Non-dimensional Thickness Research Articles

Quasi-steady Bingham plastic analysis of an electrorheological flow mode bypassdamper with piston bleed

We present an improved experimental validation of our nonlinear quasi-steadyelectrorheological (ER) and magnetorheological damper analysis, using anidealized Bingham plastic shear flow mechanism, for the flow mode of damperoperation with leakage effect. To validate the model, a double-acting ER valve orbypass damper was designed and fabricated. Both the hydraulic cylinder and thebypass duct have cylindrical geometry, and damping forces are developed in theannular bypass via Poiseuille flow. The ER fluid damper contains a controlledamount of leakage around the piston head. The leakage allows ER fluid to flowfrom one side of the piston head to the opposite side without passing through theER bypass. For this flow mode damper, the damping coefficient, defined asthe ratio of equivalent viscous damping of the Bingham plastic material,Ceq, to the Newtonianviscous damping, C,is a function of the non-dimensional plug thickness only. The damper was testedfor varying conditions of applied electric field and frequency using a mechanicaldamper dynamometer. In this analysis, the leakage damping coefficient withincorporated leakage effects, predict the amount of energy dissipated for acomplete cycle of the piston rod. Measured force verses displacement cycles formultiple frequencies and electric fields validate the ability of the non-dimensionalgroups and the leakage damping coefficient to predict the damping levels for anER bypass damper with leakage. Based on the experimental validation of themodel using these data, the Bingham plastic analysis is shown to be an effectivetool for the analysis-based design of double-acting ER bypass dampers.

Design of a High-Efficiency Magnetorheological Valve

A high efficiency design was explored for meso-scale magnetorheological (MR) valves (< 25 mm OD with an annular gap < 1 mm). The objective of this paper is to miniaturize the MR valve while maintaining the maximum performance of the MR effect in the valve. The main design issues in the MR valve involve the magnetic circuit and nonlinear fluid mechanics. The performance of the MR valve is limited by saturation phenomenon in the magnetic circuit and by the finite yield stress of the MR fluid. When field is applied to the magnetic circuit in the MR valve, a semisolid plug (as a result of particle chain formation) forms perpendicular to the flow direction through the valve, and a finite yield stress is developed as a function of field. The resulting plug thickness is used to control flow rate through, and pressure drop across, the MR valve. The nondimensional plug thickness is evaluated as a basis for evaluating valve efficiency. Design parameters of the MR valve are studied and an optimal performance was designed using steel (Permalloy) material in the magnetic circuit. A maximum magnetic flux density at the gap was achieved in the optimized valve design based on a constraint on the outer diameter limitation. Valve performance was verified with simulation. A flow mode bypass damper system was fabricated and was used to experimentally validate valve performance.

Film Thickness and Wave Velocity Measurements in a Vertical Duct

This paper describes the experimental investigation of an upward annular air-water flow in a duct with a 6.35 mm by 63.5 mm rectangular cross section. The test section was instrumented to measure the film thickness and the interfacial wave velocity. Flush-wire electrical conductivity probes were used to obtain local film thickness measurement with a spatial resolution of 200 μm or better and a temporal resolution greater than 2 kHz. Measurements of the base films range from 50 μm to ∼325 μm (2% to 10% of half-channel thickness). Statistical analysis shows that the standard deviation of the film thickness is a good measure of the film roughness. The relative roughness and the nondimensional film thickness are correlated as functions of the phasic Reynolds number ratio, R=Rel0.15/Reg0.3. It is found that at R=0.15, the relative roughness is a maximum. A simple model developed by matching the interfacial shear in the two fluids, predicts the wave velocity data very well.

Unsteady Measurement of Condensate Film Thickness for Solutal Marangoni Condensation by Using Leser Absorption Method. Measurement and Discussion.

The objective of the study is to investigate the condensate film thickness for water-ethanol solutal Marangoni dropwise condensation. The laser light absorption method for measuring the unsteady behaviors of the condensate thickness was adopted. The relation between the condensate behavior and the measuring position was specified by the observation using the high-speed digital camera. The variations of the film thickness were measured by changing the surface subcooling for various vapor concentrations. The unsteady variations of the thickness of condensate film, which was appeared after the sweeping of the heat transfer surface by the departing drop and existed among the various drops, and also the initial drop distances in Marangoni dropwise condensation cycle were measured. The dependencies of the minimum film thickness and non-dimensional condensate thickness among the drops on the initial drop distance were shown. It was clarified that the thinnest thickness of the condensate film was around 1 μm. The relation among the dominant factors determining the phenomena was discussed.

