Combined effect of viscosity variation and non-Newtonian Rabinowitsch fluid in wide parallel rectangular-porous plate with squeeze-film characteristics

The present article carries out the study of viscosity variation of non-Newtonian fluid with the homogeneous porous wall on wide parallel rectangular-plate based on the Rabinowitsch fluid model. The non-linear modified Reynolds equation is derived for the lubrication of rectangular squeeze film bearing with viscosity variation and porous parameter. Using the Morgan–Cameron approximation, the nonlinear Reynolds-type equation for squeeze-film which governs the film pressure is solved within the fundamentals of small perturbation technique. The characteristic of the wide parallel rectangular-porous plate is numerically computed for different physical quantities such as film pressure, load carrying capacity and response time. Moreover, as limiting cases some of the results from the available literature are recovered also. Further, the findings reveal that the viscosity variation of non-Newtonian fluid and the presence of porous wall lead to reduction in the load capacity and the response time respectively. Here, the porous matrix consists of a system of capillaries of very small radii with the homogeneous porous wall. The impact of porosity is incorporated as a result it acts as self-lubrication on bearing surface. Also, the effect of viscosity variation is one of the most important characteristics of fluid which helps in the design of bearings for lubrication in engineering and industrial applications.

     

Squeeze flow of a power-law fluid between non-parallel plates

Squeeze flow in the gap between non-parallel circular plates of radius R is discussed. The test material is assumed to be a power-law fluid, with a no-slip boundary condition at the plates. If the mean separation between the plates is h, and the angle of inclination between the plates is ? ? h/R, the force on the plates is perturbed only at O(?²) and is increased by less than 10% if ? < 0.35h/R. A torque O(?) tends to return the plates to a parallel configuration.     

Transient electro-osmotic and pressure driven flows of two-layer fluids through a slit microchannel

By method of the Laplace transform, this article presents semi-analytical solutions for transient electroosmotic and pressure-driven flows (EOF/PDF) of two-layer fluids between microparallel plates. The linearized Poisson-Boltzmann equation and the Cauchy momentum equation have been solved in this article. At the interface, the Maxwell stress is included as the boundary condition. By numerical computations of the inverse Laplace transform, the effects of dielectric constant ratio ε , density ratio ρ , pressure ratio p, viscosity ratio μ of layer II to layer I, interface zeta potential difference △ψ, interface charge density jump Q, the ratios of maximum electro-osmotic velocity to pressure velocity α , and the normalized pressure gradient B on transient velocity amplitude are presented.We find the velocity amplitude becomes large with the interface zeta potential difference and becomes small with the increase of the viscosity. The velocity will be large with the increases of dielectric constant ratio; the density ratio almost does not influence the EOF velocity. Larger interface charge density jump leads to a strong jump of velocity at the interface. Additionally, the effects of the thickness of fluid layers (h1 and h2 ) and pressure gradient on the velocity are also investigated.     

Chaotic thermal convection of couple-stress fluid layer

In this work we study the pattern of bifurcations and intermittent-chaos of non-Newtonian couple-stress shallow fluid layer subject to heating from below. The couple-stress parameter delays onset of convection, synchronizes chaotic behavior, and decreases the heat transfer . Some global aspects of the dynamics such as homoclinic bifurcations and transition to chaos are explored. The effects of particle size on the intermittent-chaos regime at particular normalized Rayleigh number, say \(r=166.1\), are investigated. With the increase in couple-stress parameter, the present Lorenz-like system synchronizes to a steady state via a series of periodic solutions interspersed with intervals of chaotic behaviors.     

Scattering of sound by two parallel semi-infinite screens

A plane sound wave is incident upon two semi-infinite rigid plates, lying along y = 0, x > 0 and y = -h, x < 0, respectively, where (x, y) are two-dimensional Cartesian coordinates. The problem is formulated into a matrix Wiener-Hopf equation which is uncoupled by the introduction of an infinite sum of poles. The exact solution is then easily obtained in terms of the coefficients of the poles, where these coefficients are shown to satisfy a linear system of algebraic equations. The far-field solution is obtained and an asymptotic approximation to the total potential is determined in the limit as h, the plate spacing, becomes small compared to the wavelength of the incident wave. The algebraic system is solved numerically in this limit and the results are shown to agree with those obtained by the method of matched asymptotic expansions.     

