What is the Continuity Equation?
The continuity equation describes the nature of the movement of physical quantities. The continuity equation is usually applied to the conserved quantities, but it can also be generalized for the extensive quantities. Quantities like mass, momentum, energy and electric charge are some major conserved quantities. The continuity equation can be applied to these quantities to describe nature and other physical phenomena.
The continuity equation plays a significant role while studying the movement of fluids, especially when fluid is passed through a tube of varying diameters. Normally the fluids which are taken into consideration have a constant density and are incompressible. This concept can be related to the human body in several aspects.
For example, the blood vessels or arteries are divided into several capillaries, which then join to form a vein. The continuity equation can calculate the speed of the blood flowing through the blood vessels. Since the blood vessels are elastic, several other factors are to be applied with the continuity equation precisely to make the proper calculation. This includes the elasticity and the diameter of the blood vessels.
To the understand continuity equation; let's consider the flow rate f first:
f=Av
Where,
f = flow rate
A = the cross-sectional area of a point in the pipe
v = the average speed at which a fluid is moving inside the pipe.
The flow rate is the amount of liquid that passes from a particular point in a unit of time. For example, the amount of water (in volume) coming out from a pipe per minute. The unit of flow rate is usually calculated in terms of milliliters per second.
The application of the continuity equation can be seen while calculating the amount of blood that the heart pumps into the vessels, thus determining a person's health condition. This process is also helpful in determining whether a blood vessel is clogged, and taking further measures against heart issues.
Derivation of Continuity Equation Assumption
The following points are the assumptions of the continuity equation:
The tube, which is taken into consideration, has a single entry and a single exit.
The fluid that flows in the tube is non-viscous.
The fluid is incompressible.
Fluid flow is steady.
Derivation of Continuity Equation
Let us consider the following diagram:
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Let us consider that the fluid flows in the tube for a short duration Δt. During this time, the fluid will cover a distance of Δx1, with a velocity of v1in the lower part of the pipe.
The distance covered by the fluid with speed v1 in time Δt will be given by,
Δx1 = v1Δt
Now, in the lower part of the pipe, the volume of fluid flows into the pipe is,
V = A1Δx1 = A1 v1Δt
We know that mass (m) = Density (ρ) × Volume (V). So, the mass of fluid in region Δx1 will be:
Δm1= Density × Volume
⇒ Δm1 =ρ1A1v1Δt ——–(Equation 1)
Now, we have to calculate the mass flux at the lower part of the pipe. Mass flux is the total defined mass of the fluid that flows through the given cross-sectional area per unit of time. For the lower part of the pipe, with the lower end of pipe having a cross-sectional area A1, the mass flux will be given by,
Δm1/Δt =ρ1A1v1——–(Equation 2)
Similarly, the mass flux of the fluid at the upper end of the pipe will be:
Δm2/Δt =ρ2A2v2——–(Equation 3)
Where,
v2 = velocity of the fluid flowing in the upper end of the pipe.
Δx2= distance traveled by the fluid.
Δt = time, and
A2 = area of a cross-section of the upper end of the pipe.
It is assumed that the density of the fluid in the lower end of the pipe is the same as that of the upper end. Thus, the fluid flow is said to be streamlined. Thus, the mass flux at the bottom point of the pipe will also be equal to the mass flux at the upper end of the pipe. Hence Equation 2 = Equation 3.
Thus,
ρ1A1v1 = ρ2A2v2 ——–(Equation 4)
Based on equation 4 it can be stated that:
ρ A v = constant
The above equation helps to prove the law of conservation of mass in fluid dynamics. As the fluid is taken to be incompressible, the density of the fluid will be constant for steady flow.
So, ρ1 = ρ2
Applying this to Equation 4; it can be written as:
A1v1 = A2v2
The generalized form of this equation is:
A v = constant
Now, let's consider R as the volume flow rate, hence the equation can be expressed as:
R = A v = constant
This is the derivation of the continuity equation.
