MIT Department of Mechanical Engineering 2.25 Advanced Fluid Mechanics The Continuity Equation: Conservation of Mass for a Fluid Element ( v = vx ,v y ,vz δV ) δV ′ δm δm No ﬂow across streamlines Consider a ﬂuid element with constant mass δm and volume δV moving in a velocity ﬁeld as shown above. The streamlines are converging and the ﬂuid element may be advected to a new position which has a higher speed. If we assume the most general case, in which the ﬂuid element is compressible, then δm is ﬁxed but δV changes. Note that: ρ = δm/δV (1) Hence: δV = ⇒ δm D (δV ) δm Dρ =− 2 = δmρ−1 ⇒ ρ Dt ρ Dt 1 D (δV ) 1 =− δV Dt ρ Dρ Dt (2) (3) We already know that the left hand side of (3) is \ · v, thus: 1 ρ Dρ Dt = −\ · v (4) Alternative way to reach the same thing is: D (δm) D (ρδV ) D (ρ) D (δV ) =0⇒ = 0 ⇒ δV +ρ =0 Dt Dt Dt Dt Dividing by ρδV leads to: 1 D (ρ) 1 D (δV ) + =0 ρ Dt δV Dt Again knowing that volumetric rate of strain, the second term, is equal to \ · v gives: 1 D (ρ) = −\ · v ρ Dt 1 (5) (6) (7) which is the same concluded in (4). The derived equation is mass conservation for any ﬂow (compressible or incompressible). In the case of incompressible ﬂows (or almost ”incompressible”-Mach numbers lower than 0.3), from ”incompressibility” we will have: 1 D (ρ) c0 ρ Dt (8) which by (7) means that in incompressible ﬂows (M a < 0.3): \·v c0 2 (9)