Numerical treatment and Global Error Estimation of a MHD Flows of an Oldroyd 6-Constant Nano-Fluid through a non-Darcy Porous medium with Heat and Mass Transfer
Abstract
Explicit Finite-Difference method was used to obtain the solution of the system of the non-linear ordinary differential equations which transform from the non-linear partial differential equations. These equations describe the steady magneto-hydrodynamic flow of an oldroyd 6-constant non-Newtonian nano-fluid through a non-Darcy porous medium with heat and mass transfer. The numerical formula of the velocity, the temperature, the concentration, and the nanoparticles concentration distributions of the problem were illustrated graphically. The effect of Darcy number Da, Forchheimer number Fs, magnetic field parameter M, local temperature Grashof number Gr, local nanoparticle Grashof Br, Prandtl number Pr, Dufour number Nd, Brownian motion parameter Nb, Thermophoresis parameter Nt, Lewis number Le, Sort number Ld, Chemical reaction parameter Rc, and Chemical reaction order m on those formula were discussed at the values of material parameters specially in the case of pure Coutte flow. Then, the effects of modified pressure gradients on those formulas were discussed in the case of pure Poiseuille flow and the generalized Couette flow. Also, an estimation of the global error for the numerical values of the solutions is calculated by using Zadunaisky technique.
Keywords
Download Options
Introduction
The study of non-Newtonian fluids has gained much attention recently in view of its promising applications in engineering and industry. Such fluids exhibit a non-linear relationship between the stresses and the rate of strain. Due to non-linear dependence, the analysis of the behavior of the non-Newtonian fluids tends to be much more complicated and subtle in comparison with that of Newtonian fluids. Flow of fluids with complex microstructure (e. g. molten polymer, polymer solutions, blood, paints, greases, oils, ketchup, etc.) cannot be described by a single model of non-Newtonian fluids. Many models that exist are based either on natural modifications of established macroscopic theories or molecular considerations. In general, the equations of motion for non-Newtonian fluids are of higher order than the Navier-Stokes equations and thus one need conditions in addition to the usual adherence boundary condition [28].
Guillope and Saut [2] has established existence results for some shearing motions of viscoelastic fluids of Oldroyd type. Some exact solutions of an Oldroyd 3-constant fluid are studied in [4], [5], [11], [13]. Baris [21] investigated the steady flow of an Oldroyd 6-constant fluid between intersecting planes using the series expansion method. Hayat et al. [23] studied the Couette and Poiseuille flows of an Oldroyd 6-constant fluid with magnetic field by using the Homotopy analysis method. Hayat et al. [25] studied the steady flow of a magneto-hydrodynamic Oldroyd 6-constant fluid by the motion of an infinite plate using the Homotopy analysis method. Wang et al. [30] investigated the non-linear magnetohydrodynamic problems of an Oldroyd 6-constant fluid by using analytical method and the finite-difference discretization method. Hayat et al. [24] studied the effect of the slip condition on flows of an Oldroyd 6-constant fluid. Rana et al. [14] studied the Hall effects on hydromagnetic flow of an Oldroyd 6-constant fluid between concentric cylinders by the finite difference method. Hayat et al. [26] investigated the exact solution of a thin film flow of an Oldroyd 6-constant fluid over a moving belt by the Homotopy perturbation.
Investigation of nanofluid flow has received special focus in the past due to its relevance in numerous industrial applications. The researchers not only discovered unexpected thermal properties of nanofluids but also proposed mechanisms behind the enhanced thermal properties of nanofluids and thus identified unusual opportunities to develop them as next generation coolants for computers and safe coolants for nuclear reactors. A combination of nanofluid with biotechnological components can provide potential applications in agriculture, pharmaceuticals and biological sensors. Various types of nanomaterials including nanoparticles, nanowires, nanofibers, nanostructures, and nanomachines are used in biotechnological applications. The commercialization of nano-biotechnological products seems to have a potential future and within next a few years many new products of this nature are likely to be used. Nano and micro-fluidics is a new area which has potential for engineering applications, especially for the development of new biomedical devices and procedures [1].
Conclusion
In this work, we have studied magnetohydrodynamic flows of an Oldroyd 6-constant nano-fluid through a non-Darcy porous medium with heat and mass transfer. The governing boundary value problem was solved numerically by an Explicit FiniteDifference method. We concentrated our work on obtaining the velocity, the temperature, the concentration, and the nanoparticle distributions which are illustrated graphically at different values of the parameters of the problem in three cases, pure Couette flow, pure Poiseuille flow, and generalized Couette flow. Global error estimation is also obtained using Zadunaisky technique. We used 26 points to find the interpolating polynomial of degree 25 in interval [0,1] and the results are shown in table (1). We notice that, the error in table (1) is good enough to justify the use of resulting numerical values.