Weakly nonlinear stability of viscous flow past a flexible surface

The weakly nonlinear stability of viscous fluid flow past a flexible surface is analysed in the limit of zero Reynolds number. The system consists of a Couette flow of a Newtonian fluid past a viscoelastic medium of non-dimensional thickness H (the ratio of wall thickness to the fluid thickness), and viscosity ratio μr (ratio of the viscosities of wall and fluid media). The wall medium is bounded by the fluid at one surface and two different types of boundary conditions are considered at the other surface of the wall medium – for ‘grafted’ gels zero displacement conditions are applied while for ‘adsorbed’ gels the displacement normal to the surface is zero but the surface is permitted to move in the lateral direction. The linear stability analysis reveals that for grafted gels the most unstable modes have α ∼ O(1), while for adsorbed gels the most unstable modes have α → 0, where α is the wavenumber of the perturbations. The results from the weakly nonlinear analysis indicate that the nature of the bifurcation at the linear instability is qualitatively very different for grafted and absorbed gels. The bifurcation is always subcritical for the case of flow past grafted gels. It is found, however, that relatively weak but finite-amplitude disturbances do not significantly reduce the critical velocity required to destabilize the flow from the critical velocity predicted by the linear stability theory. For the case of adsorbed gels, it is found that a supercritical equilibrium state could exist in the limit of small wavenumber for a wide range of parameters μr and H, while the bifurcation becomes subcritical at larger values of the wavenumber and there is a transition from supercritical to subcritical bifurcation as the wavenumber is increased.

Head-on Quenching of a Premixed Flame on the Single Wall Surface.

The head-on quenching of premixed flames on the single wall surface has been analyzed based on thermal theory and as results, the flame temperature, nondimensional flame thickness and a Peclet number, PQe, have been obtained at the time of quenching. Then, to confirm the adequacy of the analytical results, experiments using a combustion vessel were performed, and photographs were taken at the moment of head-on quenching of methane/air flames on a single wall surface using the Schlieren method, to measure the flame thickness at the time of quenching. As a result, the following were confirmed: regarding the nondimensional flame thickness at the time of quenching, although quantitative differences are observed between experimental values and theoretical values, it is always larger than 1 under all experimental conditions; PQe obtained from the experiments agrees qualitatively with that determined by theoretical analysis.

Natural convection in enclosures with localized heating from below and symmetrical cooling from sides

Natural convection of air in a two‐dimensional, rectangular enclosure with localized heating from below and symmetrical cooling from the sides has been numerically investigated. Localized heating is simulated by a centrally located heat source on the bottom wall, and four different values of the dimensionless heat source length, 1/5, 2/5, 3/5 and 4/5 are considered. Solutions are obtained for Rayleigh number values from 103 to 106. Local results are presented in the form of streamline and isotherm plots as well as the variation of local Nusselt number on the heated wall. Finally, the average Nusselt number at the heated part of the lower wall, \overline Nu, was shown to increase with an increase the Rayleigh number, Ra, or of the nondimensional heat source thickness, ε.

On the shape and velocity of fluid-filled fractures in the Earth

Summary In this paper I present a model to estimate the shape and velocity of slowly ascending buoyancy-driven fluid-filled fractures. The model considers elastic deformation, linear fracture mechanics and fluid flow. An attempt is made to incorporate the effect ofthe 2-D fluid pattern on the viscous pressure drop. Most other models assume that the viscous pressure drop can be approximated by flow through a channel with constant width, although the form of the fracture is known to deviate from such a simple shape. The 2-D flow in my model has an important consequence for the mechanism of buoyancy-driven fracture propagation—it predicts a large pressure gradient at the tail of the propagating fracture, indicating that the tail of the fracture is most important in hindering the fracture propagation. A singularity at the tail of the fracture can be avoided when a small amount of fluid trails in the channel left behind the propagating fracture. The trailing and decoupling of fluids at the tail seems to be accompanied by small flow and shape instabilities, which is indicated by the jerky movement of the tail observed in propagating air-filled fractures in solidified gelatine, and by numerical boundary element solutions of the coupled flow-deformation–fracturing problem. By comparing the predictions for propagation velocities with laboratory observations of buoyancy-driven fracture propagation in gelatine, I derive a non-dimensional effective thickness at the tail of the fracture for which trailing of fluids may occur. The model is applied to Earth-relevant problems such as oil- and water-filled fractures in pressurized sediments or magma-filled dykes in the lithosphere, which are discussed in the paper.

