Derivation of Continuity Equation

Edited By Vishal kumar | Updated on Jul 02, 2025 05:04 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?

Derivation of Continuity Equation - Definition, Formula, FAQs

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.

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:

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

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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.

Related Topics,

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 :

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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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)

11. How does the continuity equation differ for compressible and incompressible fluids?

For incompressible fluids, the density remains constant, simplifying the equation to focus on the relationship between velocity and cross-sectional area. For compressible fluids, the equation must account for changes in density, making it more complex and involving additional terms related to density variations.

12. What is the general form of the continuity equation?

The general form of the continuity equation is:

13. How does the continuity equation simplify for steady, incompressible flow?

For steady, incompressible flow, the continuity equation simplifies to:

14. Can you explain the concept of mass flow rate in relation to the continuity equation?

Mass flow rate is the amount of mass flowing through a cross-section per unit time. It's directly related to the continuity equation and can be expressed as:

15. How does the continuity equation relate to the concept of streamlines in fluid flow?

Streamlines are imaginary lines in a fluid that are tangent to the velocity vector at every point. The continuity equation ensures that streamlines cannot cross each other in steady flow, as this would violate mass conservation. It also dictates that streamlines become closer together where the fluid velocity increases.

16. What are the assumptions made when deriving the continuity equation?

The main assumptions in deriving the continuity equation are:

17. Can the continuity equation be applied to gases as well as liquids?

Yes, the continuity equation applies to both gases and liquids. However, for gases, which are compressible, the full form of the equation including density changes must be used. For liquids, which are generally treated as incompressible, the simplified form is often sufficient.

18. What role does the continuity equation play in designing fluid systems like pipelines?

In designing fluid systems, the continuity equation is crucial for:

19. What is the difference between the differential and integral forms of the continuity equation?

The differential form (∂ρ/∂t + ∇ · (ρv) = 0) describes mass conservation at a point, considering local changes in density and velocity. The integral form applies to a finite control volume, relating the mass flux through the boundaries to the rate of mass change within the volume. Both forms express the same principle but are used in different contexts.

20. How does the continuity equation relate to the concept of incompressibility in fluids?

For incompressible fluids, the continuity equation simplifies because density remains constant. This allows us to focus solely on the relationship between velocity and cross-sectional area. The assumption of incompressibility is often valid for liquids and gases at low speeds, making the simplified equation widely applicable.

21. What is the continuity equation in fluid dynamics?

The continuity equation in fluid dynamics is a fundamental principle that describes the conservation of mass in a flowing fluid. It states that the rate at which mass enters a system is equal to the rate at which mass leaves the system plus the rate of change of mass within the system.

22. Why is the continuity equation important in studying fluid flow?

The continuity equation is crucial because it helps us understand how the velocity, density, and cross-sectional area of a fluid change as it flows through different regions. This principle is essential for analyzing and predicting fluid behavior in various applications, from pipe systems to aerodynamics.

23. How does the continuity equation relate to the conservation of mass?

The continuity equation is a direct application of the conservation of mass principle. It ensures that mass is neither created nor destroyed during fluid flow, maintaining a balance between the mass entering and leaving a system, accounting for any changes in density or volume.

24. How does the continuity equation apply to a varying cross-sectional area, like in a nozzle?

In a nozzle or any varying cross-sectional area, the continuity equation shows that as the area decreases, the velocity must increase to maintain the same mass flow rate. This relationship is expressed as A₁v₁ = A₂v₂, where the subscripts represent two different points along the flow.

25. What is the significance of the equation A₁v₁ = A₂v₂ in fluid dynamics?

This equation, derived from the continuity principle for incompressible fluids, shows the inverse relationship between cross-sectional area and fluid velocity. It's crucial for understanding how fluids behave in pipes, nozzles, and other systems with varying cross-sections, helping to predict velocity changes.

