Incompressible Newtonian Liquid Research Articles

A generalized mathematical model for the damped free motion of a liquid column in a vertical U-tube

A one-dimensional mathematical model is developed to analyze the unsteady characteristics of arbitrary damped free motion, denoted as Z(τ), in a rigid, constant-diameter vertical column of a U-tube filled with an incompressible Newtonian liquid, where τ represents time. This model utilizes a simplified unsteady momentum equation derived from the Navier–Stokes equations in a circular coordinate system. Moreover, it incorporates assumptions about the periodicity of arbitrary damped free oscillations and employs the Fourier series representation to characterize the damped free motion. The combination of assumptions made for periodicity, the simplified momentum equation, and the Fourier series representation makes the current mathematical model unique and novel compared to prevailing models in the literature. In this model, the governing partial differential equation contains two dependent variables: Z(τ) is the known variable, as one can measure from experiments, and the instantaneous velocity uz is the unknown variable. Fitting the experimental data into the Fourier series provides the Fourier coefficients associated with the specific experiment. The Laplace transform method is used to determine the analytical solution for uz corresponding to the known Z(τ). The analytical expressions for instantaneous flow characteristics of practical importance, including area-averaged velocity, wall shear stress, and acceleration/deceleration, are deduced from uz. The analytical solutions presented are valid for generalized unsteady motions, including underdamped oscillations with varying amplitudes and periods, underdamped oscillations with varying amplitudes and constant period, and overdamped motion that does not exhibit a single oscillation. The findings from the present model offer insights for formulating a new friction model.

Propulsion of a three-sphere microrobot in a porous medium.

Microorganisms and synthetic microswimmers often encounter complex environments consisting of networks of obstacles embedded into viscous fluids. Such settings include biological media, such as mucus with filamentous networks, as well as environmental scenarios, including wet soil and aquifers. A fundamental question in studying their locomotion is how the impermeability of these porous media impacts their propulsion performance compared with the case of that in a purely viscous fluid. Previous studies showed that the additional resistance due to the embedded obstacles leads to an enhanced propulsion of different types of swimmers, including undulatory swimmers, helical swimmers, and squirmers. In this paper, we employ a canonical three-sphere swimmer model to probe the impact of propulsion in porous media. The Brinkman equationis utilized to model a sparse network of stationary obstacles embedded into an incompressible Newtonian liquid. We present both a far-field theory and numerical simulations to characterize the propulsion performance of the swimmer in such porous media. In contrast to enhanced propulsion observed in other swimmer models, our results reveal that both the propulsion speed and efficiency of the three-sphere swimmer are largely reduced by the impermeability of the porous medium. We attribute the substantial reduction in propulsion performance to the screened hydrodynamic interactions among the spheres due to the more rapid spatial decays of flows in Brinkman media. These results highlight how enhanced or hindered propulsion in porous media is largely dependent on individual propulsion mechanisms. The specific example and physical insights provided here may guide the design of synthetic microswimmers for effective locomotion in porous media in their potential biological and environmental applications.

Initial water imbibition of gas-saturated natural reservoir rock: A generalized multifactor geometry model with capillary bundles

The spontaneous imbibition in porous medium is of interest for many engineering problems particularly in the reservoir exploitation over several decades. So far, various analytical or numerical models have been proposed to simulate the imbibition, but none of them can fully take into account real microscopic geometry characteristics of pore structure. Therefore, assuming quasi-steady and completely developed laminar flow with no-slip boundary conditions when capillary bundles imbibe an incompressible Newtonian liquid, a generalized spontaneous imbibition model is proposed based on a fractal pore structure consisting of tortuous and nonuniform capillaries with different sizes and noncircular cross-sections. A theoretical analysis and experimental contrast demonstrate that the derived formulas enable the unification of several traditional and newly developed models available from the literature and have the better predicted trend with the relative error controlled within 100% in comparison to them. The obtained model is evaluated against previously reported data of water imbibition measured for various natural porous rocks in multiple directions. The validation results show that the generalized model can be utilized to characterize the imbibition behavior of porous rocks with little clay and even extend to other natural and engineering porous materials, including consolidated and unconsolidated media, and that the interconnected pores cannot always be assumed to be cylindrical or uniform. Moreover, the present model is applied to inversely estimate the pore size as a result of good agreement with experimental measurements. Simultaneously, in accordance with the new theoretical model, the difference in water imbibition is newly revealed for selected rock samples from the literature.

