Moving boundary in a non-Newtonian fluid

Moving boundary value problem in non-Newtonian fluid is considered. Exact analytical solution for the flow of second-grade fluid for a rigid moving plate oscillating in its own plane, is obtained. The Doppler effect has been observed due to the motion of the plate. The shearing stress on the plate is also calculated. It is concluded that the solutions for stationary porous boundaries can be obtained from the solutions of moving rigid boundaries.     

Dirichlet and Neumann boundary conditions for the pressure poisson equation of incompressible flow

In a recent paper Gresho and Sani showed that Dirichlet and Neumann boundary conditions for the pressure Poisson equation give the same solution. The purpose of this paper is to confirm this (for one case at least) by numerically solving the pressure equation with Dirichlet and Neumann boundary conditions for the inviscid stagnation point flow problem. The Dirichlet boundary condition is obtained by integrating the tangential component of the momentum equation along the boundary. The Neumann boundary condition is obtained by applying the normal component of the momentum equation at the boundary. In this work solutions for the Neumann problem exist only if a compatibility condition is satisfied. A consistent finite difference procedure which satisfies this condition on non-staggered grids is used for the solution of the pressure equation with Neumann conditions. Two test cases are computed. In the first case the velocity field is given from the analytical solution and the pressure is recovered from the solution of the associated Poisson equation. The computed results are identical for both Dirichlet and Neumann boundary conditions. However, the Dirichlet problem converges faster than the Neumann case. In the second test case the velocity field is computed from the momentum equations, which are solved iteratively with the pressure Poisson equation. In this case the Neumann problem converges faster than the Dirichlet problem.     

Similarity solutions for non-Newtonian power-law fluid flow

The problem of the boundary layer flow of power law non-Newtonian fluids with a novel boundary condition is studied.The existence and uniqueness of the solutions are examined,which are found to depend on the curvature of the solutions for different values of the power law index n.It is established with the aid of the Picard-Lindel¨of theorem that the nonlinear boundary value problem has a unique solution in the global domain for all values of the power law index n but with certain conditions on the curvature of the solutions.This is done after a suitable transformation of the dependent and independent variables.For 0 n 1,the solution has a positive curvature,while for n 1,the solution has a negative or zero curvature on some part of the global domain.Some solutions are presented graphically to illustrate the results and the behaviors of the solutions.     

Stability of steady-state non-Newtonian fluid layer flow

The stability of the steady-state plane-parallel flow of a non-Newtonian fluid layer in the gravity field along an inclined rigid surface is investigated. It is shown that the most dangerous are the long-wave perturbations propagating over the free surface. The stability maps are plotted for such perturbations in the Reynolds number — gravity parameter plane. With increase in the gravity number the layer flow becomes less stable. The layer deviation from the vertical lines stabilizes the flow.     

Self-similar solutions of non-Newtonian fluid boundary layer in MHD

The self-similar solutions of the boundary layer for a non-Newtonian fluid in MHD were considered in [1, 2] for a power-law velocity distribution along the outer edge of the layer and constant electrical conductivity through the entire flow. However, the MHD flows of many conducting media, which are solutions or molten metals, cannot be described by the MHD equations for non-Newtonian fluids.The self-similar solutions of the boundary layer for a non-Newtonian fluid without account for interaction with the electromagnetic field were studied in [3].In the following we present the self-similar solutions for the boundary layer of pseudoplastic and dilatant fluids with account for the interaction with an electromagnetic field for the case of a power-law velocity distribution along the outer edge of the layer, when the conductivity of the fluid is constant throughout the flow and the magnetic Reynolds number is small.Izv. AN SSSR. Mekhanika Zhidkosti i Gaza, Vol. 2, No. 6, pp. 77–82, 1967The author wishes to thank S. V. Fal'kovich for his interest in this study.     

Asymptotic Nusselt and entropy generation numbers for non-Newtonian fluid flow

The first and the second laws of thermodynamics are applied to obtain general expressions for the asymptotic Nusselt (Nu) and entropy generation (Ns) numbers in ducts of circular cross-section or a channel made of two long parallel plates. The governing momentum and energy equations are simplified with reasonable assumptions and solved to obtain closed form of analytical solutions. Results of the asymptotic Nusselt and Bejan numbers are presented graphically as a function of fluid index (n), Peclet number (Pe), and group parameter (Br/).     

Non-steady MHD flow for a non-Newtonian fluid with variable conductivity

This paper deals with a similarity solution for the unsteady flow of a conducting non-Newtonian power-law in-compressible fluid, when a porous plate is moving uniformly in the presence of a transverse magnetic field, assuming that the electrical conductivity is a function of the velocity. The aim of this analysis is to determine the velocity and the effect of variation of the electrical conductivity on the solution. The basic equations have been solved by applying the perturbation method for small and large values of the magnetic interaction parameterN. The main features of the exact solution is that it represents shear flow.     

