Piezoviscous Fluid Research Articles

A spectral collocation scheme for the flow of a piezo-viscous fluid in ducts with slip conditions

In this paper we present a numerical scheme based on spectral collocation methods to investigate the flow of a piezo-viscous fluid, i.e., a fluid in which the rheological parameters depend on the pressure. In particular, we consider an incompressible Navier–Stokes fluid with pressure dependent viscosity flowing in: (i) a two-dimensional non-symmetric planar channel; (ii) a three-dimensional axisymmetric non-straight conduit. For both cases we impose the Navier slip boundary conditions that can be reduced to the classical no-slip condition for a proper choice of the slip parameter. We assume that the dependence of the viscosity on the pressure is of exponential type (Barus law), even though the model can be replaced by any other viscosity function. We write the mathematical problem (stress based formulation) and discretize the governing equations through a spectral collocation scheme. The advantage of this numerical procedure, which to the authors’ knowledge has never been used before for this class of fluids, lies in in the ease of implementation and in the accuracy of the solution. To validate our model we compare the numerical solution with the one that can be obtained in the case of small aspect ratio, i.e., the leading order lubrication solution. We perform some numerical simulation to investigate the effects of the pressure-dependent viscosity on the flow. We consider different wall functions to gain insight also on the role played by the channel/duct geometry. In both cases (i), (ii) we find that the increase of the coefficient appearing in the viscosity function results in a global reduction of the flow, as physically expected.

Falkner–Skan boundary layer flow of a fluid with pressure-dependent viscosity past a stretching wedge with suction or injection

In this paper we study the boundary layer flow of a particular class of non-Newtonian fluids, namely piezo-viscous fluids, past a stretching wedge (Falkner–Skan flow). These fluids are essentially Navier–Stokes fluids with pressure-dependent viscosity. We consider the two-dimensional steady flow and introduce some similarity variables, so that the problem is reduced to a third-order nonlinear BVP. We solve the problem numerically by means of a shooting procedure combined with Newton’s method, and we find local similarity solutions. We investigate the behaviour of the velocity field, of the drag and of the thickness of the boundary layer for different values of the involved parameters, making a comparison between the piezo-viscous and the Newtonian case. We find that, depending on some material parameters, the effect of pressure is that of reducing or increasing the boundary layer with respect to the Newtonian case.

Elastohydrodynamic Film Formation and Sol/Gel Transition of Aqueous Fluids

One of the most widely used water soluble lubricants is PolyAlkylene Glycol (PAG). PAG aqueous solutions can form a gel depending on the concentration and temperature, which affects the formation and friction of lubricating films. This experimental work combined rheological measurements and in-situ film-forming analysis in pure rolling conditions using PAG aqueous solutions of various concentrations. It shows that the pure PAG behaved classically as a piezoviscous fluid, while its aqueous solutions behaved as isoviscous fluids. This was confirmed by the establishment of a film formation map. In addition, the aqueous solutions exhibited two behaviors, characteristic of either sol or gel. A detailed analysis of the film thickness evolution, based on Moes–Venner’s predictions, allowed us to calculate the effective viscosity in the inlet zone and to discuss the origin of the two families, sol vs gel.Graphical

On lower-dimensional models in lubrication, Part B: Derivation of a Reynolds type of equation for incompressible piezo-viscous fluids

The Reynolds equation is a lower-dimensional model for the pressure in a fluid confined between two adjacent surfaces that move relative to each other. It was originally derived under the assumption that the fluid is incompressible and has constant viscosity. In the existing literature, the lower-dimensional Reynolds equation is often employed as a model for the thin films, which lubricates interfaces in various machine components. For example, in the modelling of elastohydrodynamic lubrication (EHL) in gears and bearings, the pressure dependence of the viscosity is often considered by just replacing the constant viscosity in the Reynolds equation with a given viscosity-pressure relation. The arguments to justify this are heuristic, and in many cases, it is taken for granted that you can do so. This motivated us to make an attempt to formulate and present a rigorous derivation of a lower-dimensional model for the pressure when the fluid has pressure-dependent viscosity. The results of our study are presented in two parts. In Part A, we showed that for incompressible and piezo-viscous fluids it is not possible to obtain a lower-dimensional model for the pressure by just assuming that the film thickness is thin, as it is for incompressible fluids with constant viscosity. Here, in Part B, we present a method for deriving lower-dimensional models of thin-film flow, where the fluid has a pressure-dependent viscosity. The main idea is to rescale the generalised Navier-Stokes equation, which we obtained in Part A based on theory for implicit constitutive relations, so that we can pass to the limit as the film thickness goes to zero. If the scaling is correct, then the limit problem can be used as the dimensionally reduced model for the flow and it is possible to derive a type of Reynolds equation for the pressure.

