The compressible Newtonian extrudate swell problem

We solve the compressible Newtonian extrudate swell problem in order to investigate the effect of compressiblity on the shape of the extrudate. We employ a first-order equation of state relating the density to the pressure and use finite elements for the numerical solution of the problem. Our results show that the shape of the extrudate and the final extrudate swell ratio are not significanlty affected even at high compressibility values.     

Numerical simulation of the extrusion of strongly compressible Newtonian liquids

The axisymmetric and plane extrusion flows of a liquid foam are simulated assuming that the foam is a homogeneous compressible Newtonian fluid that slips along the walls. Compressibility effects are investigated using both a linear and an exponential equation of state. The numerical results confirm previous reports that the swelling of the extrudate decreases initially as the compressibility of the fluid is increased and then increases considerably. The latter increase is sharper in the case of the exponential equation of state. In the case of non-zero inertia, high compressibility was found to lead to a contraction of the extrudate after the initial expansion, similar to that observed experimentally with liquid foams and to decaying oscillations of the extrudate surface. The time-dependent calculations show that the oscillatory steady-state solutions are stable. These steady-state oscillatory solutions are not affected by the length of the extrudate region nor by the boundary condition along the wall.     

Perturbation solutions of weakly compressible Newtonian Poiseuille flows with Navier slip at the wall

We consider both the planar and axisymmetric steady, laminar Poiseuille flows of a weakly compressible Newtonian fluid assuming that slip occurs along the wall following Navier’s slip equation and that the density obeys a linear equation of state. A perturbation analysis is performed in terms of the primary flow variables using the dimensionless isothermal compressibility as the perturbation parameter. Solutions up to the second order are derived and compared with available analytical results. The combined effects of slip, compressibility, and inertia are discussed with emphasis on the required pressure drop and the average Darcy friction factor.     

Squeeze flow of a power-law fluid between two rigid spheres with wall slip

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The squeeze flow of a bi-viscosity fluid between two rigid spheres with wall slip

In this paper, the squeeze flow between two rigid spheres with a bi-viscosity fluid is examined. Based on lubrication theory, the squeeze force is calculated by deriving the pressure and velocity expressions. The results of the normal squeeze force are discussed, and fitting functions of the squeeze and correction coefficients are given. The squeeze force between the rigid spheres increases linearly or logarithmically with the velocity when most or part of the boundary fluid reaches the yield state, respectively. Furthermore, the slip correction coefficient decreases with the increase in the velocity. The investigation may contribute to the further study of bi-viscosity fluids between rigid spheres with wall slip.     

On the stability of the shear flow of a viscoelastic fluid with slip along the fixed wall

In this paper we solve the time-dependent shear flow of an Oldroyd-B fluid with slip along the fixed wall. We use a non-linear slip model relating the shear stress to the velocity at the wall and exhibiting a maximum and a minimum. We assume that the material parameters in the slip equation are such that multiple steady-state solutions do not exist. The stability of the steady-state solutions is investigated by means of a one-dimensional linear stability analysis and by numerical calculations. The instability regimes are always within or coincide with the negative-slope regime of the slip equation. As expected, the numerical results show that the instability regimes are much broader than those predicted by the linear stability analysis. Under our assumptions for the slip equation, the Newtonian solutions are stable everywhere. The interval of instability grows as one moves from the Newtonian to the upper-convected Maxwell model. Perturbing an unstable steady-state solution leads to periodic solutions. The amplitude and the period of the oscillations increase with elasticity.     

Newtonian flow in a triangular duct with slip at the wall

We consider the Newtonian Poiseuille flow in a tube whose cross-section is an equilateral triangle. It is assumed that boundary slip occurs only above a critical value of the wall shear stress, namely the slip yield stress. It turns out that there are three flow regimes defined by two critical values of the pressure gradient. Below the first critical value, the fluid sticks everywhere and the classical no-slip solution is recovered. In an intermediate regime the fluid slips only around the middle of each boundary side and the flow problem is not amenable to analytical solution. Above the second critical pressure gradient non-uniform slip occurs everywhere at the wall. An analytical solution is derived for this case and the results are discussed.     

Scaling group transformation for MHD boundary layer flow over permeable stretching sheet in presence of slip flow with Newtonian heating effects

Taking into account the slip flow effects, Newtonian heating, and thermal radiation, two-dimensional magnetohydrodynamic （MHD） flows and heat transfer past a permeable stretching sheet are investigated numerically. We use one parameter group transformation to develop similarity transformation. By using the similarity transformation, we transform the governing boundary layer equations along with the boundary conditions into ordinary differential equations with relevant boundary conditions. The obtained ordinary differential equations are solved with the fourth-fifth order Runge-Kutta- Fehlberg method using MAPLE 13. The present paper is compared with a published one. Good agreement is obtained. Numerical results for dimensionless velocity, temperature distributions, skin friction factor, and heat transfer rates are discussed for various values of controlling parameters.     