Three-dimensional flow of Oldroyd-B fluid over surface with convective boundary conditions

The present study addresses the three-dimensional flow of an Oldroyd-B fluid over a stretching surface with convective boundary conditions. The problem formulation is presented using the conservation laws of mass, momentum, and energy. The solutions to the dimensionless problems are computed. The convergence of series solutions by the homotopy analysis method (HAM) is discussed graphically and numerically. The graphs are plotted for various parameters of the temperature profile. The series solutions are verified by providing a comparison in a limiting case. The numerical values of the local Nusselt number are analyzed.     

Boundary layer flow of a Jeffrey fluid with convective boundary conditions

The boundary layer stretched flow of a Jeffrey fluid subject to the convective boundary conditions was investigated. The governing dimensionless problems were computed by using the homotopy analysis approach. Convergence of the derived solutions was checked and the influence of embedded parameters was analyzed by plotting graphs. It was noticed that the velocity increases with an increase in the Deborah number. Furthermore, it was found that the temperature is also an increasing function of the Biot number. We further found that for fixed values of other parameters, the local Nusselt number increases by increasing the suction parameter and Deborah number. Numerical values of the skin friction coefficient and local Nusselt numbers were computed and examined. Copyright © 2011 John Wiley & Sons, Ltd.     

Electroviscous effect on non-Newtonian fluid flow in microchannels

Understanding non-Newtonian flow in microchannels is of both fundamental and practical significance for various microfluidic devices. A numerical study of non-Newtonian flow in microchannels combined with electroviscous effect has been conducted. The electric potential in the electroviscous force term is calculated by solving a lattice Boltzmann equation. And another lattice Boltzmann equation without derivations of the velocity when calculating the shear is employed to obtain flow field. The simulation of commonly used power-law non-Newtonian flow shows that the electroviscous effect on the flow depends significantly on the fluid rheological behavior. For the shear thinning fluid of the power-law exponent n < 1, the fluid viscosity near the wall is smaller and the electroviscous effect plays a more important role. And its effect on the flow increases as the ratio of the Debye length to the channel height increases and the exponent n decreases. While the shear thickening fluid of n > 1 is less affected by the electroviscous force, it can be neglected in practical applications.     

Free-surface non-Newtonian fluid flow in a round pipe

Free-surface pseudoplastic and viscoplastic fluid flows in a round pipe were studied for the case where the direction of motion coincides with the direction of gravity. Numerical modeling was performed using a technique based on a combination of the SIMPLE algorithm and the method of invariants. Three characteristic filling regimes were found to exist: a complete filling regime, a regime characterized by air-cavity formation on the solid wall, and a jet regime. Critical parameter values separating the regions of existence of these regimes were calculated. The evolution of quasisolid cores was studied for flow of a fluid with an yield point.     

An asymptotic theory for laminar pipe flow of a non-Newtonian fluid

For a generalized Newtonian fluid the viscosity * varies with the shear rate . Instead of assuming a certain dependence like rheological models do, the viscosity is expanded in a Taylor serie with respect to . Based on this expansion a perturbation approach to laminar pipe flow withq _w = const. and viscous heating included is formulated. The basic flow (zero order solution) is that of a Newtonian fluid. Higher order terms successively account for the influence of a non-Newtonian fluid. — The asymptotic results compare reasonably well with those of specific rheological models like power law or Ellis model. — The influence of temperature dependent properties (including the viscosity) can be accounted for by the same kind of asymptotic approach. The influence of shear rate as well as temperature dependence thus can be combined in general results valid for all generalized Newtonian fluids.Für ein verallgemeinertes Newtonsches Fluid ist die Viskosität * von der Scherrate abhängig. Statt nun eine bestimmte Abhängigkeit anzunehmen, wie dies für rheologische Modelle geschieht, wird die Viskosität als Taylor-Reihe in Bezug auf entwickelt. Ausgehend von dieser Entwicklung wird eine reguläre Störungsrechnung durchgeführt. Dies schließt den Effekt der Reibungswärme ein. Die Grundströmung ist die Strömung eines Newtonschen Fluides. Terme höherer Ordnung berücksichtigen den Einfluß des nicht-Newtonschen Fluidverhaltens. — Die asymptotischen Ergebnisse stimmen gut mit denen spezifischer rheologischer Modelle (power law, Ellis) überein. — Mit der gleichen asymptotischen Methode kann auch der Einfluß der Temperaturabhängigkeit der Stoffwerte erfaßt werden. Damit kann dann der Einfluß sowohl der Scherraten- als auch der Temperaturabhängigkeit auf eine allgemeine Weise für verallgemeinerte Newtonsche Fluide formuliert werden.     

A perturbation solution for non-Newtonian fluid two-phase flow through Porous media

In this paper, the mathematical problem of weak non-Newtonian fluid two-phase flow through porous media, including the effect of capillary pressure, is solved by singular perturbation method in combination with regular perturbation method. The asymptotic analytical solutions of the fractional flow function and the wetting phase saturation are obtained. The results are verified by numerical calculations and by classical solutions for corresponding Newtonian case. The influences of the non-Newtonian exponent and capillary pressure are discussed.     

