Asymptotic Nusselt numbers for dissipative non-Newtonian flow through ducts

General expressions for fully developed temperature profiles and Nusselt numbers are obtained for heat transfer to non-Newtonian fluid flow between parallel plates and through circular pipes subjected to a uniform wall heat flux. The effect of viscous dissipation is taken into account since it may often be significant in the flow of non-Newtonian fluids. Asymptotic Nusselt numbers for three widely used models, i.e. the power law fluid, the Bingham plastic, and the Ellis fluid are obtained as specific results.     

Asymptotic Nusselt numbers for Newtonian flow through a parallel-plate channel with recycle

General expressions for evaluating the asymptotic Nusselt number for a Newtonian flow through a parallel-plate channel with recycle at the ends have been derived. Numerical results with the ratio of thicknesses as a parameter for various recycle ratios are obtained. A regression analysis shows that the results can be expressed by log Nur^0.83=0.3589 (log)² -0.2925 (log) + 0.3348 forR 3, 0.1 0.9; logNu=0.5982(log)² +0.3755 × 10^–2 (log) +0.8342 forR 10^–2, 0.1 0.9.

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Asymptotische Nusselt-Zahlen für die Newtonsche Strömung durch einen Kanal aus parallelen Platten mit Rückführung

Zusammenfassung In dieser Untersuchung wurden allgemeine Ausdrücke hergeleitet um die asymptotische Nusselt-Zahl für eine Newtonsche Strömung durch einen Kanal aus parallelen Platten mit Rückführung an den Enden berechnen zu können. Es wurden numerische Ergebnisse mit den Dicken-Verhältnissen, als Parameter für verschiedene Rückführungs-verhältnisse, erhalten. Eine Regressionsanalyse zeigt, daß die Ergebnisse wie folgt ausgedrückt werden können: log Nur^0,83=0,3589 (log)² -0,2925 (log) + 0,3348 fürR 3, 0,1 0,9; logNu=0,5982(log)² +0,3755 × 10^–2 (log) + 0,8342 fürR 10^–2, 0,1 0,9.

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Nomenclature A₁ shooting value,d(0)/d - A₂ shooting value,d(1)/d - B channel width - Gz Graetz number, U_bW²/L - h _m logarithmic average convective heat transfer coefficient - h _x average local convective heat transfer coefficient - k thermal conductivity - L channel length - Nu average local Nusselt number, 2 h_xW/k - Nu _m logarithmic average Nusselt number, 2h_mW/k - R recycle ratio, reverse volume flow rate divided by input volume flow rate - T temperature of fluid - T _m bulk temperature, Eq. (8) - T ₀ temperature of feed stream - T _s wall temperature - U velocity distribution - U _b reference velocity,V/BW - V input volume flow rate - v dimensionless velocity distribution, U/U_b - W channel thickness - x longitudinal coordinate - y transversal coordinate - Z₁-z₆ functions defined in Eq. (A1) - thermal diffusivity - least squares error, Eq. (A7) - weight, Eqs. (A8), (A9) - dimensionless coordinate,y/W - dimensionless coordinate,x/GzL - function, Eq. (7)     

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.     

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.     

Weakly-imposed Dirichlet boundary conditions for non-Newtonian fluid flow

We propose a new formulation for weakly imposing Dirichlet boundary conditions in non-Newtonian fluid flow. It is based on the Gerstenberger–Wall formulation for Newtonian fluids [1], but extended to non-Newtonian fluids. It uses a stabilization term in the weak form that is independent from the actual fluid model used, except for an adjustable parameter κ, having the physical dimension of a viscosity. The new formulation is tested, combined with an extended finite element method, for the flow past a cylinder between two walls using both a generalized Newtonian and a viscoelastic fluid. It is shown that the convergence is optimal for the generalized Newtonian fluid by comparing with a converged boundary-fitted solution using traditional strong boundary conditions. Also the solution of the viscoelastic fluid compares very well with a traditional solution using a boundary-fitted mesh and strong Dirichlet boundary conditions. For both fluid models we also test various values of the κ parameter and it turns out that a value equal to the zero-shear-viscosity gives good results. But, it is also shown that a wide range of κ values can be chosen without sacrificing accuracy.     