Transverse wave motion of piezoelectric (622) layer due to step potential

The propagation of transverse waves in a piezoelectric layer of class (622) is investigated, when excited by either (a) step electric potential, keeping zero stress or (b) step shear stress, keeping zero potential. The resulting electric potential or shear stress is presented graphically. For case (a), electric potential linearly increases as the thickness of the plate increases, whereas stress is maximum near nondimensional thickness r=0.1 and decreases gradually and becomes zero at the top surface. For case (b), electric potential becomes negative and linearly decreases up to nondimensional thickness, r=0.87 and then increases and becomes zero at the top surface with slight oscillatory nature as the thickness of the plate increases. It is also observed that mere application of time dependent electric potential in piezoelectric (622) class generates transverse waves, which is not possible in any other piezoelectric material.

Experimental free coating flows at high capillary and Reynolds number

Experimental studies are carried out to enhance the fundamental understanding of coating processes over a broad parametric range. Experiments herein identify the phenomena leading to the formation of an asymptotic meniscus profile, which eventually develops a cusp at the interface. The non-dimensional parameters that describe these phenomena are identified. In addition, flow visualization is carried out to reveal the entire flow structure using a visible laser. Two phenomena of free coating are identified depending on a parameter called the property number(Po). When Po is larger than about 0.5, the non-dimensional final film thickness (T 0) becomes constant beyond the capillary number(Ca) of about unity. When Po is less than about 0.1, T 0 depends on Ca and the Reynolds number(Re) but it becomes constant beyond the Weber number(=Ca Re) of about 0.2. In both cases T 0 becomes constant as the effect of surface tension on the meniscus becomes relatively unimportant. The cusp formation is due to the effect of inertia(Re). The effect of applicator dimensions on T 0 is also investigated for large Re flows.

Analysis and Testing of Electrorheological Bypass Dampers

We experimentally validate nonlinear quasi-steady electrorheological (ER) and magnetorheological (MR) damper models, using an idealized Bingham plastic shear flow mechanism, for the flow mode of damper operation. An electrorheological valve or bypass damper was designed, and fabricated using predominantly commercial off-the-shelf hydraulic components. Both the hydraulic cylinder and the bypass duct have cylindrical geometry, and damping forces are developed in the annular bypass via Poiseuille (flow mode) flow. Damper models assume parallel plate geometry. Three nondimensional groups are used for damper analysis, namely, the Bingham number, Bi, the nondimensional plug thickness, 6, and the area coefficient defined as the ratio of the piston head area, AP, to the cross-sectional area of the annular bypass, Ad. In the flow mode case, the damping coefficient, which is defined as the ratio of equivalent viscous damping of the Bingham plastic material, Ceq, to the Newtonian viscous damping, C, is a function only of the nondimensional plug thickness. The damper was tested using a mechanical damper dynamometer for sinusoidal stroke of 2 in., over a range of frequencies below 0.63 Hz. The damping coefficient vs. nondimensional plug thickness diagram was experimentally validated using these data over a range of damper shaft velocities or frequencies and applied electric fields. Because the behavior of ER and MR fluids are qualitatively similar, these ER damper modeling results can be extended to analysis of flow mode MR dampers.

Analysis and Testing of Electrorheological Bypass Dampers

We experimentally validate nonlinear quasi-steady electrorheological (ER) and magnetorheological (MR) damper models, using an idealized Bingham plastic shear flow mechanism, for the flow mode of damper operation. An electrorheological valve or bypass damper was designed, and fabricated using predominantly commercial off-the-shelf hydraulic components. Both the hydraulic cylinder and the bypass duct have cylindrical geometry, and damping forces are developed in the annular bypass via Poiseuille (flow mode) flow. Damper models assume parallel plate geometry. Three nondimensional groups are used for damper analysis, namely, the Bingham number, Bi, the nondimensional plug thickness, 6, and the area coefficient defined as the ratio of the piston head area, AP, to the cross-sectional area of the annular bypass, Ad. In the flow mode case, the damping coefficient, which is defined as the ratio of equivalent viscous damping of the Bingham plastic material, Ceq, to the Newtonian viscous damping, C, is a function only of the nondimensional plug thickness. The damper was tested using a mechanical damper dynamometer for sinusoidal stroke of 2 in., over a range of frequencies below 0.63 Hz. The damping coefficient vs. nondimensional plug thickness diagram was experimentally validated using these data over a range of damper shaft velocities or frequencies and applied electric fields. Because the behavior of ER and MR fluids are qualitatively similar, these ER damper modeling results can be extended to analysis of flow mode MR dampers.

