Additional: Continuity Equation
Equation of Continuity ($ A_1v_1 = A_2v_2 $)
The Equation of Continuity is a fundamental principle in fluid dynamics that expresses the conservation of mass in fluid flow. It is particularly useful for analysing the flow of incompressible fluids in pipes or tubes with varying cross-sectional areas.
Principle of Conservation of Mass in Fluid Flow
For a fluid flowing steadily through a tube, if no fluid is being added or removed along the length of the tube, the mass of fluid entering any section of the tube per unit time must be equal to the mass of fluid leaving that section per unit time. This is a statement of the conservation of mass.
Derivation of the Equation of Continuity (for Incompressible Fluid)
Consider an ideal, incompressible fluid flowing in a steady manner through a tube of flow. Let the tube have a non-uniform cross-sectional area. Consider two sections, 1 and 2, of the tube with cross-sectional areas $A_1$ and $A_2$, respectively.
(Image Placeholder: A tube with a wider section on the left (Area A1) and a narrower section on the right (Area A2). Arrows show fluid flow from left to right. Velocity v1 is shown in the wider section, and velocity v2 is shown in the narrower section. Indicate the volume of fluid entering A1 in time dt and leaving A2 in time dt.)
Let the fluid speed at section 1 be $v_1$ and the fluid speed at section 2 be $v_2$. In a small time interval $\Delta t$, the fluid particles at section 1 move a distance $\Delta x_1 = v_1 \Delta t$. The volume of fluid entering section 1 in time $\Delta t$ is $\Delta V_1 = A_1 \Delta x_1 = A_1 v_1 \Delta t$. The mass of fluid entering section 1 in time $\Delta t$ is $\Delta m_1 = \rho_1 \Delta V_1 = \rho_1 A_1 v_1 \Delta t$, where $\rho_1$ is the fluid density at section 1.
Similarly, in the same time interval $\Delta t$, the fluid particles at section 2 move a distance $\Delta x_2 = v_2 \Delta t$. The volume of fluid leaving section 2 in time $\Delta t$ is $\Delta V_2 = A_2 \Delta x_2 = A_2 v_2 \Delta t$. The mass of fluid leaving section 2 in time $\Delta t$ is $\Delta m_2 = \rho_2 \Delta V_2 = \rho_2 A_2 v_2 \Delta t$, where $\rho_2$ is the fluid density at section 2.
By the conservation of mass, the mass entering section 1 must equal the mass leaving section 2 in the same time interval:
$ \Delta m_1 = \Delta m_2 $
$ \rho_1 A_1 v_1 \Delta t = \rho_2 A_2 v_2 \Delta t $
Dividing by $\Delta t$:
$ \rho_1 A_1 v_1 = \rho_2 A_2 v_2 $
This is the general equation of continuity for the steady flow of a fluid, relating the product of density, area, and velocity at two points along the flow.
For an incompressible fluid, the density is constant throughout the fluid, i.e., $\rho_1 = \rho_2 = \rho$. In this case, the density cancels out from the equation:
$ \rho A_1 v_1 = \rho A_2 v_2 $
$ A_1 v_1 = A_2 v_2 $
This is the Equation of Continuity for the steady flow of an incompressible fluid. It states that the product of the cross-sectional area of the tube and the fluid speed is constant at every point along the tube of flow.
The product $Av$ represents the volume flow rate or discharge rate ($Q$), which is the volume of fluid passing through a given cross-section per unit time. The equation of continuity for incompressible fluid states that the volume flow rate is constant throughout the tube:
$ Q = A v = \text{constant} $
Implications of the Equation of Continuity: For incompressible fluid flow, where the tube narrows (area $A$ decreases), the fluid speed ($v$) must increase to keep the product $Av$ constant. Conversely, where the tube widens (area $A$ increases), the fluid speed must decrease. This explains why water flows faster out of a hose when you partially cover the opening with your thumb (reducing the area, increasing the speed).
Example 1. Water flows through a pipe of diameter 4 cm at a speed of 2 m/s. What is the speed of the water when it flows through a narrower section of the pipe with a diameter of 2 cm?
Answer:
Diameter of the wider section, $D_1 = 4$ cm. Radius $r_1 = D_1/2 = 2$ cm $= 0.02$ m.
Area of the wider section, $A_1 = \pi r_1^2 = \pi (0.02 \text{ m})^2 = 0.0004\pi$ m$^2$.
Speed in the wider section, $v_1 = 2$ m/s.
