Difference between integral and differentiation in cfd
Answers
Both equations tell the same tale: the conservation of momentum. One is formulated for an infinitesimal fluid particle (the differential form) whilst the other is applied to a region in space, the control volume. These two are linked by the Gauss theorem, and can be derived via almost ten different ways. Both originate from the same first principles (the Newton's 2nd law of motion). Mathematically, they are equivalent in the infinitesimal limit.
For numerical solutions, the differential form is used together with difference approximations (FDM, finite difference method). Integral form is used with the finite volume method, FVM. These are equivalent in uniform grids. The differential form does not have a solution in the classical sense in presence of discontinuities (eg. compressible flows with shocks), hence, one uses the weak form of the integral equations. A physically unique solution is sought using an entropy condition.
The conservative property has a numerical aspect, concerning the overall conservation. This is usually important in an approximate solution; FVM are better in this respect. For example, a conservative momentum flux is \nabla . \rho u u, while u . \nabla \rho u is not. In differential form these are again equivalent but one can never can shrink the grid spacing infinitely to ensure the solution of the differential equations is that of the integral equations applied to an infinitesimally small control volume.