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Extra resources for A treatise on the analytical geometry of the point, line, circle, and conical sections (1885)
For example, consider a ﬂow in a two-dimensional channel with solid boundaries on the top and the bottom, an inﬂow boundary on the left and an outﬂow boundary on the right. There will be viscous boundary layers next to the two walls, but there should not be any boundary layers at the inﬂow and outﬂow boundaries other than those near the walls. This will be achieved if the imposed boundary conditions yield a well-posed problem in the inviscid limit. See the above references for further details. κ 12 1.
This system is hyperbolic. It admits weak or discontinuous solutions (Courant and Friedrichs (1976), Lax (1973)). The order of the Euler equations is reduced by one compared with the full Navier–Stokes equations, leading to the loss of the velocity no-slip boundary condition and the boundary condition on the temperature (or other thermodynamic variable). The boundary conditions at stationary solid walls are no-ﬂux: u·n= 0. The conditions at inﬂow or outﬂow boundaries are dependent upon the characteristic directions there.
2 1. 1 Phases of Matter The phases of matter may be broadly categorized into solids and ﬂuids. Simply stated, solids resist deformation and retain a rigid shape; in particular, the stress is a function of the strain. Fluids do not resist deformation and take on the shape of its container owing to their inability to support shear stress in static equilibrium. More precisely, the stress is a function of the strain rate. The distinction between solids and ﬂuids is not so simple and is based upon both the viscosity of the matter and the time scale of interest.