FAQs on Derivation of Continuity Equation
1. What is the fundamental principle behind the derivation of the equation of continuity?
The derivation of the equation of continuity is based on the law of conservation of mass. It states that for a fluid in steady flow, the mass of the fluid entering a pipe in a specific time interval is equal to the mass of the fluid exiting the pipe in the same time interval. This implies that mass is neither created nor destroyed within the flow, so the mass flow rate (ρAv) remains constant throughout the pipe.
2. What are the key assumptions made when deriving the simplified continuity equation (A₁v₁ = A₂v₂)?
To derive the simplified form of the continuity equation, several key assumptions about the fluid and its flow are made:
- Steady Flow: The fluid velocity at any given point in the pipe does not change with time.
- Incompressible Fluid: The density (ρ) of the fluid is constant and does not change with pressure. This is a good approximation for liquids like water.
- Non-Viscous Fluid: The fluid has no internal friction (viscosity). This is an idealisation, as all real fluids have some viscosity.
- Streamline Flow: The fluid particles move in smooth paths or streamlines without crossing each other.
3. How do you apply the continuity equation in a practical example, such as water flowing through a tapering pipe?
Consider water flowing through a pipe that narrows. Let the wider end have a cross-sectional area A₁ and the water speed be v₁. The narrower end has an area A₂ and water speed v₂. According to the continuity equation, A₁v₁ = A₂v₂. If the initial area A₁ is 10 cm² and the speed v₁ is 2 m/s, and the pipe narrows to an area A₂ of 5 cm², the new speed v₂ can be calculated as: v₂ = (A₁v₁) / A₂ = (10 cm² × 2 m/s) / 5 cm² = 4 m/s. The speed of the water doubles as the area is halved.
4. Why does the speed of a river increase where it narrows?
The increased speed of a river in a narrow section is a direct real-world example of the equation of continuity. The river channel acts like a pipe. Assuming the volume of water flowing per second (the flow rate) is constant, the product of the river's cross-sectional area (A) and its flow velocity (v) must remain constant (Av = constant). When the river passes through a narrow gorge, its cross-sectional area 'A' decreases. To maintain a constant flow rate, the velocity 'v' of the water must increase.
5. What is the difference between the continuity equation for a compressible fluid versus an incompressible one?
The key difference lies in the fluid's density (ρ).
- For an incompressible fluid like water, the density is assumed to be constant (ρ₁ = ρ₂). This allows the general equation (ρ₁A₁v₁ = ρ₂A₂v₂) to be simplified to A₁v₁ = A₂v₂.
- For a compressible fluid like air, the density can change with pressure. Therefore, you must use the more general form of the equation, ρ₁A₁v₁ = ρ₂A₂v₂, as the density 'ρ' is not constant and cannot be cancelled out.
6. How does the equation of continuity relate to the circulation of blood in the human body?
The principle of continuity applies to blood circulation. The aorta, the main artery from the heart, branches into numerous smaller arteries, which in turn branch into millions of tiny capillaries. While each capillary is extremely narrow, their total combined cross-sectional area is much larger than the area of the aorta. According to the equation Av = constant, since the total area 'A' of the capillaries is very large, the velocity 'v' of blood flow through them must be very slow. This slow speed is crucial for the efficient exchange of oxygen and nutrients with the surrounding tissues.
7. Does the equation of continuity contradict Bernoulli's principle?
No, the equation of continuity and Bernoulli's principle do not contradict each other; they are complementary principles that describe fluid motion. The equation of continuity is derived from the conservation of mass, relating fluid speed to the cross-sectional area. Bernoulli's principle is derived from the conservation of energy, relating fluid speed, pressure, and height. In fact, they are often used together to solve problems in fluid dynamics. For example, the continuity equation can determine the change in speed, which can then be used in Bernoulli's equation to find the corresponding change in pressure.