Diameter of the narrower section, $D_2 = 2$ cm. Radius $r_2 = D_2/2 = 1$ cm $= 0.01$ m.
Area of the narrower section, $A_2 = \pi r_2^2 = \pi (0.01 \text{ m})^2 = 0.0001\pi$ m$^2$.
Let the speed in the narrower section be $v_2$.
Since water is incompressible, we can use the equation of continuity for incompressible fluids:
$ A_1 v_1 = A_2 v_2 $
Substitute the known values:
$ (0.0004\pi \text{ m}^2) \times (2 \text{ m/s}) = (0.0001\pi \text{ m}^2) \times v_2 $
$ 0.0008\pi \text{ m}^3\text{/s} = 0.0001\pi \text{ m}^2 \times v_2 $
Cancel $\pi$ and rearrange for $v_2$:
$ v_2 = \frac{0.0008}{0.0001} \text{ m/s} = 8 $ m/s.
Alternatively, using areas in terms of diameters: $A = \pi (D/2)^2 = \frac{\pi D^2}{4}$.
$ \frac{\pi D_1^2}{4} v_1 = \frac{\pi D_2^2}{4} v_2 $
$ D_1^2 v_1 = D_2^2 v_2 $
$ v_2 = \frac{D_1^2}{D_2^2} v_1 = \left(\frac{D_1}{D_2}\right)^2 v_1 $
$ v_2 = \left(\frac{4 \text{ cm}}{2 \text{ cm}}\right)^2 \times 2 \text{ m/s} = (2)^2 \times 2 \text{ m/s} = 4 \times 2 \text{ m/s} = 8 $ m/s.
The speed of the water in the narrower section is 8 m/s. As the area decreases (by a factor of 4), the speed increases (by a factor of 4).
Ideal Fluid Flow Assumptions
Much of the simplified analysis in fluid dynamics, including the derivation of the equation of continuity and Bernoulli's principle, relies on assuming the fluid is an ideal fluid and that the flow is of a specific type.
Understanding these assumptions is important for knowing the limitations of the derived principles when applied to real-world situations.
Assumptions for Ideal Fluid Flow
An ideal fluid is a theoretical model used to simplify fluid dynamics problems. The key assumptions about an ideal fluid and its flow are:
- Incompressible: The density ($\rho$) of the fluid remains constant throughout the flow, regardless of changes in pressure. Real liquids are nearly incompressible, but gases are highly compressible, so this assumption is generally not valid for gases unless pressure changes are small.
- Non-viscous (Invoscid): There are no internal frictional forces between adjacent fluid layers (viscosity $\eta = 0$). This means no energy is lost due to viscous dissipation during the flow. Real fluids have viscosity, and this assumption is often a major deviation from reality, especially in narrow pipes or at low speeds where viscous effects are significant.
- Steady Flow (Laminar): The velocity of the fluid at any fixed point in space does not change with time. Streamlines are smooth curves that do not cross each other. In steady flow, each fluid particle follows a smooth, predictable path. Turbulent flow (chaotic and irregular) is not steady, and ideal fluid equations do not apply directly to it without modifications or statistical analysis.
- Irrotational: Fluid elements do not rotate about their own axis as they move. If you imagine a small paddle wheel placed in the flow, it would move without rotating. This assumption simplifies the relationship between velocity and potential energy. Flows with vortices or eddies are rotational.
Implications and Limitations
These assumptions greatly simplify the mathematical analysis of fluid flow, leading to straightforward principles like the equation of continuity and Bernoulli's equation.
However, real fluids and real-world flow conditions often violate these assumptions:
- All real fluids have some viscosity. Viscous forces cause energy dissipation (loss of mechanical energy, often as heat) during flow. This means that in real fluids, the sum $P + \frac{1}{2}\rho v^2 + \rho g h$ is generally not constant along a streamline; it decreases in the direction of flow due to work done against viscous forces.
- Gases are compressible, especially at high speeds or under large pressure changes.
- Flow often becomes turbulent, particularly at high velocities or around obstacles. Turbulent flow is much more complex and requires different analytical tools.
- In some flows, fluid elements rotate (e.g., in vortices or boundary layers).
Despite these limitations, the ideal fluid model and principles derived from it provide a good first approximation for many real-world situations, especially when viscosity and compressibility effects are relatively small (e.g., flow of water at moderate speeds, flow of air over aircraft wings at cruising speeds). They also serve as a foundation for understanding more complex fluid behaviour. When deviations from ideal behaviour are significant, engineers and physicists use more advanced models and experimental data.