Exact solutions for the stability of viscoelastic rectangular plate subjected to tangential follower force

An exact solution procedure is formulated for the stability analysis of viscoelastic rectangular plate with two opposite edges simply supported and other two edges clamped as well as the viscoelastic rectangular plate with one edge clamped and other three edges simply supported under the action of tangential follower force. Firstly, by assuming the transverse displacement (W) as independent functions which automatically satisfies the simply supported boundary conditions, the governing partial differential equation is reduced to an ordinary differential equation with variable coefficients. Then, by the normalized power series method and applying the boundary conditions yield the eigenvalue problem of finding the roots of a fourth-order characteristic determinant. The results show that the aspect ratio λ and the dimensionless delay time H have great effects on the types of instability and the critical loads of the viscoelastic plates.     

An investigation of non-newtonian flow past a sphere

Any experimental work on the flow of a polymer solution or any theoretical analysis on the basis of a visoelastic constitutive equation does not always bring out viscoelastic effects but may be showing a non-Newtonian viscosity effect. Therefore, in order to obtain a clear understanding about viscoelastic effects, it is desirable to have a sufficient knowledge of the non-Newtonian viscosity effect. To facilitate this, finite-difference numerical solutions of non-Newtonian flow were carried out using a non-Newtonian viscous model for the Reynolds numbers of 0.1, 1.0, 20 and 60.Drag force measurements and flow visualization experiments were also performed over a wide range of experimental conditions using polymer solutions. The present work appears to support the following idea: When compared with the Newtonian case on the basis of DV_∞P/η₀, where η₀ is the zero shear viscosity, it is on account of the non-Newtonian viscosity that the friction and pressure drags decrease, that the separating vortices behind the sphere become larger, and that no shift occurs in the streamlines. On the other hand, it is due to viscoelasticity that the normal force drag increases, that the separating vortices behind the sphere become smaller, and that an upstream shift occurs in the streamlines.     

A thin-plate analysis and experimental evaluation of couple-stress effects

Couple-stress effects in materials have received much theoretical consideration recently but little or no experimental verification. The first part of this paper deals with a simplified elastic analysis for thin plates with couple stress which should provide a simple experimental approach to the determination of the order of magnitude of these effects. Classical concepts were utilized where applicable and many existing solutions can be modified by the present analysis to include couple stress. In the second part, experiments approximating cylindrical bending were carried out on plates of commercial-grade low-carbon steel and high-purity aluminum in the annealed and wrought conditions in order to establish the order of magnitude of the couple stress constantl as found in a simple couple-stress theory. Results were compared with classical theory and other experimental data correlated by the Neuber method to determine whether or not the couple-stress effect as predicted by a simple couple-stress theory could account for the reduction in the stress concentration found for small-radius notches and holes. It was found that the order of magnitude ofl was not greater than 0.002 in. rather than 0.02 in. as would be required to explain the above-mentioned reduction in stress concentration. Since the grain size of those materials was also ?0.002 in., it was concluded that:

1. (a)
   Couple-stress effects predicted by a simple theory were not significant enough to account for the reduction of stress concentration noted above for the material tested.     
   
   

The viscous squeeze-film behaviour of shear-thinning fluids

A finite element technique is proposed to predict the purely viscous squeeze-film behaviour of an arbitrary shear-thinning fluid confined between parallel discs and subjected to a constant load. The technique requires establishment of the distribution of viscosity in the gap. The variable viscosity is modelled by a discrete number of Newtonian fluids, with each fluid lying in a region bounded by lines of constant shear rate. Each of these Newtonian regions is further divided into regions which appear as “finite element” rectangles in the r-z plane. The equations governing squeeze-film flow are applied to this finite element network and an ordinary differential equation is ultimately derived which governs the gap decrease with time. Solving this equation is not simple because the coefficients of two terms change as the gap decreases. When the number of Newtonian fluids is sufficient, the technique predicts the squeeze-film time of a power-law fluid to within a fraction of a percent. Application of the technique to synovial fluid viscosity prevents the cartilage surfaces from touching for only a fraction of a second.     

