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1

A locally modified single-phase model for analyzing magnetohydrodynamic boundary layer flow and heat transfer of nanofluids over a nonlinearly stretching sheet with chemical reaction

Akbarzadeh P.

Journal of Theoretical and Applied Mechanics

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2018

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Vol. 56 nr 1

81--94

EN

The problem of boundary layer flow and heat transfer of nanofluids over nonlinear stretching of a flat sheet in the presence of a magnetic field and chemical reaction is investigated numerically. In this paper, a new locally modified single-phase model for the analysis is introduced. In this model, the effective viscosity, density and thermal conductivity of the solid-liquid mixtures (nanofluids) which are commonly utilized in the homogenous single-phase model, are locally combined with the prevalent single-phase model. Similarity transformation is used to convert the governing equations into three coupled nonlinear ordinary differential equations. These equations depend on five local functions of the nanoparticle volume fraction viz., local viscosity ratio, magnetic, Prandtl, Brownian motion and thermophoresis functions. The equations are solved using Newton’s method and a block tridiagonal matrix solver. The results are compared to the prevalent single-phase model. In addition, the effect of important governing parameters on the velocity, temperature, volume fraction distribution and the heat and mass transfer rates are examined.

2

Numerical solutions of unsteady boundary layer equations for a generalized second grade fluid

Kecebas A., Yurusoy M.

Journal of Theoretical and Applied Mechanics

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2011

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Vol. 49 nr 1

71-82

EN

Unsteady, incompressible boundary layer equations for a modified power- law fluid of the second grade are considered. The model is a combination of the power-law and second grade fluid in which the fluid may exhibit normal stresses, shear thinning or shear thickening behaviour. The equations of motion are formulated for two-dimensional flows, and from which the boundary layer equations are derived. By using the similarity transformation, we reduce the boundary layer equations to system of non-linear ordinary differential equation. The ordinary differential equations are numerically integrated for classical boundary layer conditions. Effects of the power-law index and second grade coefficient on the boundary layers are shown.

PL

W pracy omówiono równania niestacjonarnej i nieściśliwej warstwy przyściennej zmodyfikowanego modelu płynu drugiego rzędu typu potęgowego. Rozważany model stanowi kombinację koncepcji płynu drugiego rzędu i opisu potęgowego, która pozwala na odzwierciedlenie zjawiska występowania naprężeń normalnych w płynie oraz efektu zmiany grubości warstwy pod wpływem naprężeń stycznych. Sformułowano równania ruchu dla przepływu dwuwymiarowego i na ich podstawie wyprowadzono równania warstwy przyściennej. Używając przekształcenia przez podobieństwo, uproszczono równania warstwy do układu nieliniowych równań różniczkowych zwyczajnych. Następnie równania te scałkowano numerycznie, stosując klasyczne warunki brzegowe. W dalszej części przeanalizowano wpływ wykładnika potęgowego modelu oraz współczynnika drugiego rzędu na zachowanie się płynu w warstwie przyściennej.

3

Patel M., Timol M. G.

International Journal of Applied Mechanics and Engineering

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2010

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Vol. 15, no 4

1205-1214

EN

A steady three dimensional MHD free convection and mass transfer flow past a semi-infinite surface in the presence of heat generation has been studied numerically for non-Newtonian power law fluids. The governing partial differential equations are reduced to a system of an ordinary differential equation using similarity transformation.

4

Kaczorek T.

International Journal of Applied Mathematics and Computer Science

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2010

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Vol. 20, no 3

507-512

EN

The notion of a common canonical form for a sequence of square matrices is introduced. Necessary and sufficient conditions for the existence of a similarity transformation reducing the sequence of matrices to the common canonical form are established. It is shown that (i) using a suitable state vector linear transformation it is possible to decompose a linear 2D system into two linear 2D subsystems such that the dynamics of the second subsystem are independent of those of the first one, (ii) the reduced 2D system is positive if and only if the linear transformation matrix is monomial. Necessary and sufficient conditions are established for the existence of a gain matrix such that the matrices of the closed-loop system can be reduced to the common canonical form.