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adjoint problem assume Au<k Benard convection boundary conditions 3.5 Ch(z circular Couette flow coefficient computations consider the adjoint constant heat sources continuity equation convection with constant define the matrix denoted density diagonal matrix Difference Equation eigen eigenfunction eigenvalue problem eM.f satisfies equation 2.5 equation 2.8b EXCHANGE OF STABILITIES four—vector governing equations hence initial temperature profile inner product INSTABILITY OF COUETTE k—th eigenvalue kinematic viscosity linear operator map real vectors matrix operators max|f(r maxQ<z<i Mu<k National Physical Laboratory non—self—adjoint nondimensional nonlinear equations Oq is simple orthogonal partial differential perturbed eigenvalues Prandtl number preserves the reality PRINCIPLE OF EXCHANGE radiative transfer radius of convergence RAND Corporation real functions reduced problem rotating Schwartz's inequality self—adjoint form self—adjoint systems simple eigenvalues sinh solutions square integrable term thermal conductivity thermal diffusivity tion traveling waves Tz(z vectors into real velocity Xz exp zero