Electroviscous effect and electrokinetic energy conversion in time periodic pressure-driven flow through a parallel-plate nanochannel with surface charge-dependent slip

In this paper we analytically investigate the electroviscous effect and electrokinetic energy conversion in the time periodic pressure-driven flow of an incompressible viscous Newtonian liquid through a parallel-plate nanochannel with surface charge-dependent slip. Analytical and semi-analytical solutions for electric potential, velocity and streaming electric field are obtained and are utilized to compute electrokinetic energy conversion efficiency. The results show that velocity amplitude and energy conversion efficiency are reduced when the effect of surface charge on slip length is considered. The surface charge effect increases with zeta potential and ionic concentration. In addition, the energy conversion efficiency is large when the ratio of channel half-height to the electric double layer thickness is small. The boundary slip results in a large increase in energy conversion. Higher values of the frequency of pressure pulsation lead to higher values of the energy conversion efficiency. We also obtain the energy conversion efficiency in constant pressure-driven flow and find that the energy conversion efficiency in periodical pressure-driven flow becomes larger than that in constant pressure-driven flow when the frequency is large enough.

An investigation on the effects of the angles between the mitral and aortic orifice during diastolic period using FSI

BackgroundIt was well documented that the changes in the physiological structure of the heart can alter the hemodynamic behaviour of the human left ventricle. Although there were various imaging tools that can stipulate the blood flow characteristics inside the ventricle, but the effect of the variation in the angles between the mitral and aortic orifices were yet to be determined. Hence the primary focus of this paper was to use Fluid-Structure Interaction scheme throughout the diastolic phase to examine and determine the hemodynamic behaviour inside the cavity under various angles between the mitral and aortic orifices (50°, 55° and 60°). MethodsThe incompressible Newtonian liquid (time dependent), linear viscous liquid and stress tensor equations were combined together with the Navier-Stoke’s equations and Arbitrary Lagrangian Eulerian and for the elasticity of the structure. ResultsDuring diastasis, the position of the vortex was seen to be slightly upwards (55°) compared to 50° and 60°. Also, the influence of the wall shear stress was primarily observed to be much higher with the rise in the inlet velocity but the effect was seen reduced when the inlet velocity was minimal. Moreover, at the end of the filling phase maximum intraventricular pressure was obtained to be at the tip of LV for 50° compared to 55° and 60°. ConclusionThese findings provide significant insights on hemodynamic conditions and structural displacement, which from a clinical point of view would be useful in determining different conditions of cardiac diseases.

On the transient phase of the Faraday instability

This study pertains to the three-dimensional direct numerical simulation (DNS) of a vertically oscillating vessel containing an incompressible Newtonian liquid, surrounded by air at rest and ambient conditions. Squire's theorem was extended and shown to apply in this case, allowing for the theory of linear stability to be implemented and a comparison to be made with the DNS results. It was further discovered that the method by which a fluid instability is initiated in the numerical simulation affects the initial development of the instability. This phenomenon was confirmed through an optimal perturbations analysis. A possible physical explanation of this effect is also presented.

Modeling of inflation of liquid spherical layers under zero gravity

A mathematical model of the inflation of a spherical layer (shell) made of heat-conducting incompressible Newtonian liquid has been developed and examined. The model presents a set of differential equations, the analytical solution of which has been found in the form of the dependences of the velocity of liquid flow and the pressure in the liquid layer on the inner radius of the shell. Results of simulation of the shell inflation process under different conditions of gas pumping are presented and discussed.

Enhanced low-Reynolds-number propulsion in heterogeneous viscous environments

It has been known for some time that some microorganisms can swim faster in high-viscosity gel-forming polymer solutions. These gel-like media come to mimic highly viscous heterogeneous environment that these microorganisms encounter in-vivo. The qualitative explanation of this phenomena first offered by Berg and Turner [Nature (London) 278, 349 (1979)], suggests that propulsion enhancement is a result of flagellum pushing on quasi-rigid loose polymer network formed in some polymer solutions. Inspired by these observations, inertia-less propulsion in a heterogeneous viscous medium composed of sparse array of stationary obstacles embedded into a incompressible Newtonian liquid is considered. It is demonstrated that for prescribed propulsion gaits, including propagating surface distortions and rotating helical filament, the propulsion speed is enhanced when compared to swimming in purely viscous solvent. It is also shown that the locomotion in heterogenous viscous media is characterized by improved hydrodynamic efficiency. The results of the rigorous numerical simulation of the rotating helical filament propelled through a random sparse array of stationary obstructions are in close agreement with predictions of the proposed resistive force theory based on effective media approximation.

