Attention will be limited to incompressible fluids except when buoyancy is important, in which case the Boussinesq approximation will be made.The equations describing heat transfer are complex, having some or all of the following characteristics: they are nonlinear; they comprise algebraic, partial differential and/or integral equations; they constitute a coupled system; the properties of the substances involved are usually functions of temperature and may be functions of pressure; the solution region is usually not a simple square, circle or box; and it may (in problems involving solidification, melting, etc.) change in size and shape in a manner not known in advance.Thus analytical methods, leading to exact, closed form solutions, are almost always not available. In the first, the equations are simplified — for example, by linearization, or by the neglect of terms considered sufficiently small, or by the assumption of constant properties, or by some other technique until an equation or system of equations is obtained for which an analytical solution can be found.Thus, a full solution of the energy equation and perhaps also the equations of motion is required.These are partial differential equations, possibly coupled.Radiation is somewhat different, involving surfaces separated (in general) by a fluid which may or may not participate in the radiation.If it is transparent, and if the temperatures of the surfaces are known, the radiation and convection phenomena are uncoupled and can be solved separately.There is also an error in this approach: for example, if derivatives are replaced by finite differences, only an approximate value for the derivative will be obtained, in principle, if the problem is well-posed and if the solution method is well-designed, this error will approach zero as the grid is made increasingly fine.In practice, a fine but not infinitesimal grid must be used.If the surface temperatures are not specified, but are to be found as part of the solution, or if the fluid absorbs or emits radiant energy, then the two phenomena are coupled.In either case, the solution of the algebraic and/or integral equations for radiation is required in addition to that of the differential equations for convection and conduction.

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