Derivation of Continuity Equation - Definition, Formula, FAQs

Edited By Vishal kumar | Updated on Sep 24, 2024 09:48 PM IST

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?

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Derivation of Continuity Equation - Definition, Formula, FAQs

What is the continuity equation?/define equation of continuity

Definition: In physics, the equation of continuity is defined as a 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

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 rate of flow. The continuity eq. is representing as:

R = A v = constant

Where,

R = volume flow rate

A = flow area

v = flow velocity

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Derivation of Continuity Equation speculations (Assumption)

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 Δt. In the course of this time, the liquid (fluid) will cover an distance of Δx₁, with a velocity of v₁ in the lower fragment of the pipe.

The interval covered by the fluid with rate v₁in time Δt will be shown by,

Δx₁= v_1Δt

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

V= A₁∆x₁= A₁v₁∆t

As it is known,

m = pV

Where,

m = Mass

p = Density

V = Volume

So, the mass of liquid (fluid) in part Δx₁will be:

$\Delta m_1=p\times V$

Δm₁=p₁A₁v₁∆t………(1)

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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 time. For the lower part of the pipe, with the lower end of pipe having a cross-sectional area A₁_, the mass flux are given by,

$\frac{\Delta m_1}{\Delta_t}=p_1A_1v_1.....(2)$

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

$\frac{\Delta m_2}{\Delta t}=p_2A_2v_2......(3)$

Where,

v₂ is the velocity of the liquid flowing in the upper end of the pipe

∆x₂is the distance travelled by the fluid

Δt is time

A₂ is 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,

p₁A₁v₁= p₂A₂v₂………..(4)

From eq. (4) we can write:

ρAv=constant

This equation he;ps to prove 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, p₁=p₂

Put it this to Eq. 4; it can be written as:

A₁v₁=A₂v₂

The general form of this eq. is:

A v = constant

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

R = A v = constant

Hence, it is the derivation of the continuity equation.

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What is the continuity equation in differential form?

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

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

Where,

∇=divergence

ρ=amt. of qty.qper unit volume

j = flux of q.

t = time

σ=generation of q per unit volume per unit time

This common eqn. are 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), σ = 0 and the eqns become

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 necessarily of Maxwell's equations, even though charge conservation is more elemental than Maxwell's equations. It express that the divergence of the current density J (amperes per square metre) is equal to the negative speed of change of the charge density ρ (coulombs per cubic metre),

$\triangledown .j=-\frac{\partial p}{\partial t}$

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

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What is the continuity equation in semiconductors?

The entire current flow in the semiconductor embraces 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:

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

Where:

n = local concentration of electrons

$\mu _n$ = electron mobility

E = electric field across the depletion part

D_n = diffusion coefficient for electrons

G_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:

$\frac{\partial p}{\partial t}+\frac{1}{r}\frac{\partial rpu}{\partial r}+\frac{1}{r} \frac{\partial pv}{\partial \theta}+ \frac{\partial pw}{\partial z}=0$

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:

∇.v=0

Steady Flow Continuity Equation

The continuity equation in cylindrical coordinates is:

$\frac{\partial}{\partial x}(pu)+\frac{\partial }{\partial y}(pv)+\frac{\partial }{\partial z}(pw)=0$

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

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Frequently Asked Questions (FAQs)

1. What is the principle of the continuity equation?

Principle of continuity equation: Continuity principle, or continuity equation, The principle is a result of the law of conservation of mass. Continuity equation deals with the law of conservation of fluid mechanics.

2. What are the Applications of the continuity equation?

The primary application of the continuity equation or equation of continuity is involved in the field of Hydrodynamics, Electromagnetism, Aerodynamics, and Quantum Mechanics. The eqn of continuity forms the elemental rule of Bernoulli's Principle. It is also related to the Aerodynamics principle with its applications.

The differential form of the continuity eq. is used to find out the consistency of Maxwell's Equation. Aside from, the differential form of the continuity equation is also used in Electromagnetism.

Continuity eq. is used to examine the consistency of Schrodinger Eq.

Common and Special Theory of Relativity, Noether's Theorem, also used the Continuity equation.

3. What is the Significance of Eq. of continuity?

The equation of continuity is a applicable data to solve many characteristics of the fluid during its motion:

As flow in = flow out, we can find out some characteristics of a liquid under some situations, and then we can apply the continuity equation to measure characteristics of that fluid under other situations.

Q₁=Q₂

This can be represented as:

A₁∗v₁=A₂∗v₂

Now, the continuity equation does measure its application to any incompressible fluid. Therefore the fluid is incompressible; the amt. of fluid that flows in a surface must equal the amt. of fluid that flows out of the surface.

Physical Significance

We can see the result of the equation of continuity in our garden. Water flows through the pipe of our garden, and when it reaches the small end of the pipe or the nozzle, the pace of water increases. With the increase of rate of the fluid, the cross-sectional area decreases and with the decrease in pace of fluid decreases, the cross-sectional area increases. Therefore, This is some significance of the equation of continuity.

4. Define continuity equation.

In physics, the equation of continuity is defined as a 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.

5. Write the steady flow equation.

From the steady flow energy equation derivation, the equation is given by: Q=m(h₂-h₁)

6. What is the principle of the continuity equation?

7. What are the Applications of the continuity equation?

The differential form of the continuity eq. is used to find out the consistency of Maxwell's Equation. Aside from, the differential form of the continuity equation is also used in Electromagnetism.

Continuity eq. is used to examine the consistency of Schrodinger Eq.

Common and Special Theory of Relativity, Noether's Theorem, also used the Continuity equation.

8. What is the Significance of Eq. of continuity?

The equation of continuity is a applicable data to solve many characteristics of the fluid during its motion:

Q1=Q2

This can be represented as:

A1∗v1=A2∗v2

Physical Significance

9. Define continuity equation.

10. Write the steady flow equation.

From the steady flow energy equation derivation, the equation is given by: Q=m(h2-h1)