continuity equation


The continuity equation[1] is a mathematical statement about mass-preserving in a continuumMathworldPlanetmathPlanetmath material body ℬ. This principle indeed is but a partial re-statement of the hypothesisMathworldPlanetmath of continuity of motion11Cf. motion of continuum, plus the introduction of a new physical constant 𝐌=π‹πŸ‘β’π“-𝟐. We shall postulateMathworldPlanetmath three properties for this associate positive quantity so-called mass, m=m⁒(ℬ).

Concept of mass density

In order to define mass densityPlanetmathPlanetmath we postulate:

1. We require that mass be additive, so that the masses d⁒m of volume elements of the body ℬ add up to the total mass of the body , i.e.


where 𝒱=𝒱⁒(𝐗,t0) is the volume occupied by the body ℬ in an initial material reference configurationMathworldPlanetmathPlanetmath χ⁒(ℬ)≑χ⁒(𝐗,t0), t0≀τ≀t, being t an arbitrary β€œpresent” instant.

2. We assume that mass is a continuous functionMathworldPlanetmathPlanetmath of the volume. That is a characteristic assumptionPlanetmathPlanetmath in continuum mechanics; in the mathematical languagePlanetmathPlanetmath we say that mass m⁒(ℬ) and volume 𝒱⁒(ℬ) are in the additive set functions.

3. In a field theory we must be able to describe the mass of a portion of a body. For this reason we introduce a point function called mass density, that we shall define in the following manner. Consider the smooth mappings β„¬β†’β„œ given by

Ο‡:𝒱⁒(𝐗,t0)→𝔳⁒(𝐱,Ο„),t0≀τ≀t, (1)

where 𝐗 fix the position of any material particle P0βˆˆβ„¬ at Ο„=t0, the initial reference configuration, and 𝐱=χ⁒(𝐗,Ο„) fix the position of the same particle, at point P located in certain region β„œ embedded in the Euclidean spaceMathworldPlanetmath (ℝ3,βˆ₯β‹…βˆ₯), where the volume spatial description 𝔳⁒(𝐱,Ο„) of the body ℬ is defined. So, the indicated coordinateMathworldPlanetmathPlanetmath transformationMathworldPlanetmath allows to define the mapping of neighborhoodsMathworldPlanetmathPlanetmath


to which we can associate the pair of (mΞ΄,𝒱δ) and (mΟ΅,𝔳ϡ), respectively. (ArgumentsMathworldPlanetmath are understood) Assuming now that all of these functions yield to zero as (Ξ΄,Ο΅)β†’0, we define 22Cf. Buck [3], pp. 99-104.

ρ0≑ρ0⁒(𝐗,t⁒o):=limΞ΄β†’0⁑mδ𝒱δ,ρ≑ρ⁒(𝐱,Ο„):=limΟ΅β†’0⁑mϡ𝔳ϡ,t0≀τ≀t, (2)

if these limits exist. Moreover, we can see these densities as a mapping transformation from the material (Lagrangian) description defined by coordinates XΞ± in the initial material reference configuration at Ο„=t0, to the spatial (Eulerian) description defined by coordinates xi in any later spatial configuration at Ο„=t, i.e.

ρ0⁒(XΞ±,t0)↦ρ⁒(xi,t),ρ0=J⁒ρ,J=|βˆ‚β‘xiβˆ‚β‘XΞ±|. (3)

Therefore the mapping d⁒V↦d⁒v between the volume elements d⁒V and d⁒v corresponding to the initial material configuration and any later space configuration will also be valid, i.e. for an arbitrary mass element d⁒m in the body ℬ, d⁒m=ρ0⁒d⁒V=ρ⁒d⁒v, which is precisely the reason for the postulation of continuity equation.

Integral form of continuity equation

A set of particles V of positive mass

m≑m(ℬ):=βˆ«π’±Ο0dVβ‰‘βˆ«π”³Οdv=:m(β„œ), (4)

is a body, and such invariance is valid also for any portion of continuum, as we shall see in the differential form of that equation. By taking the material time derivative, we get

D⁒mD⁒t=mΛ™=βˆ«π’±Ο0⁒𝑑VΒ―Λ™=βˆ«π”³Οβ’π‘‘vΒ―Λ™=0. (5)

Differential form of continuity equation

From the continuity and positiveness on the integrand of the last integralPlanetmathPlanetmath in Eq.(5), we obtain the following statement.

