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 are the Applications of the Equation of Continuity?
The primary application of the Equation of Continuity is seen in the field of Hydrodynamics, Electromagnetism, Aerodynamics, and Quantum Mechanics. The equation of continuity forms the fundamental rule of Bernoulli's Principle. It is also associated with the Aerodynamics principle and its applications.
The differential form of the equation of continuity is used to determine the consistency of Maxwell's Equation. Apart from it, the differential form of the equation of continuity is also used in Electromagnetism.
Equation of continuity is used to check the consistency of the Schrodinger Equation.
General and Special Theory of Relativity, Noether's Theorem, also used the equation of continuity.
2. What is the Significance of Continuity Equation?
The equation of continuity is based on the assumption that the fluid that flows in will be equal to the fluid that flows out. This is a useful tool to solve many properties of the fluid during its motion:
Since the flow in = flows out, we can calculate some properties of a liquid under some conditions, and then we can apply the continuity equation to measure the properties of that fluid under other conditions.
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Q1=Q2
This can be expressed as:
A1∗v1=A2∗v2A
Here, the equation of continuity does find its application to any incompressible fluid. Since the fluid is incompressible, the amount of fluid that flows on a surface must equal the amount of fluid that flows out of the surface.
Significance: We can observe the effect of the equation of continuity in our garden. Water flows through the pipe of our garden, and when it reaches the narrow end of the pipe or the nozzle, the speed of water increases. With the increase of speed of the fluid, the cross-sectional area decreases and with the decrease in speed of fluid decreases, the cross-sectional area increases. This is a consequence of the equation of continuity.
3. What is flux?
Flux can be described as the phenomenon or effect that occurs when it passes or travels through a substance or surface. For defining flux, the continuity equation is required. For this, there must be quantities present like q which can move or flow. This can be a mass, electric charge, energy, number of molecules and momentum. We can consider ρ as the density of the quantity which is the amount of q per volume unit. The flux of this quantity is a vector field which is denoted by j. Thus, the dimension of flux can be defined as the amount of q per unit time by a unit area.
4. What is the use of the continuity equation?
The continuity equation has a significant importance when the movement of fluids is studied and when the fluid is passed through a tube or surface of varying diameters. Usually, the fluids taken have a constant density and are normally incompressible. This can be used for the human body in many different aspects. We can consider the example of blood vessels and arteries which are further divided into several capillaries which then further divide into veins. The continuity equation can be used to measure the speed of the blood flow through the blood vessels. The blood vessels have elasticity and various other factors which can be considered during the application of the continuity equation. The factors will help make precise calculations and results. Thus, the flow of blood can be found using the continuity equation by considering the factors like the elasticity of the blood vessels as well as the diameter.
5. What are the assumptions made during the derivation of the continuity equation?
For the derivation of the continuity equation there are certain assumptions made. These are some of the following:
The tube from which the fluid is flowing and is taken into consideration must have a single entry point as well as a single exit point
The fluid flowing inside the tube must be a non-viscous fluid
The fluid must be incompressible
The fluid flow must be steady
6. What are the uses of fluid dynamics?
Fluid dynamics can be defined as the branch of science which deals with the movement or ‘flow’ of liquids and gasses. It is one of the 2 branches of fluid mechanics which can be defined as the study of fluids and the forces that affect them. Another branch of it is fluid statics which is the study of fluids at rest. The uses or the importance of fluid dynamics can be used in the study of weather patterns, plate tectonics, currents of ocean and circulation of blood. Other uses of fluid dynamics in technology are in the applications of rocket engines, oil pipes, turbines and even air conditioners.
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