Elastico-viscous squeeze films: Part 2. superimposed rotation

The first part of this paper contains a reconsideration of the conventional squeeze-film situation. It is shown that the Part 1 concentration on the half-time t_{$\text{1}\text{2}$} sometimes hides interesting elastico-viscous effects, since these are most pronounced at short times after the load is applied.The bulk of the paper is concerned with the more general situation in which a rotational flow is superimposed on the basic squeezing flow. This is brought about by rotating the bottom plate. An approximate theoretical analysis is shown to lead to a satisfactory prediction of observed behaviour under conditions of light loading.The experiments show substantial transient stress-overshoot effects under some conditions and there is also a possible indication of long-range memory effects in some of the experiments.     

The oscillatory squeeze flow rheometer: comprehensive theory and a new experimental facility

In this paper, we give detailed attention to a relatively recent method for the determination of the linear dynamic properties of viscoelastic systems, namely, the so-called oscillatory squeeze flow (OSF) technique. We provide a comprehensive theory for the OSF rheometer, which includes a full discussion of the influence of fluid inertia. In the process, it is argued that, fortuitously perhaps, fluid inertia is more easily accommodated in the OSF rheometer than in the corresponding torsional-flow techniques. A new version of the OSF rheometer is described and experimental results on a set of viscoelastic systems are used to demonstrate the versatility of the technique. In the process, the potential use of the instrument within an industrial quality control environment is stressed.     

Technical note: correcting for shear strain in an oscillatory squeeze flow rheometer

Currently, rheologists working in the field of oscillatory squeeze flow use extensional strain to characterize the deformations. Due to the shear-dominated flow observed in low Trouton ratio fluids undergoing squeeze flow, it is proposed that an alternate geometry-dependent definition for shear strain in squeeze flow be used instead. Through the use of finite element modelling, it has been shown that this geometry-dependent strain definition allows for better comparison of measurements between both squeeze flow rheometers of different geometric configurations and rotational rheometers. This idea was then explored through laboratory experiments, further supporting this hypothesis. While this definition of strain will only hold true within the bounds of a material’s linear viscoelastic regime, it will help to determine where this boundary is, and thus allow for more accurate material characterization. This type of relationship will become increasingly important with the growing use of squeeze flow rheometers for large-amplitude oscillatory squeezing trials.     

Large deformations of viscoelastic squeeze films

The large deformation behavior of a viscoelastic squeeze film being compressed or extended between two parallel circular plates is studied from a Lagrangian viewpoint. A single integro-differential equation is shown to govern the flow of a rubberlike liquid in this geometry, and the solution is compared with results for a Newtonian squeeze film. Instantaneous solid-like response, small amplitude oscillations, and creep-recovery calculations are presented.     

Fluid inertia effects in squeeze films

Summary Fluid inertia effects in squeeze films are analyzed. Experimental results are also presented. The agreement between theory and experiment is very good.     

Micropolar squeeze films between rough rectangular plates

In this paper, the lubrication theory for squeezing with micropolar fluids in smooth surfaces has been advanced to analyze the effects arising from roughness considerations using the stochastic approach. This theory is subsequently applied to the problem of squeezing between rough rectangular plates. It is observed that the roughness effects are more pronounced for micropolar fluids as compared to the Newtonian fluids.Nomenclature a x-dimension of rectangular plate - A area of rectangular plate - b z-dimension of rectangular plate - B non-dimensional roughness parameter, c/h _n (for load capacity), c/h _n1 (for squeeze time) - c maximum asperity deviation from nominal film height - E expectancy operator - f(N, l, h) defined by equation (4) - F(N, L, H) defined by equation (31) - F ₁(N, L, B) defined by equation (29) - F ₂(N, L, B) defined by equation (30) - F ₃(N, L, H _n , B) defined by equation (34) - F ₄(N, L, H _n , B) defined by equation (35) - g probability density distribution function - h film height, h=h _n +h _s - h _n nominal film height - h _s deviation of film height from nominal level - h _n1 initial (nominal) film height - H, H _n , H _s non-dimensional forms of h, h _n , h _s respectively - l characteristic material length, (/4)^1/2 - L length ratio, h _n /l (for load capacity), h _n1/l (for squeeze time) - n integer - N coupling number, (/(2+))^1/2 - p pressure - q _x , q _z flow components in x- and z-directions, respectively - t time - T non-dimensional time - w load capacity - W non-dimensional load capacity - x, z cartesian coordinates - angular coordinate - Newtonian viscosity - , micropolar viscosity coefficients - aspect ratio, b/a - standard deviation - /h _n - random variable - defined by equation (19) - defined in equation (28) - defined in equation (33)     

Inertia effects in rheometrical flow systems Part 3: Some energy considerations with respect to the flow field in the balance rheometer

Summary Following up a previous paper by one of the presents authors on the flow field in the balance rheometer, inertia effects being included, in this paper some energy considerations with respect to this flow field are presented. It is shown that in a frame rotating with the same angular velocity as the hemispheres the power supplied by these hemispheres equals the rate of energy dissipation in the sample, i.e. in this coordinate system there is no stress power paradox. Further it is shown that the elastic couple for a Newtonian liquid, appearing in the calculations, stems from the extra kinetic energy caused by the deviation of the actual flow field from the flow field that appears when inertia effects are ignored.