On compressible and piezo-viscous flow in thin porous media

In this paper, we study flow through thin porous media as in, e.g. seals or fractures. It is often useful to know the permeability of such systems. In the context of incompressible and iso-viscous fluids, the permeability is the constant of proportionality relating the total flow through the media to the pressure drop. In this work, we show that it is also relevant to define a constant permeability when compressible and/or piezo-viscous fluids are considered. More precisely, we show that the corresponding nonlinear equation describing the flow of any compressible and piezo-viscous fluid can be transformed into a single linear equation. Indeed, this linear equation is the same as the one describing the flow of an incompressible and iso-viscous fluid. By this transformation, the total flow can be expressed as the product of the permeability and a nonlinear function of pressure, which represents a generalized pressure drop.

Numerical simulations of an incompressible piezoviscous fluid flowing in a plane slider bearing

We provide numerical simulations of an incompressible pressure-thickening and shear-thinning lubricant flowing in a plane slider bearing. We study the influence of several parameters, namely the ratio of the characteristic lengths $\varepsilon>0$ (with $\varepsilon\searrow0$ representing the Reynolds lubrication approximation); the coefficient of the exponential pressure--viscosity relation $\alpha^*\geq0$; the parameter $G^*\geq0$ related to the Carreau--Yasuda shear-thinning model and the modified Reynolds number $\mathrm{Re}_\varepsilon\geq0$. The finite element approximations to the steady isothermal flows are computed without resorting to the lubrication approximation. We obtain the numerical solutions as long as the variation of the viscous stress $\mathbf{S}=2\eta(p,\mathrm{tr}\mathbf{D}^2)\mathbf{D}$ with the pressure is limited, say $|\partial\mathbf{S}/\partial p|\leq1$. We show conclusively that the existing practice of avoiding the numerical difficulties by cutting the viscosity off for large pressures leads to results that depend sorely on the artificial cut-off parameter. We observe that the piezoviscous rheology generates pressure differences across the fluid film.

Squeeze flow of a piezoviscous fluid

We investigate the squeeze flow of a piezoviscous fluid. The classical Navier–Stokes fluid model and the no-slip boundary condition on the plate–fluid interface implies the presence of the pressure singularity at the contact line. It can be conjectured that the same behaviour is present even for piezoviscous fluids. Consequently the pressure variation in the fluid could be large, and the pressure dependent viscosity could be important even if the piezoviscous coefficient is apparently small. We investigate this conjecture by means of numerical simulation based on a variant of the spectral collocation method. The numerical simulations indicate that the effects due to pressure dependent viscosity are for certain range of parameter values significant.

Steady State EHL Line Contact Analysis with Surface Roughness and Linear Piezo-viscosity

A special form of hydrodynamic lubrication, elastohydrodynamic lubrication (EHL), is generally encountered in mechanical components involving highly stressed non-conformal contact such as gears, cams, roller bearing etc. EHL is characterized with substantial elastic deformation of the interacting surface and piezo-viscous rise in lubricant viscosity which assist largely in evolution of load carrying fluid film.In the current paper, the effect of surface roughness of different magnitude in EHL line contact on pressure distribution is considered. At the same time graph between film thickness with rolling direction is plotted which gives reader the knowledge of EHL film thickness. Steady state effect of EHL with linear piezo-viscosity response at low pressure is taken into consideration. Full isothermal, Newtonian simulation of the EHL problem is discussed. It investigate the pressure distribution of sinusoidal waviness in the line contact, more precisely it study the effect of linear piezo-viscous fluid on surface roughness of different amplitude in steady state EHL line contact. Graphs plotted gives the clear idea to the designers, the behaviour of surface roughness with several amplitude and wavelength by linear piezo-viscous liquid like 2, 3-Dimethylpentane, L7808 (high temperature lubricant) and water/glycol solution of two water percentage and exponential fluid like santotrac 40 (S 40).The Reynolds equation, which governs the generation of pressure in the lubricated contact, is discretized using finite differences and solved along with the load balance equation using Newton-Raphson technique. A computer code is developed for implementation of solution algorithm. The results are obtained to study the linear and exponential piezo-viscous effect on surface roughness on the EHL characteristics.

The motion of a piezoviscous fluid under a surface load

Using a variant of a spectral collocation method we numerically solve the problem of the motion of a highly viscous fluid with pressure dependent viscosity under a surface load, which is a problem relevant in many applications, in particular in geophysics and polymer melts processing. We compare the results with the results obtained by the classical Navier–Stokes fluid (constant viscosity). It turns out that for a realistic parameter values the two models give substantially different predictions concerning the motion of the free surface and the velocity and the pressure fields beneath the free surface.As a byproduct of the effort to test the numerical scheme we obtain an analytical solution—for the classical Navier–Stokes fluid—of the surface load problem in a layer of finite depth.

