Boundary layer equations porous media

The left-hand side of this equation is of order 1. Considering the right-hand side, 8t/L is by assumption small so, since d2B dX2 and S2B/dY2 are both of order l, the first term on the right-hand side of the above equation will be much less than the second and the first term is. therefore, negligible compared to the second. Hence, the following form of the energy equation applies in two-dimensional boundary layer flow in a porous medium ... 

The approximate integral equation method that was discussed in Chapters 2 and 3 can also be applied to the boundary layer flows on surfaces in a porous medium. As discussed in Chapters 2 and 3, this integral equation method has largely been superceded by purely numerical methods of the type discussed above. However, integral equation methods are still sometimes used and it therefore appears to be appropriate to briefly discuss the use of the method here. Attention will continue to be restricted to two-dimensional constant fluid property forced flow. 

As discussed in Chapters 2 and 3, in the integral method it is assumed that the boundary layer has a definite thickness and the overall or integrated momentum and thermal energy balances across the boundary layer are considered. In the case of flow over a body in a porous medium, if the Darcy assumptions are used, there is, as discussed before, no velocity boundary layer, the velocity parallel to the surface near the surface being essentially equal to the surface velocity given by the potential flow solution. For flow over a body in a porous medium, therefore, only the energy integral equation need be considered. This equation was shown in Chapter 2 to be ... 

Darcy-like flows obtain except in a thin boundary layer near solid walls, where Darcy s equation is unable to satisfy the no-slip boundary condition [cf. Brinkman s treatment (B35) of flow through a porous medium bounded externally by a circular cylinder with solid walls]. 

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