26. Can you explain how the continuity equation is used in weather forecasting?

In weather forecasting, the continuity equation helps model atmospheric flows. It's used to:

27. Can you explain how the continuity equation is used in designing wind turbines?

In wind turbine design, the continuity equation helps:

28. How does the continuity equation relate to the concept of fluid slip at boundaries?

The continuity equation doesn't directly address fluid slip, but it's affected by boundary conditions. In no-slip conditions (typical for viscous fluids), the velocity at the boundary is zero. With slip conditions, there's a non-zero velocity at the boundary. The continuity equation must be solved with appropriate boundary conditions to accurately model the flow, especially near surfaces.

29. How does the continuity equation help in understanding the Venturi effect?

The Venturi effect, where fluid pressure decreases as it flows through a constricted section of pipe, is a direct consequence of the continuity equation. As the pipe narrows, the equation dictates that fluid velocity must increase to maintain the same mass flow rate, leading to a pressure drop according to Bernoulli's principle.

30. Can you explain how the continuity equation is used in analyzing river flow?

In river flow analysis, the continuity equation helps understand how water velocity changes with river width and depth. It shows that in narrower or shallower sections, water velocity increases to maintain the same volumetric flow rate. This is crucial for predicting flood behavior, erosion patterns, and sediment transport.

31. How does the continuity equation apply to multiphase flows?

In multiphase flows (e.g., mixtures of gas and liquid), the continuity equation must be applied to each phase separately. The total mass conservation is then the sum of the individual phase continuity equations. This approach accounts for the different densities and velocities of each phase in the mixture.

32. Can the continuity equation be used to explain why rivers flow faster in some sections than others?

Yes, the continuity equation explains this phenomenon. In wider, deeper sections of a river, the water flows more slowly. As the river narrows or becomes shallower, the water must flow faster to maintain the same volumetric flow rate, as dictated by A₁v₁ = A₂v₂. This principle is key to understanding river dynamics and flow patterns.

33. How does the continuity equation relate to the concept of mass balance in chemical engineering?

The continuity equation is essentially a mass balance applied to fluid flow. In chemical engineering, mass balance ensures that the total mass entering a system equals the mass leaving plus any accumulation. The continuity equation applies this principle to flowing fluids, making it a fundamental tool in process design and analysis.

34. What is the significance of the divergence term in the continuity equation?

The divergence term (∇ · (ρv)) in the continuity equation represents the net outflow of mass per unit volume. It accounts for how the mass flux varies in different directions. A positive divergence indicates mass leaving a region, while a negative divergence indicates mass entering, crucial for understanding three-dimensional flow patterns.

35. How does the continuity equation apply to compressible flows in high-speed aerodynamics?

In high-speed aerodynamics, where compressibility effects are significant, the full form of the continuity equation must be used. It accounts for density changes due to pressure and temperature variations. This is crucial for analyzing phenomena like shock waves and supersonic flows, where fluid properties change rapidly across small distances.

36. How does the continuity equation relate to the concept of steady and unsteady flows?

For steady flows, where conditions don't change with time, the continuity equation simplifies as ∂ρ/∂t becomes zero. In unsteady flows, this time-dependent term remains, accounting for temporal changes in density or velocity. Understanding this distinction is crucial for analyzing time-varying fluid systems like pulsating flows or transient phenomena.

37. What is the role of the continuity equation in studying blood flow in the circulatory system?

In studying blood flow, the continuity equation helps understand how blood velocity changes as it moves through vessels of different sizes. It explains why blood flows faster in narrower vessels and slower in wider ones, maintaining constant volumetric flow rate. This is crucial for analyzing circulation, designing artificial vessels, and understanding cardiovascular diseases.

38. How does the continuity equation apply to non-Newtonian fluids?

The continuity equation applies to non-Newtonian fluids in the same way as Newtonian fluids, as it's based on mass conservation, not fluid properties. However, when analyzing flow behavior, the equation must be used in conjunction with appropriate constitutive equations that describe the non-Newtonian fluid's stress-strain relationship.

