Navier-stokes Equations for Fluid Dynamics
نویسنده
چکیده
1.1. Eulerian and Lagrangian coordinates. Let us begin with Eulerian and Lagrangian coordinates. The Eulerian coordinate (x, t) is the physical space plus time. The Eulerian description of the flow is to describe the flow using quantities as a function of a spatial location x and time t, e.g. the flow velocity u(x, t). This can be visualized by sitting on the bank of a river and watching the water pass a fixed location. The equations governing the flow will be mainly written and solved in the Eulerian coordinate. In contrast, Lagrangian description can be visualized as sitting in a boat and drifting down a river. To make it precise, let us introduce the reference coordinate ξ which can be thought as a (very fine) uniform grid. Each cell is a fluid parcel which is a very small amount of the fluid consisting of reasonable microscopic particles (molecules and atoms). The size of fluid parcels is large enough such that the averaged quantities remains meaningful. Meanwhile it is small compared to the macro length scales of the flow under consideration such that it can be thought as a point. We assume that the flow starts from a reference configuration and is mapped into its deformed configuration. Mathematically it can be described as a map, called flow map, from the reference coordinate to the Eulerian coordinate
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