Estimation of the parameters of Herschel-Bulkley fluid under wall slip using a combination of capillary and squeeze flow viscometers

The determination of the parameters of viscoplastic fluids subject to wall slip is a special challenge and accurate results are generally obtained only when a number of viscometers are utilized concomitantly. Here the characterization of the parameters of the Herschel-Bulkley fluid and its non-linear wall slip behavior is formulated as an inverse problem which utilizes the data emanating from capillary and squeeze flow rheometers. A finite element method of the squeeze flow problem is employed in conjunction with the analytical solution of the capillary data collected following Mooneys procedure, which uses dies with differing surface to volume ratios. The uniqueness of the solution is recognized as a major problem which limits the accuracy of the solution, suggesting that the search methodology should be carefully selected.     

Linear stability diagrams for the shear flow of an Oldroyd-B fluid with slip along the fixed wall

We consider the time-dependent shear flow of an Oldroyd-B fluid with slip along the fixed wall. Slip is allowed by means of a generic slip equation predicting that the shear stress is a non-monotonic function of the velocity at the wall. The complete one-dimensional stability analysis to one-dimensional disturbances is carried out and the corresponding neutral stability diagrams are constructed. Asymptotic results for large values of the elasticity number and finite element calculations are also presented. The instability regimes are within or coincide with the negative-slope regime of the slip equation. The numerical calculations agree with the linear stability results when the size of the initial perturbation is small. Large perturbations may destabilize a linearly stable steady state, leading to a periodic solution. The period and the amplitude of the periodic solutions increase with elasticity. Received: 19 June 1997 Accepted: 22 September 1997     

Wave regimes on a film of generalized Newtonian fluid flowing down a vertical plane

The flow of a thin film of generalized Newtonian fluid down a vertical wall in the gravity field is considered. For small flow-rates, in the long-wave approximation, an equation describing the evolution of the surface perturbations is obtained. Depending on the signs of the coefficients, this equation is equivalent to one of four equations with solutions significantly different in evolutionary behavior. For the most interesting case, soliton solutions are numerically found.     

On flow and stability of a Newtonian fluid past a rotating plane

An exact solution is given for the steady flow of a Newtonian fluid occupying the halfspace past the plane z=0 uniformly rotating about a fixed normal axis (Oz). This solution is obtained in a velocity field of the form considered by Berker [2] and can be deduced as a limiting case, as h+, of the solution to the problem relative to the strip 0zh imposing at z=h either the adherence boundary conditions or the free surface conditions. Furthermore, the stability of this flow, subject to periodic disturbances of finite amplitude, is studied using the energy method and the result is compared with those corresponding to stability of flows in the strip 0zh.

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Sommario In questa nota si mostra che-oltre alla calssica soluzione di von Karman [1] — esiste, per opportuni valori del gradiente di pressione all'infinito, una soluzione esatta per il moto stazionario di un fluido Newtoniano posto nel semispazio limitato dal piano z=0 uniformemente rotante attorno ad un asse ad esso perpendicolare (Oz). Tale soluzione, ottenuta sulla scia del lavoro di Berker [2], si può dedurre anche come limite, per h+, della soluzione del problema relativo alla striscia 0zh quando sul piano z=h si assegnano o le condizioni di aderenza o le condizioni di frontiera libera. Si studia poi la stabilità di tale moto rispetto a perturbazioni spazialmente periodiche di ampiezza finita col metodo dell'energia e si confronta il risultato ottenuto con quelli relativi alla stabilità dei moti nella striscia 0zh.

     

Unsteady compressible flow due to relative rotation and translation of a viscous fluid on a disk

Summary The modification of the axisymmetric viscous flow due to relative rotation of the disk or fluid by a translation of the boundary is studied. The fluid is taken to be compressible, and the relative rotation and translation velocity of the disk or fluid are time-dependent. The nonlinear partial differential equations governing the motion are solved numerically using an implicit finite difference scheme and Newton's linearisation technique. Numerical solutions are obtained at various non-dimensional times and disk temperatures. The non-symmetric part of the flow (secondary flow) describing the translation effect generates a velocity field at each plane parallel to the disk. The cartesian components of velocity due to secondary flow exhibit oscillations when the motion is due to rotation of the fluid on a translating disk. Increase in translation velocity produces an increment in the radial skin friction but reduces the tangential skin friction.     