Piezo-viscous flows over an inclined surface

Stokes was the first to recognize that the viscosity of many fluids varies significantly with pressure. Later, several experimental studies showed that such a variation may be exponential. Here, by using the lubrication theory as revised by Rajagopal and Szeri, we study the flow of a piezo-viscous fluid down an incline in various flow regimes.

Stability of plane Poiseuille–Couette flows of a piezo-viscous fluid

We examine stability of fully developed isothermal unidirectional plane Poiseuille–Couette flows of an incompressible fluid whose viscosity depends linearly on the pressure as previously considered in Hron et al. [J. Hron, J. Málek, K.R. Rajagopal, Simple flows of fluids with pressure-dependent viscosities, Proc. R. Soc. Lond. A 457 (2001) 1603–1622] and Suslov and Tran [S.A. Suslov, T.D. Tran, Revisiting plane Couette–Poiseuille flows of a piezo-viscous fluid, J. Non-Newtonian Fluid Mech. 154 (2008) 170–178]. Stability results for a piezo-viscous fluid are compared with those for a Newtonian fluid with constant viscosity. We show that piezo-viscous effects generally lead to stabilisation of a primary flow when the applied pressure gradient is increased. We also show that the flow becomes less stable as the pressure and therefore the fluid viscosity decrease downstream. These features drastically distinguish flows of a piezo-viscous fluid from those of its constant-viscosity counterpart. At the same time the increase in the boundary velocity results in a flow stabilisation which is similar to that observed in Newtonian fluids with constant viscosity.

Revisiting plane Couette–Poiseuille flows of a piezo-viscous fluid

We re-examine fully developed isothermal unidirectional plane Couette–Poiseuille flows of an incompressible fluid whose viscosity depends linearly on the pressure as previously considered in [J. Hron, J. Málek, K.R. Rajagopal, Simple flows of fluids with pressure-dependent viscosities, Proc. R. Soc. Lond. A 457 (2001) 1603–1622]. We show that the conclusion made there that, in contrast to Newtonian and power-law fluids, piezo-viscous fluids allow multiple solutions is not justified, and that the inflection velocity profiles reported in [J. Hron, J. Málek, K.R. Rajagopal, Simple flows of fluids with pressure-dependent viscosities, Proc. R. Soc. Lond. A 457 (2001) 1603–1622] cannot exist. Subsequently, we undertake a systematic parametric study of these flows and identify three distinct families of solutions which can exist in the considered geometry. One of these families has no similar counterpart for fluids with pressure-independent viscosity. We also show that the critical wall speed exists beyond which Poiseuille-type flows are impossible regardless of the magnitude of the applied pressure gradient. For smaller wall speeds channel choking occurs for Poiseuille-type flows at large pressure gradients. These features distinguish drastically piezo-viscous fluids from their Newtonian and power-law counterparts.

On the Choking of the Flow of Piezoviscous Liquids

Experimental evidence for choking of a piezoviscous fluid in a capillary tube is presented. It is shown that under a nearly isothermal condition, it is possible for the flow to experience choking whereby increasing the inlet pressure ceases to affect the flow rate. The occurrence of this phenomenon can be predicted by noting a singularity in the flowrate equation in a capillary tube under typical conditions encountered in injection molding (cf. Denn, 1981). Further examination of the general equations governing the shear flow of laminar, incompressible, linearly viscous (Newtonian) liquids under high pressures reveals a singularity in the solution for the pressure gradient. This apparently unreported phenomenon is shown to occur in liquids whose viscosity, μ, increases exponentially with pressure, p, as μ = μ0eαp, where α denotes the pressure-viscosity coefficient. For a two-dimensional configuration with the shearing action taking place in the x direction and with p = p(x, y), the singularity is shown to take place when: τyx=±(1/α)1+α2τxxτyy, at which case both δp/δx and δp/δy →∞. The occurrence of such a singularity in the pressure gradient may have an important physical implication in lubrication flows particularly in concentrated contacts such as elastohydrodynamic lubrication of rolling element bearings and gears.

The Piezo-Viscous Fluid, Rigid Solid Regime of Lubrication

A general analysis of the lubrication of rigid-ellipsoidal solids by a piezo-viscous fluid is reported. Solutions based upon the Reynolds cavitation boundary condition are presented and details are given of the resulting pressure distributions and film thickness predictions. Four distinct forms of the fluid film lubrication of a general curved solid near a plane have previously been noted and described as: (i) rigid-isoviscous, (ii) elastic-isoviscous, (iii) elastic-variable viscosity (piezo-viscous), (iv) rigid-variable viscosity (piezo-viscous). Solutions already exist for the first three forms of lubrication for conjunctions formed between solids having curvatures in two principal directions, but the rigid-variable viscosity regime has not previously been analysed within the same general set of conditions. The results not only enable the film thickness between curved solids in nominal point contact to be predicted with reasonable confidence, but they indicate quite clearly the dominant physical effects governing the hydrodynamic performance of such conjunctions.