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The heat and mass transfer characteristics of unsteady flow of incompressible chemically reactive upper-convected Maxwell fluid along a stretching surface in the presence of temperature-dependent viscosity has been studied. The theoretical analysis on heat and mass transfer over a stretching sheet is investigated for numerical analysis. With the use of stream function formulation, the governing boundary layer equations of momentum, energy, and concentration are reduced to a set of linked ordinary differential equations. The nonlinear ordinary differential equations are then solved numerically by using the Keller Box method. The physical behavior of governing parameters on velocity, temperature, and concentration profiles and the local skin friction coefficient and heat and mass transfer rates are graphed and tabulated. The physical impact of Maxwell parameter β, unsteadiness parameter M, Schmidt number Sc, Prandtl number Pr, reaction rate parameter γ, and variable viscosity parameter ε on the heat and mass transfer has been examined along the stretching surface numerically. The novelty of the present work is to examine the importance of destructive reaction and contractive reaction on the dynamics of upper-convected Maxwell fluid flow in the presence of temperature-dependent viscosity effects. It is observed an interesting behavior of temperature distribution and concentration profile is noted for lower value of viscosity parameter ε in the presence of chemical reaction. It is also found that skin friction and the rate of heat transfer are decreased by increasing the variable viscosity parameter ε .

magnetic field and rotation. However when e * 0 the inhibiting effect of both magnetic field and rotation increases as the parameter c increases. An energy analysis of the Benard problem is considered when the magnetic field is in the direction of the vertical. The stabilizing effect of the magnetic field is investigated via a generalized energy functional for a non-linear magnetohydrodynamic fluid. It is shown that the magnetic field has a stabilizing effect for this model. The Compound Matrix Technique and the Chebyshev polynomials method have been used to solve the related eigenvalue problems. These methods shall be described in detail in chapter three and four respectively-Chapter Two Minimisation / Maximisation Procedures The Compound Matrix Method Let us examine the problem of determining an eigenvalue X of the nth order system-A (X , x) Y (3.1) subject to the boundary conditions B Y-0 on x-0 , (3.2) CY-0 on x-1 (3.3) where Y is an n vector, A(X,x) is an nxn matrix and B and C are respectively (n-m)xn and mxn matrices of full rank. i.e. (n-m) conditions are given at x-0 and m conditions at x = 1. We may assume without loss of generality that m < n/2. Notionally the general solution of the nth order system (3.1) has n degrees of freedom. Clearly (n-m) of these degrees of freedom can be removed by requiring that conditions (3.2) be satisfied and so we may conceptually construct m functionally independent solutions of (3.1) which also satisfy conditions (3.2). Let these solutions be w-i (X ,x) , w 2(X,x)........ wm (X,x) where n / ' (n" 0 n T Wi(X,x) = (wif w 1(W i ....... , w I) L Let us define the nxm matrix M to be the matrix whose rth column is w r (X,x) i.e. M = [w, , w 2, * 1 3-(2,3,6), d> 1 4 = (2,4,5), *1 5 = (2,4,6), $ 6 = (2,5,6),

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This paper has been prompted by some recent computations of eigenvalues of the Orr-Sommerfeld equation for very high Reynolds numbers. We have used a spectral analysis to emulate these calculations and our results have motivated some general remarks on the suitability of tracking and spectral methods as numerical eigenvalue schemes in the context of stability theory. Our remarks are further supported by illustrating the development of eigenvalues in the Magnetic Benard.