Nondimensional analysis of semi-active electrorheological and magnetorheological dampers using approximate parallel plate models

We develop nonlinear quasi-steady electrorheological (ER) and magnetorheological (MR) damper models using an idealized Bingham plastic shear flow mechanism. Dampers with cylindrical geometry are investigated, where damping forces are developed in an annular bypass via Couette (shear mode), Poiseuille (flow mode) flow, or combined Couette and Poiseiulle flow (mixed mode). Models are based on parallel plate or rectangular duct geometry, and are compared to our prior 1D axisymmetric models. Three nondimensional groups are introduced for damper analysis, namely, the Bingham number, , the nondimensional plug thickness, , and the area coefficient defined as the ratio of the piston head area, , to the cross-sectional area of the annular bypass, . The approximate parallel plate analysis compares well with the 1D axisymmetric analysis when the Bingham number is small, or , or the nondimensional plug thickness is small, . Damper performance is characterized in terms of the damping coefficient, which is the ratio of the equivalent viscous damping constant, , to the Newtonian viscous damping constant, C. In shear mode, the damping coefficient is a linear function of the Bingham number. In flow mode, the damping coefficient is a function of the nondimensional plug thickness only. For the mixed mode damper, the damping coefficient reduces to that for the flow mode case when the area coefficient is large. The quasi-steady damping coefficient versus nondimensional plug thickness diagram is experimentally validated using measured 10 Hz hysteresis cycles for a electrorheological mixed mode damper.

The Free Vibration of a Cylindrical Shell on an Elastic Foundation

The frequency equation for a thin circular cylindrical shell resting on an elastic foundation is developed by using the first order shell theory of Sanders and eigenfrequencies are calculated. These eigenfrequencies are plotted against the axial wave parameter. Effects of the axial wave parameter, circumferential wave number, non-dimensional thickness and foundation parameters on eigenfrequencies are investigated. It is found that the foundation modulus chiefly affects the radial mode eigenfrequency and has no effect on torsional and longitudinal modes. On the otherhand, shear modulus does have influence on radial as well as tangential modes of vibrations. Though the effect on radial mode frequency is more pronounced.

A THREE-DIMENSIONAL ANALYSIS OF THE SPHEROIDAL AND TOROIDAL ELASTIC VIBRATIONS OF THICK-WALLED SPHERICAL BODIES OF REVOLUTION

This paper addresses the spheroidal (i.e. coupled bending-stretching) and toroidal (i.e. torsional or equivoluminal) elastic vibrations of thick-walled, spherical bodies of revolution by means of the three-dimensional theory of elasticity in curvilinear (spherical) co-ordinates. Stationary values of the dynamical energies of the spherical body are obtained by the Ritz method using a complete set of algebraic-trigonometric polynomials to approximate the radial, meridional, and circumferential displacements. Extensive convergence studies of non-dimensional frequencies are presented for the spheroidal and toroidal modes of thin-walled spherical bodies of revolution. Results include all possible 3-D modes, i.e. radial stretching, combined bending-stretching, pure torsion, and shear deformable flexure through the wall thickness (including thickness-shear, thickness-stretch, and thickness-twist). It is shown that the assumed displacement polynomials yield a strictly upper-bound convergence to exact solutions of the title problem, as a sufficient number of terms is retained. Since the effects of transverse shear and rotary inertia are inherent to the present 3-D formulation, an examination is made of the variation of non-dimensional frequencies with non-dimensional wall thickness, h/R ranging from thin-walled (h/R=0⋅05) to thick-walled (h/R=0⋅5) spherical bodies. The findings confirm that the variation of the spheroidal frequencies increases with increasing h/R and mode number, whereas the variation of the toroidal frequencies decreases with increasing h/R and mode number. This work offers some accurate 3-D reference data for the title problem with which refined solutions drawn from thin and thick shell theories and sophisticated finite element techniques may be compared. © 1997 by John Wiley & Sons, Ltd.

