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. D

    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. H

    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...
  3. ConjugatedClanger

    I What does ## \psi^{*}\hat{H}\psi ## mean?

    I was reading a section of The Physics off Quantum Mechanics by James Binney and David Skinner. On page 45, when discussing the probability current (in the wave mechanics formalism) in calculating it they state: I.e.$$ i \hbar \left( \psi^{*} \frac{\partial}{\partial t} \psi + \psi...
  4. D

    Simple question regarding weight flow rate and specific gravity

    i’m studying my textbook in hydraulics and i’d like to know the reason why specific gravity is used in one of the problems. it’s a pretty simple problem and i understood it mostly except why specific gravity is multiplied to 9810 (the unit weight of water). if the SG is given in the problem, is...
  5. ergospherical

    Linearisation of continuity equation (cosmology)

    After expanding to first order in ##\epsilon## and subtracting off the unperturbed equation, I get\begin{align*} \frac{\partial \delta \rho}{\partial t} + 3H \delta \rho + \frac{\bar{\rho}}{a} \nabla \cdot \delta \mathbf{v}=0 \end{align*}I'm not sure how to deal with the ##3H \delta \rho## term...
  6. huangdaiyu

    A Steady state confined flow field: Is it cyclic?

    For a fluid that is confined to a finite region with no sources and sinks, are the only options for the flow field a) static, and b) cyclic? The example I have in mind is Rayleigh convection in a shallow dish heated from below, where convection cells are formed beyond a certain temperature...
  7. G

    Fluid mechanics: water jet impacting an inclined plane

    I was looking at an example of fluid mechanics and I don't understand this. Statement figures: CONTINUITY EQUATION $$\left. \dfrac{dm}{dt}\right]_{MC}=(\dot{m}_2+\dot{m}_3)-\dot{m}_1=0$$ $$\dot{m}_1=\dot{m}_2+\dot{m}_3$$ $$\rho c_1A_1=\rho c_2A_2+\rho c_3A_3$$ $$\rho c_1 h1=\rho c_2 a1+\rho...
  8. V

    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. F

    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. lahanadar

    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...
  11. hagopbul

    Question about this Continuity Equation (electromagnetism)

    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...
  12. E

    B How to derive the continuity equation for a perfect fluid?

    Stress tensor for the fluid is ##T_{ab} = \rho u_a u_b + P(\eta_{ab} + u_a u_b)##, whilst the equation of motion (assuming the system is isolated) is given by ##\partial^a T_{ab} = 0##. So I tried$$\begin{align*} \partial^a T_{ab} &= \partial^a \rho u_a u_b + \partial^a P(\eta_{ab} + u_a u_b)...
  13. B

    Fluid velocity and pipe diameter using the continuity equation

    Hi, Can anyone let me know if I had done this Q correctly? Thanks for any help!
  14. speaknow

    Continuity equation of the electric field

    According to the continuity equation of the electric field (i.e., ▽·Ε = 0) a decrease in cross-section area will increase the electric field strength, Why is that?
  15. brotherbobby

    Water flowing out of a tank through a hole

    I assume the water to start flowing from rest at position 1, hence ##v_1 = 0##. Applying the continuity equation, ##A_1 v_1 = A_2 v_2##, we find the (wrong) result that the velocity at position 2 is ##v_2 = 0## also! (We assume that ##A_2## is small but finite) Hence, to answer the question...
  16. A

    A Continuity Equation for fluid in a curved spacetime

    The current of fluid is the vector J^{\nu}. In free-falling laboratory due to Equivalence principle holds the know Continuity Equation J^{\nu}_{, \nu}=0, where the ordinary 4-divergence is used. Latter equation was derived in Minkowski spacetime, thus the Christoffel Symbols are all zero for...
  17. D

    I Deciphering the Units of Continuity Equation in GR

    Taking the time derivative of the energy density of the energy momentum tensor should equal to the spatial derivatives or divergence of the momentum density components. How do the units work out though? shouldn't a time derivative of the energy density be in kg/xt^3 and the spatial derivative of...
  18. Pushoam

    A Understanding Continuity Equation & Conservation Laws

    I understand that from local conservation of charge, we get eqn. 8.4. I don't get why it is called continuity eqn. What is continuous in it? Conservation of momentum gives us equation, ## \frac {d\vec p }{dt} = \vec F ##. This equation is not called continuity equation. Can we get a continuity...
  19. W

    I The continuity equation and the divergence

    according to continuity equation (partial ρ)/(partial t) +divergence J = 0 . there is such a situation that there is continuous water spreads out from the center of a sphere with unchanged density ρ, and at the center dm/dt = C(a constant), divergence of J = ρv should be 0 anywhere except the...
  20. M

    Continuity equation and mass flowing through two pipes

    Can we apply continuity equation between the given two cases? The only difference in the second case is that the pipe of diameter d2 is replaced by a pipe of diameter d3. Will the mass flow rate be same for both the cases.
  21. I

    Bernoulli's Equation, Find gauge pressure at 2nd point

    Homework Statement At a certain point in a pipeline, the velocity is 1 m/s and the gauge pressure is 3 x 105 N/m2. Find the gauge pressure at a second point in the line 20 m lower than the first if the cross-section at the second point is one half that at the first. The liquid in the pipe is...
  22. J

    Why Does Zero Divergence of Current Density Imply Zero Charge Density?

