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38 shows that the divergence of the velocity of a flowing, incompressible fluid is zero. 4 Three-Dimensional (3D) Convection/Dispersion Equation. The convection/dispersion → equation in three dimensions is obtained by writing flux ( J ) in the multidimensional form → → J = C v − D ∇ C. 39) Substituting Eq. 39 into the 3D continuity equation gives → ∂C . 40) −∇ · C v + ∇ · D ∇ C − q = ∂t → If we assume that v and D are constant, we can simplify Eq. 40 to the form of → ∂C D ∇2C − v · ∇ C − q = . 41) ∂t Eq.
12 is a nonlinear, second-order PDE. It is second order because of the second-order partial derivative of pressure with respect to x, and it is nonlinear because of the square of the pressure gradient term. Solutions of PDEs depend on the form of the PDEs and their associated boundary conditions. An important class of second-order PDEs has the form Print Search Chapter 1 Home Chapter 3 Bookmarks Help I-48 Petroleum Engineering Handbook—Vol. I ∂2 ψ( x, y ) ∂2 ψ( x, y ) ∂2 ψ( x, y ) A + 2B + C = G, ...............................
Fig. 13—SDOF beating phenomenon. and cωf C1 + (k − mωf 2)C2 = 0. 90) Solving for the constants gives: ( ωf 2 k 1− C1 = ( k 1− Print Search Contents ωn 2 ωf 2 ωn 2 ) 2 Home ) F0 .................................................. 91) 2 + c ωf 2 Chapter 2 Bookmarks Help I-24 Petroleum Engineering Handbook—Vol. I Fig. 14—SDOF forced and damped CLO system. and C2 = −cωf F0 ( ωf 2 k 1− ωn 2 ) , ................................................ 92) 2 2 + c ωf 2 which gives the particular solution: x p = F0 {( ( k 1− k 1− + ( k 1− ωf 2 ωn 2 ωf 2 ωn 2 ) 2 −cωf ωf 2 ωn 2 ) 2 2 + c ωf 2 ) 2 + c ωf 2 sin ωf t } cos ωf t .