Derivation of Continuity Equation - Definition, Formula, FAQs

In physics, Derivation of the continuity equation is one of the most supreme derivations in fluid dynamics. In this article, we will discuss the continuity equation. What is the formula for the continuity equation? What is continuity equation derivation? Derive steady flow energy equation. What is the continuity equation in differential form? What is the continuity equation in semiconductors? What is the equation of continuity in electromagnetism? What is the law of continuity?

What is the continuity equation?

Definition: In physics, the equation of continuity is defined as the mass balance of a fluid flowing via a stationary volume segment. It states that the speed or rate of mass accumulation in this volume element equals the speed of mass in minus the rate of mass out.

Continuity Equation Derivation

The continuity equation shows that the material of the cross-sectional area of the pipe and the fluid rate at any particular point across the pipe is consistently constant. This material is equal to the volume flow per sec. or simply the rate of flow. The continuity eq. is represented as:

$$
\mathrm{R}=\mathrm{A} \vee=\text { constant }
$$

Where,
$R=$ volume flow rate
$A=$ flow area
$v=$ flow velocity

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Assumptions in the continuity equation:

The below keys are the assumptions of continuity equation:

• The tube, which is taken into contemplation, has a single entry and a single exit.

• The fluid that influx (flows) in the tube is non-viscous fluid.

• The fluid is close or in-compressible.

• Fluid influx (flow) is steady.

Derivation of Continuity Equation

Let us see the following diagram:
Let us assume that the fluid flows in the tube for a short period $\Delta t$. In the course of this time, the liquid (fluid) will cover a distance of $\Delta x_1$, with a velocity of $v_1$ in the lower fragment of the pipe.

The interval covered by the fluid with rate $v_1$ in time $\Delta t$ will be shown by,

$$
\Delta x_1=v_{1 \Delta} t
$$

So, in the lower part of the pipe, the volume of fluid move (flow) into the pipe is,

$$
\mathrm{V}=\mathrm{A}_1 \Delta \mathrm{x}_1=\mathrm{A}_1 \mathrm{v}_1 \Delta \mathrm{t}
$$

As it is known,

$$
\mathrm{m}=\mathrm{pV}
$$

Where,
$\mathrm{m}=$ Mass
$\mathrm{p}=$ Density
$V=$ Volume

So, the mass of liquid (fluid) in part $\Delta \mathrm{x}_1$ will be:

$$
\begin{aligned}
& \Delta m_1=p \times V \\
& \Delta \mathrm{~m}_1=\mathrm{p}_1 \mathrm{~A}_1 \mathrm{v}_1 \Delta \mathrm{t} \ldots \ldots (1)
\end{aligned}
$$

Also read :

• NCERT solutions for Class 11 Physics Chapter 10 Mechanical Properties of Fluids
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Therefore, at the lower region of the pipe, we have to calculate the mass flux. Mass flux is the 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 the pipe having a cross-sectional area $A_1$_, the mass flux is given by:

$$
\frac{\Delta m_1}{\Delta_t}=p_1 A_1 v_1 \ldots . .(2)
$$

Likewise, the mass flux of the liquid (fluid) at the upper end of the pipe will be:

$$
\frac{\Delta m_2}{\Delta t}=p_2 A_2 v_2 \ldots \ldots (3)
$$

Where,

$v_2$ is the velocity of the liquid flowing in the upper end of the pipe
$\Delta \mathrm{x}_2$ is the distance travelled by the fluid
$\Delta t$ is time
$\mathrm{A}_2$ is an area of cross-section of the upper end of the pipe
It is speculated (assumed) that the density of the liquid (fluid) in the lower end of the pipe is the same as that of the upper end. Then the fluid flux (flow) is said to be streamlined.

So 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. Then

Equation $2=$ Equation 3
Therefore,

$$
p_1 A_1 v_1=p_2 A_2 v_2 . (4)
$$

From eq. (4) we can write:

$\rho A v=$ constant
This equation proves the law of conservation of mass in liquid (fluid) dynamics. As the liquid (fluid) is taken to be compact, the density of the liquid (fluid) will be constant for steady flow.

