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Unformatted text preview: 1 Conservation Equations 1.1 Conservation of mass Mass conservation states that the net rate of change of the mass of a control volume (CV) is equal to the net rate of transport across the boundary of the CV. In vector form, d dt Z V dV + Z A n u dA = 0 (1) in which n is the outward pointing normal along the surface of the control volume. The quantity u is the mass flux vector, and since the normal is defined as pointing out, the second term will be positive/negative if a net amount of mass is leaving/entering the CV. The divergence theorem of vector calculus transforms the surface integral into a volume integral, and the order of differentiation and integration in the first term can be switched. This gives Z V t + ( u ) dV = 0 (2) in which is the divergence operator; the specific form of this function will depend on the coordinate system. The above relation must hold for any subvolume of the CV, and thus the integrand must be zero: t + ( u ) = 0 (3) This is the divergent form of the continuity equation. The chain rule of calculus can be applied to the second term, ( u ) = u + u (4) Note that the density gradient, , is a vector, and u gives a scalar. When replaced into Eq. ( 3 ) the result is D Dt + u = 0 (5) in which the material derivative is defined as D Dt = t + u (6) Equation ( 5 ) is the conservative form of the continuity equation. The divergent and conservative forms are completely equivalent; their distinction becomes more relevant when developing numerical methods for computing flowfields. 1.2 Momentum conservation Newtons law states that the rate change of linear momentum of a system is equal to the net force acting on a system. When applied to a CV, the rate of change of the CV momentum is given by the 1) the instantaneous time derivative of the CV momentum, plus 2) the net rate of transport of momentum across the CV boundaries. Note that these two contributions are analogous to the two sources of mass change in a CV, i.e., the rate of storage and the rate of transport terms. Unlike mass, however, momentum can be created or destroyed by application of forces to the CV. The forces arise from two contributions: 1) surface forces which act on the surface of the control volume, and 2) body forces which act directly on the mass...
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This note was uploaded on 09/24/2011 for the course MECH 7220 taught by Professor Staff during the Fall '10 term at Auburn University.
- Fall '10
- Heat Transfer