Navier Stoke's Equation

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Derivation of the Navier-Stokes equations, and implications for groundwater flow Note on terminology: This essay is a compilation of material taken from books on physics, fluid mechanics and hydrogeology. One of the challenges was to produce a consistent terminology from start (properties of fluids) to finish (Darcy’s Law and its implications). In many texts vectors are indicated by bold letters (v), while in others they are indicated by a small arrow above the symbol ([pic]). I will always indicate vectors with the little arrow and scalars without one. Thus [pic] is a vector but vx is a scalar. The unit vectors in the x-, y-, and z-directions are [pic], [pic], and [pic] respectively so that the velocity vector [pic] can be written as: [pic] Fluid mechanics books and articles always refer to the x-, y, and z components of the velocity as u, v, and w; i.e., [pic] Groundwater texts do not use these symbols, so I won’t either. I will always use the symbols vx, vy, and vz as the scalar magnitudes of the vector components and the vectors [pic]and [pic] as the vector components themselves. In addition, it is common in fluid mechanics to refer to the x-, y- and z-directions as x1, x2, and x3; this is useful, for example, when working with subscripts in tensor notation. In general I’ve tried to avoid tensor notation so I will retain the usual x, y, and z notation for the axes. The only exception is for the symbol for a 2nd-order tensor itself, which I will indicate with two subscripts i and j; i.e., the stress tensor [pic] This symbol should be taken to mean a 3 × 3 tensor with entries: [pic] Finally, for inexplicable reasons it is common in physics and fluid mechanics to use [pic] to refer to both the strain tensor (for elastic solids) and strain-rate tensor (for fluids). This seems to be an unnecessary source of confusion, and I
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