The continuity equation

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1 Chapter 6 The continuity equation 61 The equation of continuity It is evient that in a certain region of space the matter entering it must be equal to the matter leaving it Let us consier an infinitesimal volume of rectangular parallelepipe form with sies δx, δy an δz (Fig 61) The flui mass that enters the elementary volume in the unit time through the surface of equation x δx/2 = 0 is δm x δx/2 = ρ, y, z, t u, y, z, t δyδz = (ρu), y, z, t δyδz, where the last step serves only to introuce a more concise notation Similarly, the flui mass that leaves the volume in the same time through the surface of equation x + δx/2 = 0 is δm x+δx/2 = (ρu) (x + δx2 ), y, z, t δyδz Hence, the total mass entering the volume in the unit time in the x-irection is equal to δm x = δm x δx/2 δm x+δx/2 37

2 38 Franco Mattioli (University of Bologna) (x, y, z) (ρu)(x δx/2, y, z) δx δy δz (ρu)(x + δx/2, y, z) Fig 61: The flux of matter entering the infinitesimal parallepipe can be ecompose into the sum of three inepenent contributions along the coorinate irections = [ (ρu), y, z, t (ρu) (x+ δx2 )] [, y, z, t δyδz = (ρu) ] x δx δyδz, where in the last step we have ivie an multiplie by δx Similar contributions arise in the y an z-irections, so that the mass of the matter entering from all the irections becomes [ (ρu) δm T = δm x + δm y + δm z = x + (ρv) + (ρw) ] δxδyδz y z On the other han, the same mass can be evaluate on the basis of the variation of the ensity of the parcel δm T = t δxδyδz Equating the two expressions of M T an iviing by the elementary volume δxδyδz, we obtain t + (ρu) x + (ρv) + (ρw) = 0, y z which in vector form can also be written as + (ρu) = 0 (61) t

3 Principles of Flui Dynamics (wwwfluiynamicsit) 39 This equation is known as the continuity equation, or also law of the mass conservation The continuity equation can be written in another equivalent form by expaning the ivergence term We obtain that is, + u ρ + ρ u = 0, t ρ + ρ u = 0 (62) t In an incompressible flui, for which ρ/t = 0, this relationship is further simplifie, thus reucing to u = 0 (63) Problem 61 Use the continuity equation to erive the meaning of the ivergence term alreay seen in problem [D4] Solution Consier a parcel of mass δm = ρδv Since the mass of the parcel oes not vary in time, one has ( ) t δm = t (ρδv ) = ρ δv + ρ δv = 0, t t from which, iviing by δv, one obtains ρ t + ρ 1 δv δv = 0 t A comparison with the continuity equation shows that the ivergence of the velocity fiel in a given point at a given instant is proportional to the fractional variation of volume of the parcels passing through the point in that instant, as in (D2) 62 Integral form of the continuity equation Let us integrate the continuity equation (61) over a certain volume V, boune by the surface S We obtain V t V + (ρu) V = 0 (64) V

4 40 Franco Mattioli (University of Bologna) In the first term the time erivative can be move outsie the integral symbol Since it applies to a quantity that oes not epen on space variables, then it becomes an orinary erivative of the first kin (see section [C1]) By further applying Gauss theorem to the secon term, the equation can be rewritten as ρ V + ρu S = 0 (65) t V S The first integral clearly represents the mass enclose within the volume V The secon integral represents the flux of matter per unit time (see Fig 62) δs α n u ut Fig 62: After an interval of time t, the flui that crosse the infinitesimal surface S occupies an oblique cyliner of length u t an base S The scalar prouct u, t cos α represents the height of the cyliner, so that tu n S represents its volume The multiplication of this quantity by the ensity provies the mass that enters or leaves the volume V through the surface S in the unit time Therefore, (65) states that the variation of the total mass within a certain finite volume equals the quantity of matter flowing through its elimiting surface This is equivalent to saying that there cannot exist either sources nor sinks of matter Let us exten in (65) the volume integral to all the space available to the flui, assuming that its bounaries are either rigi an motionless or place at infinity Along such bounaries in the former case the normal velocity an in the latter the velocity itself are zero Thus, the surface integral in (65) vanishes, an it follows that the total quantity of matter is conserve over time The integral formulation (65) is equivalent to the ifferential formulation (61) In fact, from (65) we can erive (64), an this equation must hol whatever the volume Therefore it must hol also in any infinitesimal region of the space, where the integran can be assume as constant This implies the ifferential equation (61)

5 Principles of Flui Dynamics (wwwfluiynamicsit) 41 Inee, there is a subtle ifference In the integral formulation the fiels can also be iscontinuous The step between (65) an (64) requires the continuity an ifferentiability of the integran with respect to time Thus the integral formulation is slightly more general than the ifferential formulation On the other han the transport theorem (57) allows us to write (64) as ρ V = M t V t = 0, where V(t) is the volume occupie by a certain set of parcels of total mass M(t) as they move in the space In this form the meaning of the continuity equation as the law of the conservation of the mass is absolutely clear 63 Particularly simple flows In a flui with a ensity that is constant in time an uniform over each horizontal plane an moves with a horizontal velocity equally uniform over each horizontal plane, all the terms of the continuity equation separately vanish In fact, in this case, the local erivative of the ensity is zero because of stationarity, the first two components of the momentum o not epen on the horizontal coorinates, while the thir one is zero because the vertical velocity is zero Such flows, efine only by the vertical ensity stratification an by the horizontal components of the velocity as a function of the time an the vertical coorinate, are particularly simple to stuy They can be ealt with by not consiering the continuity equation, which is automatically satisfie because all of its terms vanish Such flows are briefly calle plane-parallel flows Another kin of flow satisfying a similar property is a stationary raially symmetric flow In this case, both the ensity an the horizontal velocity epen only on the vertical coorinate z an on the istance r = (x 2 + y 2 ) 1/2 from the vertical coorinate axis We have shown in problem [E2] that such a flow is not ivergent Furthermore, since the circular motion of the parcels is not able to moify the structure of the ensity fiel, the total erivative of ensity also vanishes 64 Historical notes an essential bibliography The continuity equation for the particularly simple case of the flow in a channel was qualitatively unerstoo by many authors of the XVI an XVII century, but a ifferential form of the equation appeare much later in a work by Jean

6 42 Franco Mattioli (University of Bologna) Le Ron Alembert [11] in 1747 an in the basic work by Leonhar Euler [15] in 1755

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