    The question is about this equation: Divergence of J= - ∂ρ/∂t (equation 1) Where ρ is the density of electric charges/ volume J= the current density = Amperes/m2 I understand that if the divergence is not zero, the rate of change of the amount of charge is changing inside a closed control...
  23. binbagsss

    Deriving continuity equation of phase space in Statistical Mechanics

    Hi, So I am aiming to derive the continuity equation using the fact that phase space points are not created/destroyed. So I am going to use the Leibiniz rule for integration extended to 3-d: ## d/dt \int\limits_{v(t)} F dv = \int\limits_{v(t)} \frac{\partial F}{\partial t} dV +...
  24. It's me

    Using Noether's Theorem find a continuity equation for KG

    Homework Statement Consider the Klein-Gordon equation ##(\partial_\mu \partial^{\mu}+m^2)\varphi(x)=0##. Using Noether's theorem, find a continuity equation of the form ##\partial_\mu j^{\mu}=0##. Homework Equations ##(\partial_\mu \partial^{\mu}+m^2)\varphi(x)=0## The Attempt at a Solution...
  25. N

    Fluid dynamics continuity equation

    Homework Statement Through a (cylindrical) heating tube, warm air has to flow with a velocity of 3m/s in order to heat a rectangular spacewhich is 12 m long, 10 m wide and 2.5 m high. How large the radius of the tube should be if the air in the room get refreshed every 15 minutes ? Assume that...
  26. C

    A Continuity equation in Lagrangian coordinates

    From the Eulerian form of the continuity equation, where x is the Eulerian coordinate: \frac {\partial \rho}{ \partial t } + u \frac {\partial \rho}{\partial x} + \rho \frac { \partial u}{\partial x} = 0 The incremental change in mass is, where m is the Lagrangian coordinate: dm = \rho dx...
  27. E

    Electromagnetic continuity equation

    Hi, this looks stupid and simple, but I just can't get my head around it. Assuming a homogeneous medium. The electromagnetic continuity equation goes as ∇⋅J + ∂ρ/∂t = 0 since J = σE, ρ = ɛ∇⋅E, and assuming the time dependence exp(-iωt) we have σ∇⋅E - iωɛ∇⋅E = 0 (σ - iωɛ)∇⋅E = 0 So, σ - iωɛ = 0...
  28. H

    Understanding the Continuity Equation for Mass Flow in Pipes

    Homework Statement can someone about the continuity equation ? i only know that for a pipe , Q1 = Q2 , where Q1 and Q2 represent the rate of mass flow , Q= Av , where A= area , v = velocity Homework EquationsThe Attempt at a Solution
  29. nirmaljoshi

    Energy equation and continuity equation mismatch

    A pipe is discharging from H=100m to an open atmosphere. The available discharge at inlet is 0.4m3/s 1. The bernauli's equation tells v=sqrt(2*g*H), neglecting any losses in the pipe, v=~44m/s 2. continuity equation tells v=Q/A, say diameter of pipe=0.3m, v=~5.65m/s which is correct?
  30. S

    MHB PDE Solving Continuity Equation

    Hi, I am trying to find the exact solution of the Continuity Equation. Any Idea how can i start solving it, i need it for some calculation in Image Processing. $$\pd{C}{t}+\pd{UC}{x}+\pd{VC}{y}=0$$ Where $U$ and $V$ is velocity in $X$ and $Y$ direction. The initial condition is as...
  31. bananabandana

    Derivation of the Continuity Equation for Fluids

    Homework Statement Derive a mathematical relationship which encapsulates the principle of continuity in fluid flow. Homework EquationsThe Attempt at a Solution Imagine we have a mass of fluid ## M##, of volume ##V##, bounded by a surface ##S##. If we take a small element of this volume...
  32. DontDeimos

    Using Maxwell's equations to derive the continuity equation

    Homework Statement Use Maxwell's equations to derive the continuity equation. B=Magnetic Field E=Electric Field ρ=Charge Density J=Current Density Homework Equations Maxwell's Equations: ∇⋅E=ρ/ε0, ∇×E=-∂B/∂t ∇⋅B=0 ∇×B=ε0μ0(∂E/∂t)+μ0J Continuity Equation: ∇⋅J +∂ρ/∂t = 0 The Attempt at...
  33. P

    Interpretation for continuity equation with complex potential.

    For the potential V(x)=V1(x)+iV2(x) the continuity equation yields: ∇⋅j=-∂ρ/∂t + 2*ρ*V2/ħ (unless I am mistaken). What is the interpretation of this result?
  34. P

    The continuity equation in electromagnetism

    I understand the reasoning behind the equations ∫SJ.dS=-dQ/dt and thus ∇.J=-∂ρ/∂t. where the integral is taken over the closed surface S. However I'm a little confused about the conditions of steady currents: The book I'm using sets dQ/dt=0 and ∂ρ/∂t=0 in these cases. I don't understand this...
  35. C

    Continuity equation (charge vs matter) in SR

    If we consider a perfect relativistic fluid it has energy momentum tensor $$T^{\mu \nu} = (\rho + p) U^\mu U^\nu + p\eta^{\mu \nu} $$ where ##U^\mu## is the four-velocity field of the fluid. ##\partial_\mu T^{\mu \nu} = 0## then implies the relativistic continuity equation...
  36. C

    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. H

    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. C

    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. B

    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. H

    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. P

    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. D

    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. C

    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. A

    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. R

    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. S

    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. B

    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. S

    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. A

    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. P

    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...