1 MAE 5130: VISCOUS FLOWS Conservation of Mass September 2, 2010 Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk

2 KINEMATIC PROPERTIES: TWO ‘VIEWS’ OF MOTION 1.Lagrangian Description –Follow individual particle trajectories –Choice in solid mechanics –Control mass analyses –Mass, momentum, and energy usually formulated for particles or systems of fixed identity (ex., F=d/dt(mV) is Lagrandian in nature) 2.Eulerian Description –Study field as a function of position and time; not follow any specific particle paths –Usually choice in fluid mechanics –Control volume analyses –Eulerian velocity vector field: –Knowing scalars u, v, w as f(x,y,z,t) is a solution

3 CONSERVATION OF MASS: DERIVATION 1 Recall that all conservation laws (mass, momentum, energy) are all Lagrangian in nature, and apply to a fixed system (fixed mass) of particles We can go between Lagrangian and Eulerian descriptions of fluid motion using substantial derivative, D( )/Dt In Lagrangian terms, mass is constant Can relate substantial derivative of volume, V, to fluid velocity Note that total dilatation (or normal strain) is equal to rate of volume increase Two equivalent forms of continuity equation

4 CONSERVATION OF MASS: DERIVATION 2 Infinitesimal fixed control volume, V=dxdydz Flow through each side (each face) is 1-D Mass flow occurs on all 6 faces (3 in and 3 out) Fluid properties,  =  (x, y, z, t) x z y dz dy dx

5 CONSERVATION OF MASS: DERIVATION 2 x-direction IN OUT y-direction z-direction Storage or change in mass (change in density) of volume with time

6 SUMMARY: INCOMPRESSIBLE VS. CONSTANT DENSITY Two equivalent statements of conservation of mass in differential form In an incompressible flow Says particles are constant volume, but not necessarily constant shape Density of a fluid particle does not change as it moves through the flow field INCOMPRESSIBLE: Density may change within the flow field but may not change along a particle path CONSTANT DENSITY: Density is the same everywhere in the flow field

7 CONSERVATION OF MASS: INTEGRAL FORM This is a single scalar equation –Velocity doted with normal unit vector results in a scalar 1 st Term: Rate of change of mass inside CV –If steady d/dt( ) = 0 –Velocity, density, etc. at any point in space do not change with time, but may vary from point to point 2 nd Term: Rate of convection of mass into and out of CV through bounding surface, S 3 rd Term (=0): Production or source terms Relative to CS Inertial

8 INTEGRAL FORM VS. DIFFERENTIAL FORM (2-13) Integral form of mass conservation Apply Divergence (Gauss’) Theorem Transform both terms to volume integrals Results in continuity equation in the form of a partial differential equation Applies to a fixed point in the flow Only assumption is that fluid is a continuum –Steady vs. unsteady –Viscous vs. inviscid –Compressible vs. incompressible

9 EXAMPLE The density decreases inversely with time because the fluid particles are being stretched in the x-direction in proportion to time

10 EXAMPLE: COMBUSTION APPLICATION A combustible mixture of air and fuel is ignited at the point O at t=0 by a spark which initiates a spherical flame front moving outward from the point O at a constant speed V f The combustible mixture, which has a constant density,  c, ahead of the flame front, is converted in the very thin flame front to a hot product of combustion which has a much lower density  p Because the volume of a fluid element is increased as it is enveloped by the flame front, the combustible mixture is pushed radially outward by the expanding flame front, although the combustion products remain stationary (V p =0; the gas inside the flame front region is stationary Determine the radial outflow velocity U(r,t) of the combustible mixture at r for any time t prior to the instant when the flame front reaches r (r > V f t) Control Volume Hot, stationary combustion products,  p Unburned combustible gas mixture,  c Flame front O VftVft VfVf U(r,t) r