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.     

Mixed convection similarity solutions of a non-Newtonian fluid on a horizontal surface

The mixed convection boundary layer flow on a horizontal stationary or moving plate to a power-law non-Newtonian fluid is analyzed. An exact similarity solution is given for the case of a wall temperature that is inversely proportional to the power-law of the distance from the leading edge of the plate. It is shown that such a solution does not exist if the boundary parameter is smaller than a certain critical value. The effects of the flow index, buoyancy parameter and modified Prandtl number on the velocity and temperature profiles as well as on the global heat transfer and displacement thickness are discussed.

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Ähnlichkeitslösungen für die Mischkonvektion von einem Nicht-Newton'schen Fluid auf einer horizontalen Oberfläche

Zusammenfassung Hier ist die Mischkonvektion von Grenzschichtströmungen auf einer horizontal stillstehenden oder beweglichen Platte für einen Potenzansatz der Nicht-Newton'schen Fluide analysiert worden. Eine genaue Änlichkeitslösung ist für den Fall, daß die Wandtemperatur entgegengesetzt proportional zum Potenzansatz des Abstandes von der Vorderkante der Platte ist, gegeben worden. Es ist gezeigt worden, daß eine solche Lösung nicht existiert, wenn der Grenzparameter kleiner als ein bestimmter kritischer Wert ist. Die Einwirkungen des Strömungsindex, Auftriebsparameter und modifizierter Prantlzahl auf die Geschwindigkeitsund Temperaturprofile sind genauso intensiv besprochen worden, wie die globale Wärmeübertragung und die Verdrängungsdickte.

     

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.     

Analytical solutions for non-Newtonian fluid flows in pipe-like domains

This paper deals with some unsteady unidirectional transient flows of an Oldroyd-B fluid in unbounded domains which geometrically are axisymmetric pipe-like. An expansion theorem of Steklov is used to obtain exact solutions for flows satisfying no-slip boundary conditions. The well known solutions for a Navier-Stokes fluid, as well as those corresponding to a Maxwell fluid and a second grade one, appear as limiting cases of our solutions. The steady state solutions are also obtained for t→∞.     

Extended thermodynamics of a non-Newtonian fluid

Thermodynamics restrictions are calculated upon the constitutive equations of a non-Newtonian fluid. The fluid is of the rate type and the proper thermodynamic theory for such materials is seen to be extended thermodynamics. Thermodynamic stability conditions lead to the proper sign of the normal-stress coefficient, i.e. the sign that is compatible with experiment. Wave speeds for shear waves are calculated and the treatment of incompressible fluids is discussed.     

Peristaltic motion of a non-Newtonian fluid

Summary To understand theoretically the flow properties of physiological fluids, we have considered as a model the peristaltic motion of a power law fluid in a tube, with a sinusoidal wave of small amplitude travelling down its wall. The solution for the stream function is obtained as a power series in terms of the amplitude of the wave. The stream function and the velocity components are evaluated by solving numerically two point boundary value problems with a singular point at the origin. The influence of the applied pressure gradient along with non-Newtonian parameters on the streamlines and velocity profiles are discussed in detail.     

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.     

SIMILARITY SOLUTIONS OF BOUNDARY LAYER EQUATIONS FOR A SPECIAL NON-NEWTONIAN FLUID IN A SPECIAL COORDINATE SYSTME

Two dimensional equations of steady motion for third order fluids are expressed in a special coordinate system generated by the potential flow corresponding to an inviscid fluid. For the inviscid flow around an arbitrary object, the streamlines are the phicoordinates and velocity potential lines are psi-coordinates which form an orthogonal curvilinear set of coordinates. The outcome, boundary layer equations, is then shown to be independent of the body shape immersed into the flow. As a first approximation, assumption that second grade terms are negligible compared to viscous and third grade terms. Second grade terms spoil scaling transformation which is only transformation leading to similarity solutions for third grade fluid. By ~sing Lie group methods, infinitesimal generators of boundary layer equations are calculated. The equations are transformed into an ordinary differential system. Numerical solutions of outcoming nonlinear differential equations are found by using combination of a Runge-Kutta algorithm and shooting technique.     

