Stability Equations and Numerical Method

Here the stability equations for two dimensional plane flows have been derived, by starting from the non-dimensional Eqns. (6.2.4) -(6.2.6) given 

For the stability analysis, all the physical variables would be split into the mean part, derived in section 6.3, and a disturbance component indicated by a caret in the following. 

The quantities with the over-bar are related to the mean flow solutions of the previous section, via the appropriate transformations. The stability equations are obtained by making the additional parallel flow assumption, U = U y), V = 0 and T = T y) so that a normal mode spatial instability analysis is possible by looking for a solution of the linearized equations of the following form  

After substituting (6.4.5) into (6.4.1) to (6.4.4), one can obtain the system of equation governing the disturbance amplitude functions as,  

In these equations, prime once again denote differentiation with respect to y. One can eliminate tt and / from these equations to obtain, 

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