant ( $ρ_{1} = ρ_{2}$ ), so it cancels out, leaving the simplified version: $A_{1} v_{1} = A_{2} v_{2}$

Possible Questions and Answers

Q: What is mass flow rate?

A: It is the amount of mass of a substance that passes per unit of time. Its formula is $ρ A v$ , and according to the equation of continuity, it is constant in a closed system.

Q: Does the equation of continuity apply to blood flow in the human body?

A: Yes, it is a fundamental principle in hemodynamics. The total cross-sectional area of the capillaries is much larger than that of the aorta, so the blood flows much more slowly in the capillaries, which is essential for nutrient exchange.

Q: Why does fluid velocity increase when the pipe is squeezed?

A: When the cross-sectional area of a pipe decreases (squeezing), the fluid must increase its velocity to maintain a constant mass flow rate. This is directly given by $A_{1} v_{1} = A_{2} v_{2}$ ; if $A$ decreases, $v$ must increase proportionally.

Ideal Fluid

The simplified version of the equation (

A v = constant

) assumes the fluid is "ideal." An ideal fluid is a theoretical concept with the following properties:

Property	Description
Incompressible	The density ( $ρ$ ) of the fluid remains constant.
Non-viscous	There is no internal friction (viscosity) within the fluid.
Steady Flow	The velocity of the fluid at any given point does not change over time.
Irrotational Flow	The fluid flows without rotation or turbulence.

Real vs. Ideal Fluids

For real fluids (like gases), density can change. The equation of continuity is therefore generalized to include density:

ρ_{1} A_{1} v_{1} = ρ_{2} A_{2} v_{2} or ρ A v = constant

This is the most complete form of the equation, as it accounts for fluids that can be compressed.

Fluid flow through a pipe of varying cross-section showing velocity changes

Mathematical Derivation

The equation is derived directly from the principle of mass conservation.

Mass Conservation: Consider a fluid flowing through a pipe. The mass entering at point 1 ( $Δ m_{1}$ ) in a time interval $Δ t$ must equal the mass exiting at point 2 ( $Δ m_{2}$ ) in the same time. $Δ m_{1} = Δ m_{2}$

Relating Mass, Density, and Volume: Mass is the product of density ( $ρ$ ) and volume ( $V_{vol}$ ). $Δ m = ρ \cdot Δ V_{vol}$

Expressing Volume in Terms of Flow: The volume of a fluid segment passing a point is its cross-sectional area ( $A$ ) times the distance it travels ( $Δ x$ ). $Δ V_{vol} = A \cdot Δ x$ Since distance is velocity times time ( $Δ x = v \cdot Δ t$ ), the volume is: $Δ V_{vol} = A \cdot v \cdot Δ t$

Formulating the Mass Flow Rate: Substituting this into the mass equation gives the mass passing a point in time $Δ t$ : $Δ m = ρ \cdot A \cdot v \cdot Δ t$

Equating Mass Flow at Two Points: Applying the conservation of mass ( $Δ m_{1} = Δ m_{2}$ ): $ρ_{1} A_{1} v_{1} Δ t = ρ_{2} A_{2} v_{2} Δ t$

Final Equation: After canceling the time interval $Δ t$ , we get the general form of the equation of continuity: $ρ_{1} A_{1} v_{1} = ρ_{2} A_{2} v_{2}$ For an ideal, incompressible fluid, the density is constant ( $ρ_{1} = ρ_{2}$ ), so it cancels out, leaving the simplified version: $A_{1} v_{1} = A_{2} v_{2}$

Possible Questions and Answers

Q: What is mass flow rate?

A: It is the amount of mass of a substance that passes per unit of time. Its formula is

ρ A v

, and according to the equation of continuity, it is constant in a closed system.

Q: Does the equation of continuity apply to blood flow in the human body?

Q: Why does fluid velocity increase when the pipe is squeezed?

A: When the cross-sectional area of a pipe decreases (squeezing), the fluid must increase its velocity to maintain a constant mass flow rate. This is directly given by

A_{1} v_{1} = A_{2} v_{2}

; if

A

decreases,

v

must increase proportionally.