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Zusammenfassung Als Fortsetzung des früheren Beitrages eines der hier genannten Autoren über das Strömungsfeld in einem Képès-Rheometer unter Berücksichtigung der Flüssigkeitsträgheit werden in diesem Beitrag einige Energiebetrachtungen angestellt. Es wird gezeigt, daß in einem Koordinatensystem, das mit gleicher Winkelgeschwindigkeit wie die Halbkugeln rotiert, die durch diese Halbkugeln zugeführte Leistung der in der Probe dissipierten Leistung gleich ist, d. h. daß in diesem Koordinatensystem das sogenannte Spannungsenergieparadox nicht vorliegt. Es wird weiter gezeigt, daß das bei einer newtonschen Flüssigkeit auftretende elastische Drehmoment seinen Ursprung in der zusätzlichen kinetischen Energie hat, die der Abweichung des tatsächlichen Strömungsfeldes von dem unter Vernachlässigung der Flüssigkeitsträgheit berechneten Strömungsfeld entspricht.

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a distance between rotation axes of orthogonal rheometer - b, c positive numbers (see eq. [29]) - f _n (z) j _n (z),y _n (z),h _n ⁽¹⁾ (z) orh _n ⁽²⁾ (z) - h distance between discs of orthogonal rheometer - h _n ⁽¹⁾ (z) =j _n (z) +iy _n (z) spherical Bessel functions of the third kind and ordern - h _n ⁽²⁾ (z) =j _n (z) –iy _n (z) spherical Bessel functions of the third kind and ordern - i - j _n (z) spherical Bessel function of the first kind and ordern - k – (/2)^1/2 - n integer - p ₁ j ₁(r ₂)y ₁(r ₁) –j ₁(r ₁)y ₁(r ₂) - q ₁ - r spherical polar coordinate - r ₁(r ₂) radius of inner (outer) hemisphere - s ₁ - t time - u _r ,u ,u physical components of displacement - x, z variables - x, y, z cartesian coordinates in eq. [1] - y _n (z) spherical Bessel function of the second kind and ordern - E _kin kinetic energy - E _s stored energy - F force - F _n (z) J _n (z),Y _n (z),H _n ⁽¹⁾ (z) orH _n ⁽²⁾ (z) - G ^* =G + iG complex shear modulus - H _n ⁽¹⁾ (z) =J _n (z) +iY _n (z) Bessel functions of the third kind and ordern - H _n ⁽²⁾ (z) =J _n (z) –iY _n (z) Bessel functions of the third kind and ordern - Im imaginary part of - J _n (z) Bessel function of the first kind and ordern - N - P energy supplied to the sample during one cycle - Re real part of - S strain - U - W energy dissipated in the sample during one cycle - Y _n Bessel function of the second kind and ordern - (/G ^*)^1/2-complex shear wave factor - /(r ₁ +r ₂) - loss angle - angle between rotation axes of balance rheometer - viscosity - spherical polar coordinate - order of cylinder function (see eq. [24]) - linear combination of spherical Bessel functions of first, second, and third kind, in which the coefficients are independent of the argument and the order - µ, v constants (see eq. [24]) - see - density - shear stress - spherical polar coordinate - cylinder functions - angular velocity With 3 figures     

Modelling of ER squeeze films at low amplitude oscillations

An experimental investigation of electrorheological squeeze film dynamics is presented for constant applied voltage and low strain amplitude. Both broadband random and sinusoidal motion are examined to explain complex film dynamics. Spectral results indicate a primarily elastic response with slip at the plate boundaries. By examining the evolution of an effective shear modulus over time, sinusoidal results show that slip at the boundaries is due to a solvent layer which may be modelled as a separate variable thickness squeeze film.     

Micro-Fourier rheometer: Inertial effects

The inertial effects in a random squeezing rheometer are examined, both theoretically and experimentally. The rheometer is based on small amplitude random squeezing between two parallel plates, where the upper plate is driven by a random displacement with a broad band spectrum. A fast Fourier transform is used to deliver the complex modulus (or viscosity) of the fluid in a single brief test, over more than two decades of frequency. The inertia of the fluid is shown to produce an error factor, which is also a function of the frequency. The correction factor can be well approximated by a first-order correction in the Reynolds number, for a very large range of Reynolds number, making the inertial correction a very simple procedure for light fluids.     

Variable viscosity squeeze films and bubble removal in the manufacture of panel material

A mathematical model is developed for the two-dimensional flow of an incompressible Newtonian fluid which is being squeezed between two rigid, impermeable plates. The fluid viscosity varies linearly with temperature. The model is intended to elucidate a problem of air bubble entrapment which arises in the manufacture of panel material, particularly in the manufacture of ‘corner’ profiles. The results show that when the two plates have the same shape, the pressure gradient is such that bubbles are likely to be expelled throughout the contact, except in the central region. When a corner profile is manufactured it is likely that bubbles will be retained, thereby leading to flaws in the final product.     

