# What is Continuity equation: Definition and 87 Discussions

A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described using continuity equations.
Continuity equations are a stronger, local form of conservation laws. For example, a weak version of the law of conservation of energy states that energy can neither be created nor destroyed—i.e., the total amount of energy in the universe is fixed. This statement does not rule out the possibility that a quantity of energy could disappear from one point while simultaneously appearing at another point. A stronger statement is that energy is locally conserved: energy can neither be created nor destroyed, nor can it "teleport" from one place to another—it can only move by a continuous flow. A continuity equation is the mathematical way to express this kind of statement. For example, the continuity equation for electric charge states that the amount of electric charge in any volume of space can only change by the amount of electric current flowing into or out of that volume through its boundaries.
Continuity equations more generally can include "source" and "sink" terms, which allow them to describe quantities that are often but not always conserved, such as the density of a molecular species which can be created or destroyed by chemical reactions. In an everyday example, there is a continuity equation for the number of people alive; it has a "source term" to account for people being born, and a "sink term" to account for people dying.
Any continuity equation can be expressed in an "integral form" (in terms of a flux integral), which applies to any finite region, or in a "differential form" (in terms of the divergence operator) which applies at a point.
Continuity equations underlie more specific transport equations such as the convection–diffusion equation, Boltzmann transport equation, and Navier–Stokes equations.
Flows governed by continuity equations can be visualized using a Sankey diagram.

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1. ### I Klein-Gordon equation and continuity equation

Hi I am using the textbook "Modern Particle Physics" by Thomson. Working from the K-G equation and comparing with the continuity equation he states that the probability density is given by ρ = i ( ψ*(∂ψ/∂t) - ψ(∂ψ*/∂t) ) He then states that the factor of i is included to ensure that the...
2. ### From fluid energy conservation equation to the continuity equation

Hey there, First of all, all energy conservation equations for a fluid I found on google hadn't the ##\gamma## coefficient. What exactly is the difference? Secondly, by substituting e by ##e = \frac{1}{\gamma -1} \frac{p}{\rho}## in the following equation ##\frac{De}{Dt} + (\gamma - 1)e \nabla...

8. ### B Fluid Continuity Equation in different reference frame

If I have fluid with area 10 and velocity 10, if the velocity increases to 20 the area will become 5. But if we switch to a reference frame moving at velocity 1 opposite this motion, then it would be 10 and 11 to 5 and 21, violating the continuity equation. What is wrong?
9. ### Conservation of charge with Dirac delta

Hello, I was reviewing a part related to electromagnetism in which the charge and current densities are defined by the Dirac delta: ##\rho(\underline{x}, t)=\sum_n e_n \delta^3(\underline{x} - \underline{x}_n(t))## ##\underline{J}(\underline{x}, t)=\sum_n e_n \delta^3(\underline{x} -...
10. ### Cause-effect relation between pressure & velocity

For a steady, non-viscous and incompressible flow, one can apply both Bernoulli's principle (no potentials) as $$p+\frac{\rho v^2}{2} = p_t$$ where ##p##, ##\rho,##, ##v##, and ##p_t## are static pressure, density, flow velocity, and total pressure, respectively, and continuitiy principle as...

Hello All : reading the Bo Thide book in electromagnetism , downloaded the draft copy from the following link http://www.plasma.uu.se/ , i reached the chapter 4 now and a section in that chapter (section 4.3) have few lines that i coudnt understand (mathematically speaking) the writer conclude...
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36. ### Relationship between Principle of Least Action and Continuity Equation

Is there a profound relationship between Principle of Least Action and Continuity Equation? Can we derive one from another?
37. ### Continuity equation derivation in Griffiths - why partial derivative?

Greetings, In Griffiths E&M, 3rd. Ed., on page 214, the following is part of the derivation of the continuity equation (the same derivation is shown on the Wikipedia article for the current density, under the continuity equation section: http://en.wikipedia.org/wiki/Current_density)...
38. ### What was the first continuity equation?

Does anyone happen to know who wrote down the first continuity equation and with regard to what? I know it shows up everywhere but was it originally a fluid dynamics equation? I've been trying to research this but I'm not coming up with much history on it. Thanks!
39. ### Continuity equation and air flow

Although continuity equation is often part of fluid mechanics, does it have an application in air flow? For example, let's assume we have a frictionless air duct where air is introduced at a constant velocity and temperature. If the air duct varies in dimensions will the flow rate at the end...
40. ### Continuity Equation - For a vertical pipe

I don't understand the ideas behind the continuity equation when applied to a vertical pipe. In all the questions I see regarding a vertical pipe of constant diameter, I see that the fluid's velocity will remain constant while traveling through the pipe. Common sense will tell you this isn't...
41. ### Continuity equation, partial derivative and differential operators

Hi all! I have the following slide, and whilst I understand that the original point is "the rate of density, ρ, in each volume element is equal to the mass flux"...i am totally lost on the mathematics! (And I am meant to be teaching this tomorrow). I do not have any information on what the...
42. ### Relationship Between the Probability Current and Continuity Equation

I'm currently reading through a textbook by David Miller and attempting to teach myself quantum mechanics to assist with my electrical engineering. I have run into a little trouble trying to understand how the probability current satisfies the continuity equation with a probability distribution...
43. ### Electrodynamics Continuity Equation

Homework Statement I am currently studying for a quiz and then following a test in my Electrodynamics test. Right now I am struggling to define the following: Continuity equation and its physical meaningHomework Equations The Continuity Equation is given as the following: ∇J=-∂ρ/∂t The Attempt...
44. ### Continuity equation, cartisan to polar

Hello, I've allways wondered how to get to polar coordinates from cartisan coordinates. I took a course in fluid mechanics but we never learned how to get the continuity equation from cartisan to polar. I know you can use physics to derive the polar equation, but I want to do it just by using...
45. ### Continuity Equation in an Electromagnetic Field

Homework Statement Derive the continuity equation for a charged particle in an electromagnetic field Homework Equations The time-dependent Schrodinger equation and its complex conjugate are i\hbar\frac{\partial \psi}{\partial t}=\frac{1}{2m}(-i\hbar \vec{\nabla} - \frac{e}{c}...
46. ### Klein-Gordon Equation & Continuity Equation

Hello, My question is on the Klein-Gordon equation and it's relation to the continuity equation, so for a Klein-Gordon equation & continuity equation of the following form, I have attained the following probability density and probability current relations (although not normalised correctly...
47. ### Continuity Equation: Relationship between vA and vB in terms of d and D

Homework Statement The continuity equation provides a second relation between the vA and vB, this time in terms of the diameters d and D. Numerical check: If the diameters are d = 1 cm and D = 10 cm, what is the ratio of the speeds, vB/vA? Homework Equations To clarify, is both d and D...
48. ### Derivations for Continuity equation of Fluid & Euler's Equation of Fluid Motion

Will anyone give me the derivations for continuty equation of fluid and euler's equation of fluid motion .
49. ### Fluid dynamics: Knowledge continuity equation

Homework Statement Please click on the link for the question. http://i1154.photobucket.com/albums/p526/cathy446/physicsquestion_zps49e16ab1.jpg Assume that air spreads out after coming out from the tube at 2. The speed over tube 1 is almost zero. Homework Equations Knowledge problem on...
50. ### Confusion regarding continuity equation in electrodynamics

Suppose I have two charged particles with charge densities ρ1(r,t) and ρ2 (r,t) with corresponding velocity fields V1(r,t) and V2(r,t). Can I write continuity equation for the combined system? Wouldn't charges moving with different velocities would contribute differently to the current which...