Heat transfer and entropy generation analysis of non-Newtonian fluid flow through vertical microchannel with convective boundary condition

The entropy generation and heat transfer characteristics of magnetohydrodynamic(MHD) third-grade fluid flow through a vertical porous microchannel with a convective boundary condition are analyzed. Entropy generation due to flow of MHD non-Newtonian third-grade fluid within a microchannel and temperature-dependent viscosity is studied using the entropy generation rate and Vogel's model. The equations describing flow and heat transport along with boundary conditions are first made dimensionless using proper non-dimensional transformations and then solved numerically via the finite element method(FEM). An appropriate comparison is made with the previously published results in the literature as a limiting case of the considered problem.The comparison confirms excellent agreement. The effects of the Grashof number, the Hartmann number, the Biot number, the exponential space-and thermal-dependent heat source(ESHS/THS) parameters, and the viscous dissipation parameter on the temperature and velocity are studied and presented graphically. The entropy generation and the Bejan number are also calculated. From the comprehensive parametric study, it is recognized that the production of entropy can be improved with convective heating and viscous dissipation aspects. It is also found that the ESHS aspect dominates the THS aspect.     

Analyses of pseudo-similarity and boundary layer thickness for non-Newtonian falling film flow

One aim of this paper is to provide an extensive study on pseudo-similarity solutions of thermal boundary layer in falling film flow with non-Newtonian power-law fluids. The related mathematical models and solution approach are systematically presented. Another aim of this work is to further investigate the momentum boundary layer and the thermal boundary layer. Based on a newly defined local Prandtl number, the dependence of the thickness of the momentum boundary layer and thermal boundary layer on the power-law index is discussed. It is found that the momentum boundary layer thickness decreases monotonically with power-law index; while the thermal boundary layer thickness decreases slightly with power-law index and increases significantly with the decrease of the local Prandtl number. This study shows that the adopted pseudo-similarity approach is capable of solving the problem of non-similarity thermal boundary layer in the falling film of a non-Newtonian power-law fluid.     

The extended Graetz problem with Dirichlet wall boundary conditions

The extension of the Graetz problem to include axial conduction has been of interest in view of its application to a number of low Peclet number heat or mass transfer situations. Past efforts in dealing with this problem have been plagued with uncertainties arising from expansion in terms of “eigenfunctions” and “eigenvalues” belonging to a nonselfadjoint operator. The uncertainties spring from a lack of basis for the assumptions that no complex eigenvalues exist and that the calculated eigenvectors originate from a complete set. Other methods have been entirely numerical. The present work produces an entirelyanalytical solution to the Graetz problem for the Dirichlet boundary condition based on a selfadjoint formalism resulting from a decomposition of the convective diffusion equation into a pair of first order partial differential equations. Physically, the decomposition views the convective diffusion process as a pair of stipulations on how the temperature (or concentration) and theaxial energy (or mass) flow through a partial tube cross-section vary with radial and axial distances. The solution obtained is simple, and readily computed. To whom correspondence may be addressed     

Generalized flow analysis of non-Newtonian visco-elastic fluid flow through fractal reservoir

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Peristaltic flow of Johnson-Segalman fluid in asymmetric channel with convective boundary conditions

This work is concerned with the peristaltic transport of the Johnson-Segalman fluid in an asymmetric channel with convective boundary conditions. The mathematical modeling is based upon the conservation laws of mass, linear momentum, and energy. The resulting equations are solved after long wavelength and low Reynolds number are used. The results for the axial pressure gradient, velocity, and temperature profiles are obtained for small Weissenberg number. The expressions of the pressure gra-dient, velocity, and temperature are analyzed for various embedded parameters. Pumping and trapping phenomena are also explored.     

Slow flow of a non-Newtonian fluid past a micro-capsule

The streaming motion past a spherical microcapsule is studied. The particle consists of a thin elastic membrane enclosing an incompressible fluid. Since the problem is highly nonlinear, a perturbation solution is sought in the limiting case where the deviation from sphericity is small. Obviously, the capsule remains nearly spherical when λ, the ratio of viscous forces to elastic (shape-restoring) membrane forces is small. In this limit, the rheology of the inside fluid is immaterial and the problem is essentially characterized by three parameters: λ, the Reynolds number Re (interia effect), and the Weissenberg number We (non-newtonian effect). The deformation is obtained explicitly under the restriction We<1, Re<1. It is shown that to leading order, the capsule deforms exactly into a spheroid which can be either oblate or prolate, depending mainly upon the elasticity number We/Re: for We/Re<0.57 the spheroid is oblate, while for We/Re>0.81 a prolate spheroid results. For 0.57<We/Re<0.81 additional details of the rheology of the membrane and of the suspending fluid are needed. The degree of the deformation is governed by the parameters λ Re. All parameters of the problem enter into the expression of the drag force. On a qualitative basis, these results are similar to those for droplets although major differences exist quantitatively.