From Smoothed Particle Hydrodynamics: A Meshfree Particle Method
4.2 NavierStokes Equations in Lagrangian Form
The basic governing equations of fluid dynamics are based on the following three fundamental physical laws of conservation.

conservation of mass

conservation of momentum

conservation of energy
Different forms of equations can be employed to describe the fluid flows, depending on the specific circumstances (Anderson, 1995; Hirsch, 1988). As discussed in Chapter 1, there are two approaches for describing the physical governing equations, the Eulerian description and Lagrangian description. The Eulerian description is a spatial description, whereas the Lagrangian description is a material description. The fundamental difference of these two descriptions is that the Lagrangian description employs the total time derivative as the combination of local derivative and convective derivative. In conjunction with the Lagrangian nature of the SPH method, the governing equations in Lagrangian form will be discussed and employed in this section. The SPH equations of motion will be derived based on these governing equations in Lagrangian form.
4.2.1 Finite control volume and infinitesimal fluid cell
Consider a closed volume with finite dimensions in a fluid flow system as shown in Figure 4.1. This volume defines a control volume V associated with a closed control surface S which bounds the control volume. In the Lagrangian description, this control volume can move with the fluid flow such that the same material of the fluid is always staying inside the control volume. Therefore, though the fluid flow may result in expansion, compression, and deformation of the Lagrangian control volume, the mass of...
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