39. Can you explain how the continuity equation is used in studying groundwater flow?

In groundwater studies, the continuity equation helps model fluid movement through porous media. It's used to:

40. How does the continuity equation relate to the concept of flow rate in fluid dynamics?

The continuity equation directly relates to flow rate. For incompressible flow, the volumetric flow rate Q = Av remains constant. This means that as the cross-sectional area changes, the velocity must adjust to maintain a constant flow rate, which is a key application of the continuity principle in fluid systems.

41. What is the importance of the continuity equation in studying ocean currents?

In oceanography, the continuity equation is crucial for:

42. How does the continuity equation apply to sound waves in fluids?

For sound waves in fluids, the continuity equation describes how density fluctuations propagate. It relates the rate of change of density to the divergence of particle velocity, helping to model sound wave behavior, including reflection, refraction, and diffraction in various media.

43. How does the continuity equation relate to the concept of mass diffusion in fluids?

While the continuity equation primarily deals with bulk fluid motion, it can be extended to include mass diffusion. In this case, an additional term is added to account for the flux due to concentration gradients, allowing for the analysis of combined convection and diffusion processes in fluid systems.

44. What is the significance of the continuity equation in studying atmospheric pollution dispersion?

In atmospheric pollution studies, the continuity equation helps:

45. How does the continuity equation apply to multiscale fluid dynamics problems?

In multiscale problems, the continuity equation is applied at different scales, from microscopic to macroscopic. At smaller scales, it might involve molecular dynamics, while at larger scales, it deals with bulk fluid properties. Bridging these scales often requires careful consideration of how mass conservation principles apply across different levels of fluid behavior.

46. Can you explain how the continuity equation is used in studying volcanic eruptions?

In volcanology, the continuity equation helps:

47. How does the continuity equation relate to the concept of fluid compressibility?

The continuity equation accounts for fluid compressibility through the density term. For incompressible fluids, density remains constant, simplifying the equation. For compressible fluids, density changes must be considered, making the equation more complex. This distinction is crucial in analyzing flows where pressure changes significantly affect fluid density.

48. What is the role of the continuity equation in studying tsunami propagation?

In tsunami studies, the continuity equation is essential for:

49. How does the continuity equation apply to non-uniform flows?

For non-uniform flows, where fluid properties vary spatially, the full form of the continuity equation must be used. It accounts for changes in velocity, density, and cross-sectional area along the flow path. This is crucial for analyzing complex flow patterns in natural systems or engineered devices with varying geometries.

50. Can you explain how the continuity equation is used in studying heat transfer in fluids?

While the continuity equation itself doesn't directly involve heat transfer, it's often used alongside energy equations in heat transfer problems. It ensures mass conservation while energy equations account for heat transfer. Together, they help model how fluid flow and heat transfer interact in systems like heat exchangers or natural convection processes.

51. What is the importance of the continuity equation in studying fluid-structure interactions?

In fluid-structure interactions, the continuity equation helps:

52. How does the continuity equation apply to porous media flow?

In porous media, the continuity equation is modified to account for the porosity of the material. It describes how fluid moves through the interconnected pores, considering both the fluid properties and the characteristics of the porous medium. This is crucial in fields like hydrogeology, petroleum engineering, and filtration systems.

53. Can you explain how the continuity equation is used in studying planetary atmospheres?

For planetary atmospheres, the continuity equation helps:

54. How does the continuity equation relate to the concept of streamfunction in 2D flows?

The streamfunction is a mathematical tool that automatically satisfies the continuity equation for 2D incompressible flows. It's defined such that its partial derivatives give the velocity components, ensuring mass conservation. This relationship simplifies the analysis of 2D flow problems by reducing the number of variables needed to describe the flow.

55. What is the role of the continuity equation in studying sediment transport in rivers?

In sediment transport studies, the continuity equation is