Enhanced melt strength and stretching of linear low-density polyethylene extruded under strong slip conditions

The influence of extrusion under strong slip conditions on the extensional properties of linear low-density polyethylene was studied in this work. The material was extruded at two different temperatures under strong slip and no slip conditions, and was subsequently subjected to uniaxial elongational flow by means of a Rheotens device. Strong slip was evident through the elimination of sharkskin distortions and the stick-slip instability, as well as by the electrification of the extrudates. The extrudate swell was smaller in the presence of slip when comparing with no slip conditions at constant apparent shear rate, but it was found to be a unique function of the shear stress if comparison was performed at constant stress. The draw ratio and melt strength of the filaments obtained under slip conditions were larger compared to those without slip. In addition, draw resonance was postponed to higher draw ratios during the extrusion with strong slip at constant apparent shear rate. It is suggested that slip of the polymer at the die wall decreases the shear stress in the bulk, and therefore, restricts the disentanglement and orientation of macromolecules during flow, which subsequently produces the increase in draw ratio and melt strength during stretching.     

UNIFIED COMPUTATION OF FLOW WITH COMPRESSIBLE AND INCOMPRESSIBLE FLUID BASED ON ROE＇S SCHEME

A unified numerical scheme for the solutions of the compressible and incompressible Navier-Stokes equations is investigated based on a time-derivative preconditioning algorithm. The primitive variables are pressure, velocities and temperature. The time integration scheme is used in conjunction with a finite volume discretization. The preconditioning is coupled with a high order implicit upwind scheme based on the definition of a Roe＇s type matrix. Computational capabilities are demonstrated through computations of high Mach number, middle Mach number, very low Mach number, and incompressible flow. It has also been demonstrated that the discontinuous surface in flow field can be captured for the implementation Roe＇s scheme.     

Modelling high speed flow of a compressible fluid in a saturated porous medium

A modification to the Forchheimer-Brinkman equation, for the modelling of high speed flow of a compressible fluid in a dense saturated porous medium, is proposed. The modified equation is applied to a flow in which choking can occur.     

Rheo-PIV of a yield-stress fluid in a capillary with slip at the wall

An analysis of the yielding and flow behavior of a model yield-stress fluid, 0.2 wt% Carbopol gel, in a capillary with slip at the wall has been carried out in the present work. For this, a study of the flow kinematics in a capillary rheometer was performed with a two-dimensional particle image velocimetry (PIV) system. Besides, a stress-controlled rotational rheometer with a vane rotor was used as an independent way to measure the yield stress. The results in this work show that in the limit of resolution of the PIV technique, the flow behavior agrees with the existence of a yield stress, but there is a smooth solid?Cliquid transition in the capillary flow curve, which complicates the determination of the yield stress from rheometrical data. This complication, however, is overcome by using the solely velocity profiles and the measured wall shear stresses, from which the yield-stress value is reliably determined. The main details of the kinematics in the presence of slip were all captured during the experiments, namely, a purely plug flow before yielding, the solid?Cliquid transition, as well as the behavior under flow, respectively. Finally, it was found that the slip velocity increases in a power-law way with the shear stress.     

Blasius flow and heat transfer of fourth-grade fluid with slip

This investigation deals with the effects of slip, magnetic field, and non- Newtonian flow parameters on the flow and heat transfer of an incompressible, electrically conducting fourth-grade fluid past an infinite porous plate. The heat transfer analysis is carried out for two heating processes. The system of highly non-linear differential equations is solved by the shooting method with the fourth-order Runge-Kutta method for moderate values of the parameters. The effective Broyden technique is adopted in order to improve the initial guesses and to satisfy the boundary conditions at infinity. An exceptional cross-over is obtained in the velocity profile in the presence of slip. The fourth-grade fluid parameter is found to increase the momentum boundary layer thickness, whereas the slip parameter substantially decreases it. Similarly, the non-Newtonian fluid parameters and the slip have opposite effects on the thermal boundary layer thickness.     

Experimental estimation of wall slip for filled rubber compounds

The phenomenon of wall slip during flow of rubber compounds through capillaries is investigated for a typical styrene-butadiene elastomer with carbon black. It was found that at low temperature (110°C) the dependencies of slip velocity V _c on shear stress are described by the power law but, additionally, V _c depends on radius of a channel. At high temperatures there is a critical shear stress below which sliding is absent. Sliding appears only at higher shear stresses where, again, V _c depends on shear stress and the radius of a channel.     

Unsteady stagnation-point flow of a Newtonian fluid in the presence of a magnetic field

The unsteady stagnation-point flow of a viscous fluid impinging on an infinite plate in the presence of a transverse magnetic field is examined and solutions are obtained. It is assumed that the infinite plate at y=0 is making harmonic oscillations in its own plane. A finite difference technique is employed and solutions for small and large frequencies of the oscillations are obtained for various values of the Hartmann's number.