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.     

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.     

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.     

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

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An investigation on internal thermal flow of non-Newtonian fluid

In the present paper an unsteady thermal flow of non-Newtonian fluid is investigated which is of the fiow into axisymmetric mould cavity. In the second part an unsteady thermal flow of upper-convected Maxwell fluid is studied, For the flow into mould cavity the constitutive equation of power-law fluid is used as a rheological model of polymer fluid. The apparent viscosity is considered as a function of shear rate and temperature. A characteristic viscosity is introduced in order to avoid the nonlinearity due to the temperature dependence of the apparent viscosity. As the viscosity of the fluid is relatively high the flow of the thermal fluid can be considered as a flow of fully developed velocity field. However, the temperature field of the fluid fiow is considered as an unsteady one. The governing equations are constitutive equation, momentum equation of steady flow and energy conservation equation of non-steady form. The present system of equations has been solved numerically by the splitting differen     

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.     

An efficient matrix solver for finite-element analysis of non-Newtonian fluid flow problems

When the Galerkin finite-element method with a nine-node isoparametric Lagrangian element is applied to solve non-Newtonian fluid flow problems, a considerable amount of computing time is required to solve the discretized non-linear system of equations by Newton's method. A method proposed by Broyden has been modified to compute the Jacobian matrix associated with Newton's method. This modified Broyden's method can be combined with the frontal method to efficiently solve the linearized finite-element equations during the iteration. Numerical results of a sample problem concerning the determination of the pressure-drop/flow-rate relationship for power-law fluids in rectangular ducts show that the application of this new method can reduce computing time substantially.     

Jeffery-Hamel flow of non-Newtonian fluid with nonlinear viscosity and wall friction

A Jeffery-Hamel(J-H) flow model of the non-Newtonian fluid type inside a convergent wedge(inclined walls) with a wall friction is derived by a nonlinear ordinary differential equation with appropriate boundary conditions based on similarity relationships. Unlike the usual power law model, this paper develops nonlinear viscosity based only on a tangential coordinate function due to the radial geometry shape. Two kinds of solutions are developed, i.e., analytical and semi-analytical(numerical) solutions with suitable assumptions. As a result of the parametric examination, it has been found that the Newtonian normalized velocity gradually decreases with the tangential direction progress. Also, an increase in the friction coefficient leads to a decrease in the normalized Newtonian velocity profile values. However, an increase in the Reynolds number causes an increase in the normalized velocity function values. Additionally, for the small values of wedge semi-angle, the present solutions are in good agreement with the previous results in the literature.     

Hydrodynamic stability of Poiseuille flow of a viscoplastic non-Newtonian fluid

The hydrodynamic stability of Poiseuille flow of a viscoplastic fluid is investigated. The flow is shown to be stable for infinitesimal disturbances.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 152–154, November–December, 1974.In conclusion the authors thank S. A. Regirer for critical remarks.     

Asymptotic theory of flow attachment for a viscous incompressible fluid

A study is made of the two-dimensional laminar flow of a viscous incompressible fluid in the neighborhood of the point of attachment of the flow to a solid surface. The case of large Reynolds numbers is considered. It is assumed that the dimensions of the separation region are of the same order of magnitude as the characteristic dimension of the body around which the flow takes place. The asymptotic theory of such flow is constructed by applying the method of matched asymptotic expansions to the analysis of the Navier-Stokes equations. It is shown that in the neighborhood of the attachment point the flow is locally inviscid and can be described by the complete system of Euler equations. A solution to the corresponding boundary-value problem is constructed numerically.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 63–71, November–December, 1980.