The impact of a single drop on a wetted solid surface

The impact of single drops on a thin liquid film was studied to understand the mechanism of secondary atomisation of sprays colliding on a wetted, cold, solid surface. To span a wide range of conditions various mixtures of water and glycerol were used. The use of Weber number, Ohnesorge number and non-dimensional film thickness to describe the peculiarities of the phenomenon allowed to carry out the experiments under appropriate similarity conditions. The impact of millimetric drops was analysed in detail by photographic means, using both still photography to study impact morphology, and laser sheet visualisation to investigate secondary droplet formation. Two mechanisms of splash were identified, depending essentially on the liquid viscosity (Ohnesorge number), a parameter which appears to play an important role also in defining the splash morphology. A photographic documentation is annexed. The characteristic times of the crown formation, the non-linear evolution of cusps (jet formation) and the surface roughness influence are further discussed. The experimental results allow to propose an empirical correlation for the splashing/deposition limit, for a wide range of conditions, and a comparison to available previous works is presented. The influence of the film thickness and liquid viscosity on the splash is confirmed and quantified.

Finite element analysis of conjugate natural convection in a square enclosure with a conducting vertical wall

The scope of the finite element method in solving conjugate conduction with natural convection in enclosure has been investigated. For this purpose, the set of governing equations have been reframed and put into a form that can easily fit into the Galerkin formulation. Easy accommodation of the material properties and global solution of the same set of conservation equations constitute the salient features of this formulation. Conduction in the solid zone is simulated by assigning a high value of dimensionless viscosity in this zone. With this formulation, natural convection is studied in a square enclosure, with a conducting vertical wall. In addition to highlighting the operational aspects of the scheme, a detailed investigation has been carried out on the associated flow structure and heat transfer. Results are presented for Rayleigh number in the range of 10 3 to 10 6. The effects of conduction in the wall is studied for three different values of the non-dimensional wall thickness (0.1, 0.2 and 0.4) with a wide range of dimensionless conductivity of the wall.

Roughness Effects on the Dynamic Coefficients of Ultra-Thin Gas Film in Magnetic Recording

The stiffness and damping coefficients of a slider gas bearing operating under arbitrary Knudsen number are calculated. A perfect gas is used as the lubricant, and its behavior is described by the modified average Reynolds equation proposed by Makino et al. (1993). The effects of molecular mean free-path on the roughness-induced flow factors are included. The effects of the nondimensional film thickness, Hs0, the surface characteristics, (γ1, γ2). the inverse Knudsen number, D0, the nondimensional frequency, Ω, and the modified bearing number, Λb, on the dynamic coefficients are discussed in this paper. As expected, the values of dynamic coefficients for various roughness orientations approach the smooth value as the ratio, Hs0, becomes greater and greater, and thus the roughness effect is getting smaller and smaller. The air film of two-sided longitudinal oriented roughness is stiffer than the other two-sided oriented cases. The effect on translational damping coefficients for various two-sided oriented roughnesses is reversed as D0 greater than some value, and this value is affected significantly by Λb. Transversely rough stationary case has the lowest critical value of Λb, at which the negative translational damping appears. The results show that the roughness effect and rarefaction effect on dynamic coefficients are significant, and they cannot be ignored for stability analysis.

A quasi-three-dimensional crack problem of an elastic plate subjected to in-plane shear loading

A quasi-three-dimensional analysis for a through crack in an elastic plate subjected to in-plane shear loading is conducted. The analysis shows that the quasi-three-dimensional crack tip stress fields are composed of a plane strain Mode II singular field and an antiplane Mode III singular field. The Mode III problem is coupled with the Mode II one. The antiplane stress near the crack tip is significant within the scale of one plate thickness. The Mode III stress intensity factor tends to zero when the nondimensional thickness h a , where a is half the crack length and h is half the plate thickness, becomes vanishingly small and reaches a maximum value around h a = 0.1 . The antiplane shear effect may not be neglected for plates which are not very thin.

Investigation of the Failure of Heavily Loaded Journal Bearings

This paper gives details of a combined theoretical and experimental investigation of a plain journal bearing under heavily loaded conditions together with a metrological study of the bearing geometry. It was found that under high loading conditions a simplified analytical expression relating the Sommerfeld number to the non-dimensional minimum film thickness, using a hydrodynamic solution of the isoviscous form of the Reynolds equation, could be developed. An alternative theoretical solution based on elastohydrodynamic lubrication was also considered. In addition, experimental work determined a variety of operating conditions that produced metal-to-metal contact. These operating conditions were then compared with the theoretical minimum film thickness calculations and bearing manufacturing data. This process was used to determine combined failure criteria based on operating conditions and machining capability.