Therefore, $\mathbf{p}_1=\mathbf{p}_2$
Put it this to Eq. 4; it can be written as:

$$
\mathrm{A}_1 \mathrm{v}_1=\mathrm{A}_2 \mathrm{v}_2
$$

The general form of this eq. is:

$$
A v=\text { constant }
$$

Now, let's assume R as the volume flow rate, consequently the eq. can be stated as:

$$
\mathrm{R}=\mathrm{A} \vee=\text { constant }
$$

Hence, it is the derivation of the continuity equation.

What is the continuity equation in differential form?

By the divergence theorem, a common continuity equation can also be expressed as in a "differential form":

$$
\frac{\partial p}{\partial t}+\nabla \cdot j=\sigma
$$

Where,
$\nabla=$ divergence
$\rho=a m t$. of quantity as per unit volume
$j=$ flux of $q$.
$\mathrm{t}=$ time
$\sigma=$ generation of q per unit volume per unit time

This common equation is used to derive any continuity equation, varying from as uncomplicated as the volume continuity equation to as complex as the Navier–Stokes equations. This eqn. also generalizes the parameterization equation.

In the condition that $q$ is a conserved quantity that cannot be created or destroyed (like energy), $\sigma=0$ and the equation become

$$
\frac{\partial \rho}{\partial t}+\nabla \cdot \mathbf{j}=0
$$

What is the equation of continuity in electromagnetism?

In electromagnetism, the continuity equation is an empirical law demonstrating charge conservation. Mathematically it is an automatic necessity of Maxwell's equations, even though charge conservation is more elemental than Maxwell's equations. It expresses that the divergence of the current density J (amperes per square meter) is equal to the negative speed of change of the charge density $\rho$ (coulombs per cubic meter),

$$
\nabla \cdot j=-\frac{\partial p}{\partial t}
$$

Current is the flow of charge. The continuity equation says that if a charge discharges out of a differential volume (i.e., a divergence of current density is positive) then the number of charges within that volume decreases. So, the rate of modification of charge density is negative. Therefore, the continuity equation accounts for the conservation of charge.

NCERT Physics Notes :

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What is the continuity equation in semiconductors?

The entire current flow in the semiconductor embraces the drift current and diffusion current of both the electrons in the conduction band and holes in the valence band. Generalize form for electrons in 1 (one) dimension:

$$
\frac{\partial n}{\partial t}=n \mu_n \frac{\partial E}{\partial x}+\mu_n E \frac{\partial n}{\partial x}+D_n \frac{\partial^2 n}{\partial x^2}+\left(G_n-R_n\right)
$$

Where:
$\mathrm{n}=$ local concentration of electrons
$\mu_n=$ electron mobility
$E=$ electric field across the depletion part
$D_n=$ diffusion coefficient for electrons
$\mathrm{G}_{\mathrm{n}}=$ rate of generation of electrons
$R_n=$ rate of recombination of electrons

Continuity Equation in Cylindrical Coordinates

The continuity equation in cylindrical coordinates is:

$$
\frac{\partial p}{\partial t}+\frac{1}{r} \frac{\partial r p u}{\partial r}+\frac{1}{r} \frac{\partial p v}{\partial \theta}+\frac{\partial p w}{\partial z}=0
$$

Incompressible Flow Continuity Equation/ continuity equation incompressible flow

The continuity equation for incompressible flow as the density, $p=$ constant and is independent of distance and time, the equation is:

$$
\nabla . v=0
$$

Steady Flow Continuity Equation

The continuity equation in cylindrical coordinates is:

$$
\frac{\partial}{\partial x}(p u)+\frac{\partial}{\partial y}(p v)+\frac{\partial}{\partial z}(p w)=0
$$

All these are included in the equation of continuity class 11/continuity equation in three dimensions in fluid mechanics

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