Performance of hydromagnetic squeeze films between conducting porous rough conical plates

An endeavor has been made to discuss the behavior of hydromagnetic squeeze film between two conducting rough porous conical plates. The plates are considered to be electrically conducting and the clearance space between them is filled by an electrically conducting lubricant. A transverse magnetic field is applied between the plates. Efforts have been made to solve the concerned Reynolds’ equation with the associated boundary conditions to get the pressure distribution. This in turn, is used to obtain the expression for load carrying capacity leading to the calculation of the response time. The results are presented graphically as well as in tabular form. It is suggested by the results that the bearing system records an enhanced performance as compared to that of a bearing system working with a conventional lubricant. It is noticed that the pressure, load carrying capacity and the response time increase steadily with increasing values of the magnetization parameter. In general, the bearing suffers owing to transverse surface roughness. However, the negatively skewed roughness tends to better the performance of the bearing system marginally. This performance gets further improved especially, when the negative variance is involved. It is observed that the semi-vertical angle increases the load carrying capacity. Besides, the conductivity also increases the load carrying capacity significantly. In addition, it is revealed that the negative effect induced by the porosity can be neutralized to a nominal extent by the positive effect of the magnetization parameter in the case of negatively skewed roughness in the presence of negative variance. Thus, this study provides ample scopes for improving the performance of the bearing system considerably by choosing a suitable combination of magnetization parameter, semi-vertical angle and the conductivities of the plates.     

Effect of magnetic field and velocity slip on porous-walled rectangular squeeze films

This is a study of an electrically conducting flow in a squeeze film between two infinite strips where one of the strips has a porous bounding surface backed by a solid wall. The analysis is directed to study the interaction of a transverse magnetic field with the coupled flows in the squeeze film and the porous medium including the slip velocity at the porous bounding surface. Expressions for load capacity and thickness-time are obtained. It is observed that the magnetic field increases the load capacity and response times of squeeze films. This effect is more marked for small values of the permeability K.     

Effect of slip on porous-walled squeeze films in the presence of a transverse magnetic field

The present investigation is devoted to study the effect of viscous resistance, arising due to sparse distribution of particles in porous media, on the load capacity and thickness time response of porous-walled squeeze films in the presence of a uniform magnetic field. The results of the analysis obtained by using Beavers and Joseph [1] slip-boundary condition show that the viscous resistance increases the load capacity and thickness time response of squeeze films when compared with the results of Chandrasekhara [2] obtained in the absence of viscous resistance. Hence, for efficient performance of a porous walled squeeze film a suitable porous media in which the material is loosely packed may be used.Nomenclature p pressure in the squeeze film - h thickness of the squeeze film at time t - h ₀ thickness of the film at t=0 - u streamwise velocity component in the squeeze film - v transverse velocity component in the squeeze film - P pressure in the porous material - H thickness of the porous material - U streamwise velocity component in the porous material - V transverse velocity component in the porous material - B B ₀+b - B ₀ impressed uniform magnetic field - b induced magnetic field - E electric field vector (E _x , E _y , E _z ) - m ₀ constant defined in (6), (B ₀ ² / _{m f m})^1/2 - v _h value of v at y=h - h/h ₀, the non-dimensional variable - _n eigen values - _f viscosity - _m magnetic permeability - density - _m magnetic diffusivity, 1/ _{m e} - dimensionless parameter, - _e electrical conductivity - q velocity vector (u, v) - L load capacity - I _n integral defined in (37) - M Hartmann number defined in (7), (m ₀ ² h ²)^1/2 - l length of the strips in x-direction - K permeability of the porous material - J current density vector (J _x , J _y , J _z ) - t time - G _n series coefficient appearing in equation (27)     

An experimental test of equations predicting the load bearing capacity of squeeze films of elastico-viscous liquid

A form of squeeze film apparatus was recently described in which the movement of one plate towards the other was simulated by the continuous volume generation of liquid over the plate area. The liquid exuded from a large number of holes in the lower plate surface and formed a “continous flow” version of squeeze film apparatus with no moving parts [1]. A later paper gave derivations of equations from which squeeze film load bearing capacity could be evaluated, taking into account viscous, inertial and normal stress effects in the liquid film [2].In order to find the total load in a squeeze film system, it was necessary to obtain the relationship between the first normal stress difference and shear rate for the liquid in use, using an experimental method. At high shear rates, the jet thrust method provided these data [3,4] and from them the load bearing capacity of squeeze films of hot, polymer-thickened oil were predicted [2].A more complete test of the method is possible with a highly elastic liquid because considerable load enhancement due to extra stress is present at moderate deformation rates in squeeze film systems [1,5,6,7]. Thus a 0.1 per cent aqueous polyacrylamide solution gives well-defined load enhancement and (quite independently) the jet thrust method gives the relationship between normal stress and shear rate from which predictions of load enhancement may be made. Furthermore, convergent nozzles may be used in the jet thrust apparatus [3] to measure the stress development in an elastic liquid which is being simulateneously sheared and stretched, a situation which more closely resembles the squeeze film case than that of steady shear.