Effect of lubricant inertia in bearings with a non-Newtonian lubricant

Summary The inertia effects in externally pressurized and squeeze film bearings with lubricants obeying a power law are considered. It is found that the inertia forces decrease the load capacity of the externally pressurized bearing with a given flow rate and the inertia effect increases with the flow behaviour index. At a given feeding pressure, on the other hand, the inertia increases or decreases the load capacity when the flow behaviour index is smaller than or greater than 3, respectively. For squeeze films between circular plates and rectangular plates, the rate of squeeze is slowed down by the inertia and the inertia effect is larger in dilatant lubricants than in pseudoplastic lubricants.Nomenclature 2a diameter of the bearing, width of rectangular plates - 2b diameter of the recess - 2h film thickness - 2h ₀ initial thickness of squeeze films - l length of the rectangular plates - m consistency index - n flow behaviour index - p pressure - p _e external pressure - p _i feeding pressure - q flow rate - r radial distance - t time - u velocity of the lubricant - v squeeze velocity - w load capacity - W dimensionless load capacity - axial distance - viscosity of the lubricant - density of the lubricant     

The size-dependent elastohydrodynamic lubrication contact of a coated half-plane with non-Newtonian fluid

Based on the couple-stress theory, the elastohydrodynamic lubrication(EHL)contact is analyzed with a consideration of the size effect. The lubricant between the contact surface of a homogeneous coated half-plane and a rigid punch is supposed to be the non-Newtonian fluid. The density and viscosity of the lubricant are dependent on fluid pressure. Distributions of film thickness, in-plane stress, and fluid pressure are calculated by solving the nonlinear fluid-solid coupled equations with an iterative method. The effects of the punch radius, size parameter, coating thickness, slide/roll ratio, entraining velocity, resultant normal load, and stiffness ratio on lubricant film thickness, in-plane stress, and fluid pressure are investigated. The results demonstrate that fluid pressure and film thickness are obviously dependent on the size parameter, stiffness ratio, and coating thickness.     

Instantaneous Squeeze-Film Force Between a Heat Exchanger Tube and a Support Plate for Arbitrary Tube Motion

The instantaneous squeeze-film force between a heat exchanger tube and a support plate is studied. Based on a two-dimensional rectangular plate model, a short-sleeve squeeze-film model for arbitrary tube motion is developed. The instantaneous squeeze-film force is expressed in normal and tangential directions. The normal squeeze-film force consists of four nonlinear terms, the viscous, unsteady inertia, convective inertia and centripetal inertia terms. Three nonlinear terms, the viscous, unsteady inertia and Coriolis inertia terms, make up the tangential squeeze-film force. An experimental apparatus was developed in order to evaluate the theoretical models against measurements of a finite length squeeze film. A modified model based on the experimental data is obtained where the viscous terms for both directions are multiplied by the instantaneous Reynolds number. All the inertia terms are multiplied by constant coefficients. The modified model is in good agreement with most experimental cases for unsymmetrical linear motion, approximate circular motion and elliptical motion. The form of the modified model is suitable for predicting instantaneous squeeze-film forces in the simulation of heat exchanger tube vibration. Further work using different sized components and fluid properties is required in order to finalize coefficient values.     

Small scale,grain size and substrate effects in nano-indentation experiment of film–substrate systems

The mechanical properties of film–substrate systems have been investigated through nano-indentation experiments in our former paper (Chen, S.H., Liu, L., Wang, T.C., 2005. Investigation of the mechanical properties of thin films by nano-indentation, considering the effects of thickness and different coating–substrate combinations. Surf. Coat. Technol., 191, 25–32), in which Al–Glass with three different film thicknesses are adopted and it is found that the relation between the hardness H and normalized indentation depth h/t, where t denotes the film thickness, exhibits three different regimes: (i) the hardness decreases obviously with increasing indentation depth; (ii) then, the hardness keeps an almost constant value in the range of 0.1–0.7 of the normalized indentation depth h/t; (iii) after that, the hardness increases with increasing indentation depth. In this paper, the indentation image is further investigated and finite element method is used to analyze the nano-indentation phenomena with both classical plasticity and strain gradient plasticity theories. Not only the case with an ideal sharp indenter tip but also that with a round one is considered in both theories. Finally, we find that the classical plasticity theory can not predict the experimental results, even considering the indenter tip curvature. However, the strain gradient plasticity theory can describe the experimental data very well not only at a shallow indentation depth but also at a deep depth. Strain gradient and substrate effects are proved to coexist in film–substrate nano-indentation experiments.     

Nonlinear flutter of viscoelastic rectangular plates and cylindrical panels of a composite with a concentrated masses

The problem of flutter of viscoelastic rectangular plates and cylindrical panels with concentrated masses is studied in a geometrically nonlinear formulation. In the equation of motion of the plate and panel, the effect of concentrated masses is accounted for using the δ-Dirac function. The problem is reduced to a system of nonlinear ordinary integrodifferential equations by using the Bubnov-Galerkin method. The resulting system with a weakly singular Koltunov-Rzhanitsyn kernel is solved by employing a numerical method based on quadrature formulas. The behavior of viscoelastic rectangular plates and cylindrical panels is studied and the critical flow velocities are determined for real composite materials over wide ranges of physicomechanical and geometrical parameters.     