Magnetohydrodynamic transient flows of a non-Newtonian fluid

Some exact solutions of the time-dependent partial differential equations are discussed for flows of an Oldroyd-B fluid. The fluid is electrically conducting and incompressible. The flows are generated by the impulsive motion of a boundary or by application of a constant pressure gradient. The method of Laplace transform is applied to obtain exact solutions. It is observed from the analysis that the governing differential equation for steady flow in an Oldroyd-B fluid is identical to that of the viscous fluid. Several results of interest are obtained as special cases of the presented analysis.     

Laminar length of a non-Newtonian fluid jet

The laminar length of a submerged jet of a non-Newtonian fluid is measured employing a flow visualization technique, which makes use of the birefringent property of the fluid and a circular polariscope. Fluids of two different concentrations are studied. The results indicate that the concentration, hence the non-Newtonian nature of the fluid, has no influence on the laminar length of the jet for 600 〈 Re 〈 1100. In the Reynolds number range of 50 to 200 the laminar length is affected by the viscous properties of the fluid. The results also indicate that there is not much difference in the laminar length of a Newtonian and non-Newtonian jet for 600 〈 Re 〈 1100.The experimental facility is also used as a “falling head” capillary viscometer. The laminar length data and the viscosity data are taken simultaneously. The viscosity obtained from the present test facility is found to agree with that obtained using a standard rotary viscometer in trend only.     

Variational principles in hydrodynamics of a non-Newtonian fluid

I.IntroductionTileresearchworkonvariationalprincipleinhydrodynamicsstartedasearlyas1940's.Referencesl:~6]aretheoneswhichmakemoreimportantcolltributions.l-lowevcr,mostoftheseworksstudied11on-viscousflow,putemphasisontheproblemsofexternalllcld,andworkedontilebasisofBernoulli'sequation.Prot'essorChienWeizang11I.slesttlblishcdvariationalprincipleforhydrodynalllicpl-oblemofviscousfluid,thatistheprillciplcofmaxilnLllllpowerlosses,alldgeneralizedvariational(stationary)pl.inciplesoiltilebiLsisofNav…     

Displacement of a Newtonian fluid by a non-Newtonian fluid in a porous medium

This paper presents an analytical Buckley-Leverett-type solution for one-dimensibnal immiscible displacement of a Newtonian fluid by a non-Newtonian fluid in porous media. The non-Newtonian fluid viscosity is assumed to be a function of the flow potential gradient and the non-Newtonian phase saturation. To apply this method to field problems a practical procedure has been developed which is based on the analytical solution and is similar to the graphic technique of Welge. Our solution can be regarded as an extension of the Buckley-Leverett method to Non-Newtonian fluids. The analytical result reveals how the saturation profile and the displacement efficiency are controlled not only by the relative permeabilities, as in the Buckley-Leverett solution, but also by the inherent complexities of the non-Newtonian fluid. Two examples of the application of the solution are given. One application is the verification of a numerical model, which has been developed for simulation of flow of immiscible non-Newtonian and Newtonian fluids in porous media. Excellent agreement between the numerical and analytical results has been obtained using a power-law non-Newtonian fluid. Another application is to examine the effects of non-Newtonian behavior on immiscible displacement of a Newtonian fluid by a power-law non-Newtonian fluid.     

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.     

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.     

Steady-state convection of a non-Newtonian fluid in a vertical layer

Steady-state thermal convection is investigated for a non-Newtonian fluid with a power-law rheological characteristic in a plane vertical channel bounded by parallel planes x=±h maintained at different temperatures.perm'. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 137–139, July–August, 1972.     

Flow of a non-Newtonian fluid between heated parallel plates

The flow of a fluid of grade three between heated parallel plates is examined for two cases. In the first instance we postulate constant heat flux at the walls and via a similarity transformation calculate the Nusselt number as a function of both Γ, the parameter controlling viscous dissipation, and Λ, the non-Newtonian parameter. In the second case we restrict the temperature to change only normal to the plates; solutions in this case are obtained for two temperature-viscosity models, μ = μ(θ).