Continuity Equation

An equation that describes the transport of some physical quantity is a continuity equation. When applied to any conservative quantity, the continuity equation is quite simple and powerful. This equation can also be commonly used to apply after some generalization to any physical quantity extensive in nature. Many quantities are conserved under their respective appropriate conditions. For example, energy, momentum, mass, electric charge, etc. We may also explain other natural quantities using continuity equations.

Continuity equations describe problems that are often but not always conserved. Thus, they usually include terms like ‘source’ and ‘sink’. These terms allow continuity equations to describe quantities like the density of a molecular species. This species can be created or destroyed in a chemical reaction. Thus, its density is generally conserved until the molecule itself disappears or adds up in a chemical reaction.

Continuity Equation

Continuity Equation in Daily Life

A continuity equation can be defined in everyday life. For example, if the continuity equation defines the number of people alive, then the source to be accounted in this case is the number of births and the sink to be accounted for is the number of deaths.

The representation of any continuity equation can be done in an integral form which is applicable on finite regions known as the flux integral. Another form of representation is the differential form which is applicable on a point. Continuity equations represent peculiar transport equations. These transport equations are the convection-diffusion equation, Navier-Stokes equation, and the Boltzmann transport equation. A Sankey diagram is used to visualize these kinds of equations.

Conservation laws can be represented by stronger, more local forms which are the continuity equations. Energy can neither be created nor destroyed. This is a weak version of the conservation of energy which proposes that the total energy in the universe is constant.

But the issue with this statement is that it does not include the possibility that a state of energy can suddenly disappear from one point and can simultaneously appear at some other point. This refers to the need for a stronger statement that says that energy is locally conserved. Energy can neither be created nor destroyed nor is it able to teleport from one place to some other place. Its movement is restricted to a continuous flow. Thus, to express these types of statements, continuity equations are used.

Integral Form

Some of the statements of the integral form of the continuity equation are given as:

• Whenever an additional q flows inward through the surface of any region, the amount of q in that region increases and decreases when the flow of q is outward.
• Whenever a new q is introduced in the region, the amount of q in that region increases and destroying q results in a decrease in the amount of q.
• To change the amount of q in the region, there is no other way than these two.

The rate of increase of q within a volume V is given by the continuity equation:

$$\frac{dq}{dt}+\oint j.dS=\sum$$

Where

q is the total amount of quantity

S is the surface

$$\oint j.dS$$ is the surface integral over the closed surface

j denotes the flux of q

t denotes the time

$$\sum$$ is the net rate of generation of q inside the volume V

$$\sum$$ becomes more positive when q is being generated and thus it is called the source of q. On the other hand, $$\sum$$ becomes more negative when q is being destroyed. Thus it is called the sink of q. This term is also denoted as $$\frac{dq}{dt}|_{gen}$$. This is the total change of q from its destruction or generation inside the volume. To understand in an easier way, let’s suppose the volume to be a building and q the number of people in the building. The walls, roof, doors and the foundation of the building will rightly act as the enclosing surface.

Here the continuity equation is stated in a way that the number of building increase on the entering of people in the building. When people exit this building, the number of people decreases. These suggest the inward and outward flux. When someone gives birth in the building, it refers to generation. Destruction is referred to as to whenever someone dies in the building.

Differential Form

We can express a general continuity equation in the form of a differential equation by the divergence theorem:

$$\frac{\partial \rho }{\partial t}+\bigtriangledown .j=\sigma$$

Where,

j denotes the flux of the quantity

t denotes the time

$$\bigtriangledown$$ denotes the divergence

$$\rho$$ is the quantity per unit volume

$$\sigma$$ denotes the generation of quantity per unit volume per unit time

When $$\sigma$$ is less than zero, it indicates the terms that generate q. When $$\sigma$$ is more than zero, it indicates the terms that remove q. The equations like the volume continuity equation and the complicated forms like the Navier-Stokes equation can be derived from this general differential equation.

The advection equation can also be generalized by this equation. Some other physical equations have the same form to the continuity equation. The examples are Gauss’ law of the electric field and Gauss’ law for gravity. But these equations are not generally referred to as continuity equations. This is because j in these equations does not represent the flow of an actual physical quantity.

This makes q a conserved quantity. This means that this quantity can neither be created nor destroyed. $$\sigma$$ becomes zero and thus, the equation gets modified. The final equation becomes:

$$\frac{\partial \rho }{\partial t}+\bigtriangledown .j =0$$

Where,

j denotes the flux of the quantity

t denotes the time

$$\bigtriangledown$$ denotes the divergence

$$\rho$$ is the quantity per unit volume

Q.1. How do we express the continuity equation with respect to electromagnetism?

Answer. The continuity equation is more of an empirical law which expresses charge conservation in the field of electromagnetism. It is the consequence of Maxwell’s equations but more fundamental. The equation states that the current density J is the negative rate of change of the charge density $$\rho$$.

$$\bigtriangledown .J =-\frac{\partial \rho }{\partial t}$$

Current is basically charge per unit time. The equation states that if there is the outward movement of the charge from the differential volume, then the amount of charge in the volume decreases. Thus, the rate of change of charge density is negative. This verifies that the continuity equation tends to state the conservation of charge.

Q.2. How do we write the continuity equation for a particle that is constantly moving?

Answer. For a quantity moving continuously due to a random process, we write the continuity equation for its probability distribution. An example of this case could be the location of a single dissolved molecule under Brownian motion. The flux of this continuity equation could be given as the probability per unit area per unit time of the particle. The negative divergence of this flux is equal to the rate of change of the probability density. This is stated by the continuity equation. This continuity equation verifies that the particle is always somewhere. This is because the integration of its probability distribution always gives 1. It also states that it moves under continuous motion and does not teleport from one point to another.

Q.3. Which continuity equation is prevalent in the fluid dynamics?

The continuity equation in the case of fluid dynamics states that the rate of the mass entering a system equals the rate of the mass leaving the system. This also takes into account the amount of mass accumulat3ed within the system. The equation is:

$$\frac{\partial \rho }{\partial t}+\bigtriangledown .(\rho u) =0$$

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