Variational analysis of gradient elastic flexural plates under static loading

Gradient elastic flexural Kirchhoff plates under static loading are considered. Their governing equation of equilibrium in terms of their lateral deflection is a sixth order partial differential equation instead of the fourth order one for the classical case. A variational formulation of the problem is established with the aid of the principle of virtual work and used to determine all possible boundary conditions, classical and non-classical ones. Two circular gradient elastic plates, clamped or simply supported at their boundaries, are analyzed analytically and the gradient effect on their static response is assessed in detail. A rectangular gradient elastic plate, simply supported at its boundaries, is also analyzed analytically and its rationally obtained boundary conditions are compared with the heuristically obtained ones in a previous publication of the authors. Finally, a plate with two opposite sides clamped experiencing cylindrical bending is also analyzed and its response compared against that for the cases of micropolar and couple-stress elasticity theories.     

Squeeze flow of soft solids between rough surfaces

Newtonian liquids and non-Newtonian soft solids were squeezed between parallel glass plates by a constant force F applied at time t=0. The plate separation h(t) and the squeeze-rate were measured for different amplitudes of plate roughness in the range 0.3–31 m. Newtonian liquids obeyed the relation Vh ³ of Stephan (1874) for large plate separations. Departures from this relation that occurred when h approached the roughness amplitude were attributed to radial liquid permeation through the rough region. Most non-Newtonian materials showed boundary-slip that varied with roughness amplitude. Some showed slip that varied strongly during the squeezing process. Perfect slip (zero boundary shear stress) was not approached by any material, even when squeezed by optically-polished plates. If the plates had sufficient roughness amplitude (e.g. about 30 m), boundary slip was practically absent, and the dependence of V on h was close to that predicted by no-slip theory of a Herschel-Bulkley fluid in squeeze flow (Covey and Stanmore 1981, Adams et al. 1994).     

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)     

Stability of extensional flows and extending viscoelastic fluid sheets

The theory of nearly-extensional flow is developed to study the stability of extensional flow. For such flows a simple constitutive equation is derived for slightly disturbed extensional flow when a ‘short memory’ assumption is admissible.Following Minoshima and White and utilizing the constitutive equation obtained, a stability analysis for non-Newtonian fluid sheets is presented. The theoretical analysis presented is specific for an integral consitutive equation. The influence of the fluid elasticity on the stability behaviour is investigated. It is shown that the fluid sheet stability depends upon λk, where λ is the relaxation time and K is the elongation rate.     

A transition phenomenon of ice formation in water flow between two horizontal parallel plates

Experiments have been performed to investigate the icetransition profiles and heat-transfer characteristics for water flows between two horizontal parallel plates. The experiments are carried out under the condition that upper plate is cooled at uniform temperature kept less than freezing temperature of water, while the lower plate is heated at uniform temperature kept higher than the temperature of water flow. The temperatures of the upper and lower plates range from ?8 to ?14°C and from 10 to 60 °C, respectively, with inlet-water temperature varied from 1.5 to 4.5 °C. The cooling and heating temperature ratios, θ_c and θ_h, are ranging from 1.78 to 9.33 and from 1.22 to 39, respectively. By using three kinds of heightH of 16, 30 and 40 mm between the horizontal parallel plates, the Reynolds and Grashof numbers are varied from 3.2 × 10² to 1.5 × 10⁴ and from 3.4 × 10³ to 8.97 × 10⁶, respectively. As a result of this investigation two ice-transition modes are observed. The first ice-transition mode is due to an interruption of upper and lower thermal boundary layers, while the second mode is due to an instability of laminar boundary layer formed on water-ice interface. In order to determine the kind of ice-transition mode, criterion correlation formulas including the Reynolds numberRe _H , Grashof numberGr _H , and heating temperature ratio θ_h are determined and may be written as follows: For thermal icetransition mode (th.I.T.M.)Re _H /(Gr _H ·θ _h )^0.23<1.6×10^?3 and for hydrodynamical ice-transition mode (hy.I.T.M.)Re _H /(Gr _H ·θ _h )^0.23>2.3×10^?3 By introducing the freezing parameterB _f , correlation equations for local and mean Nusselt numbers along the water-ice interface at steady-state condition are determined. From the current experimental results it is found that the local Nusselt number may be described as the following equation:Nu _x =0.835 Re _H ^0.278 · B _f ^0.834 ·x/H)^?0.139