[Physics Class Notes] on Derivation of Continuity Equation Pdf for Exam

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:

 

 

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.