In regions where the density is continuous 33The mappings 𝐱=𝐱⁒(𝐗,t), 𝐗=𝐗⁒(𝐱,t), are assumed to be single-valued and continuously differentiable with respect to each of their variables, except possibly at certain points, curves or surfaces.the principle of mass conservation may be expressed in the equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath forms44Cf. the remarks of Hilbert[2], pp. 100-108.

ρ⁒d⁒vΒ―Λ™=0,βˆ‡x⋅𝐯=-log⁑ρ¯˙,βˆ‚β‘Οβˆ‚β‘t+βˆ‡xβ‹…(ρ⁒𝐯)=0,ρ˙+Οβ’βˆ‡x⋅𝐯=0, (6)

the derivativesPlanetmathPlanetmath standing for material rate of change55Cf. time dilatation of a volume element. These equations are essentialy due to Euler [4]. For isochoric motions


I𝐝 being the first invariantMathworldPlanetmath of Cauchy’s rate strain tensor 𝐝.

Integral vs. differential approach

The basic laws that we apply in fluid mechanics and thermodynamics studies can be formulated in terms of finite (global) or infinitesimalMathworldPlanetmathPlanetmath (local) approach depending on the nature of the problem. In the former case we apply equations governing the gross behavior of the flow through an integral formulation which is usually easier to treat analytically. In the latter we deal with differential equations providing a means of determining the detailed behavior of the flow. Since the mentioned disciplines deal with the formulation of the basic laws in terms of finite systems, such formulations are the basis for deriving the control volume equations, concept that we shall develop to continuation.

Additional definitions

1. Control volume (CV). Is an arbitrary but convenient volume spatial description 66It is usual to consider fixed the control volume with respect to the chosen coordinate systemMathworldPlanetmath. through which the fluid flows. In fact , we are normally concerned with flows through devices such as pipelines, turbines, compressors, nozzles, etc. In these cases it is difficult to focus attention on a fixed identifiable quantity of mass. So it is much more convenient, for purpose of analysisMathworldPlanetmath, to focus attention on a fixed volume through which the fluid flows, a fact that plenty justifies to use the control volume approach. It will be denoted as 𝔳⁒(𝐱,Ο„). Since the mass is varying inside of the control volume , we have

βˆ‚βˆ‚β‘tβ’βˆ«π”³Οβ’(𝐱,t)⁒𝑑v=βˆ«π”³(βˆ‚β‘Οβˆ‚β‘t)⁒𝑑vβ‰ 0.

In this case the material rate derivative must be considered as a local rate, because the spatial coordinates xi are held constant for a fixed control volume.

2. Control surface (CS). This is how we call the geometric boundary of the control volume. We shall denote it as βˆ‚β‘π”³. So, if we define a unit vectorMathworldPlanetmath 𝐧 always chosen to be the outward normal of the boundary βˆ‚β‘π”³ and d⁒a is an element of area on the control surface, we have 𝐝𝐚=𝐧⁒d⁒a. Eventually this surface may be real or imaginary; it may be at rest or in motion. For instance, in pipeflow the lateral wall is a real physical boundary that comprises part of the control surface but the two transversal circular portions of it are imaginary, i.e. there is no corresponding physical surface. These imaginary boundaries are selected arbitrarily for accounting purposes. Since the location of the control surface has a direct effect on the accounting procedure in applying the basic laws, it is extremely important that the control surface be clearly defined before beginning any form of analysis.

3. Mass flow rate.77So-called also mass flux. Let us suppose that a velocity field 𝐯⁒(𝐱,t) is defined over a region β„œ of the space such that a control volume 𝔳⁒(𝐱,t)βŠ‚β„œ. Let βˆ‚β‘π”³ be its control surface. We define the mass flow rate across an area element 𝐝𝐚 on the control surface, as


and the net mass flow rate or efflux across the control surface βˆ‚β‘π”³, as

π”ͺΛ™:=βˆ«βˆ‚β‘π”³Οβ’π―β‹…π§β’π‘‘a, (7)

where the variable arguments are understood. Let now Ο‘=ϑ⁒(𝐱,t) be the angle between the velocity field 𝐯 (tangentPlanetmathPlanetmathPlanetmath to the flow streamlines) and the outward normal 𝐧; we have

d⁒π”ͺΛ™=ρ⁒𝐯⋅𝐧⁒d⁒a=ρ⁒βˆ₯𝐯βˆ₯⁒d⁒a⁒cos⁑ϑ={an outflow,if⁒  0≀ϑ<Ο€/2an inflow,if⁒π/2<ϑ≀π.

It is clear that mass flow rate cannot cross the streamlines. The introduction of the definitions that we have done, allows to stablish the following useful propositionPlanetmathPlanetmath.

Continuity equation for a control volume.

Proposition 1.

β€œThe rate of mass change within a stationary control volume is equal to the net mass flow rate across its control surface”.


From Eqs.(5)-(6), and assuming diffeomorfic the coordinates transformation,


and by applying the Gauss-Green divergence theoremMathworldPlanetmathPlanetmath, we obtain

βˆ‚βˆ‚β‘tβ’βˆ«π”³Οβ’π‘‘v=-βˆ«βˆ‚β‘π”³Οβ’π―β‹…π§β’π‘‘a, (8)

as desired. ∎

Remark. In Eq.(8), the velocity field 𝐯 is relative to the CS. So, the formulaMathworldPlanetmath is still valid for CV rigid motionsMathworldPlanetmath (i.e. as a whole motions) as we may substitute the field 𝐯 by the relative velocity field 𝐰=𝐯-𝐯C⁒V.

Steady-state flow

In this case88Cf.[5], for instance., the fluid properties are independent on time. Therefore Eq.(8) reduces to

βˆ«βˆ‚β‘π”³Οβ’π―β‹…π§β’π‘‘a=0, (9)

thus the mass flow rate π”ͺΛ™i⁒n, across the CS βˆ‚β‘π”³i⁒n, entering the CV is equal to the mass flow rate π”ͺΛ™o⁒u⁒t, across the CS βˆ‚β‘π”³o⁒u⁒t, leaving the CV i.e.

π”ͺΛ™=π”ͺΛ™i⁒n:=βˆ«βˆ‚β‘π”³i⁒nρ|𝐯⋅𝐧|da=βˆ«βˆ‚β‘π”³o⁒u⁒tρ𝐯⋅𝐧da=:π”ͺΛ™o⁒u⁒t. (10)

That equation is extensively used in applications such as internal combustion engines, reciprocating machines, turbomachinery, etc. On the other hand, if it is possible to define concrete inlet/outlet cross sections A on the CS, the flow is normal (i.e. cos⁑ϑ=Β±1) and uniform (non-viscous flow, for example) and assuming constant density in each sectionMathworldPlanetmathPlanetmath (a reasonable supposition), we get


where v is the normal speed flow at the respective section. For multiple inlets and outlets, we have the obvious formulas



  • 1 L. Euler, Principes généraux du mouvement des fluides, Hist. Acad. Berlin 1755, 217-273, 1757.
  • 2 D. Hilbert, Mechanik der Continua, lectures of 1906-1907; MS notes by A.R. Crathorne in Univ. Illinois Library, 1907.
  • 3 R. C. Buck, Advanced Calculus, New York: McGraw-Hill Co., 1965.
  • 4 L. Euler, Sectio secunda de principiis motus fluidorum, Novi Comm. Petrop. 14 (1769), 270-386, 1770.
  • 5 R. W. Fox and A. T. McDonald, Introduction to fluid mechanics, John Wiley & Sons, Inc., 7-8, 1973.
Title continuity equation
Canonical name ContinuityEquation
Date of creation 2013-03-22 15:56:05
Last modified on 2013-03-22 15:56:05
Owner perucho (2192)
Last modified by perucho (2192)
Numerical id 8
Author perucho (2192)
Entry type Application
Classification msc 53A45