A study of transient flows of Newtonian fluids through micro-annuals with a slip boundary

In this paper, we study the pressure gradient driven transient flow of an incompressible Newtonian liquid in micro-annuals under a Navier slip boundary condition. By using the Fourier series expansion in time and Bessel functions in space, an exact solution is derived and is shown to include some existing known results as special cases. By analysing the exact solution, it is found that the influences of boundary slip on the flow behaviour are qualitatively different for different types of pressure fields driving the flow. For pressure fields with a constant pressure gradient, the flow rate increases with the increase in the slip parameter l almost linearly when l is large; while, for pressure fields with a wave form pressure gradient within a certain frequency range, as the slip parameter l increases, the amplitude of the flow rate increases first and then approaches a constant value when l becomes sufficiently large. It is also found that to achieve a given flow rate, one could have different designs and the graphs for the design are presented and discussed in this paper.

An integrated simulation for the whole forming process of the picture tube panel

An integrated model for the whole forming operation of the picture tube panel is developed in this paper, which couples the behaviors of glass and mold. The molten glass is modeled by an incompressible Newtonian liquid undergoing flow. And a three-dimensional finite element method is used to perform the simulation of the fluid and heat flow. A local one-dimensional transient analysis in the thickness direction is adopted for the part cooling stage after pressing, which employs the finite-difference method. The mold heat transfer is established by boundary conditions analysis and its numerical implementation is a three-dimensional boundary element method. The glass and mold simulations are coupled by matching the temperature and heat flux on the glass-mold interface. For residual stresses analysis, a thermo-rheologically simple viscoelastic material model is introduced to consider the stresses relaxation effect and to describe the mechanical behavior according to the temperature change. The shrinkage of formed parts induced by the residual stresses is calculated based on the theory of shells, represented as an assembly of flat elements formed by combining the constant strain and the discrete Kirchhoff triangular elements. A thermoelastic model is presented to predict the deformation of the mold blocks during pressing, which is based on the steady mold temperature field and thermoelastic boundary element method.

Breakup of electrified jets

Breakup of electrified jets is important in applications as diverse as electrospraying, electroseparations and electrospray mass spectrometry. Breakup of a perfectly conducting, incompressible Newtonian liquid jet surrounded by a passive insulating gas that is stressed by a radial electric field is studied by a temporal analysis. An initially quiescent jet is subjected to axially periodic shape perturbations and the ensuing dynamics are followed numerically until pinch-off by both a three-dimensional but axisymmetric (two-dimensional) and a one-dimensional slender-jet algorithm. Results computed with these algorithms are verified against predictions from linear theory for short times. Breakup times, ratios of the sizes of the primary to satellite drops formed at pinch-off, and the Coulombic stability of these drops are reported over a wide range of electrical Bond numbers, NE (ratio of electric to surface tension force), Ohnesorge numbers, NOh (ratio of viscous to surface tension force), and disturbance wavenumbers, k. Effect of surface charge on interface overturning is investigated. Furthermore, the influence of electrostatic stresses on the dynamics of pinch-off and the mechanisms of satellite drop formation is also addressed.

Integrated simulation of the glass and mould in the TV panel pressing process

An integrated model of the behaviour of molten glass and mould during the pressing operation of TV panel is developed in this paper. The molten glass is modelled by an incompressible Newtonian liquid undergoing flow. And a 3D finite element method is used to perform the simulation of the fluid and heat flow, which employs an equal-order velocity-pressure interpolation method. Mould cooling system design is important and significantly affects productivity and the quality of the final part. The mould heat transfer during the TV panel pressing process is established by boundary conditions analysis. Numerical implementation is a 3D boundary element method. These two simulations are coupled by matching the temperature and heat flux at the glass-mould interface. As compared with the experimental data, the presented model and developed system are sufficiently accurate.

Three-Dimensional Numerical Simulation of the Pressing Process in TV Panel Production

A mathematical model of the behavior of molten glass during the pressing operation of a TV panel is developed in this article. The molten glass is modeled by an incompressible Newtonian liquid undergoing flow. Then a three-dimensional finite element method is used to perform the simulations of the fluid and heat flow. The simulation employs an equal-order velocity-pressure interpolation method. The relation between velocity and pressure is obtained from the discretized momentum equations in order to derive the pressure equation. The glass simulation is coupled with the mold temperature simulation by matching the temperature and heat flux on the glass-mold interface.

Simplicity and complexity in a dripping faucet

Continuous emission of drops of an incompressible Newtonian liquid from a tube–dripping–is a much studied problem because it is important in applications as diverse as inkjet printing, microarraying, and microencapsulation, and recognized as the prototypical nonlinear dynamical system, viz., the leaky faucet. The faucet’s dynamics are studied in this paper by a combination of experiment, using high-speed imaging, and computation, in which the one-dimensional slender-jet equations are solved numerically by finite element analysis, over ranges of the governing parameters that have heretofore been unexplored. Previous studies when the Bond number G that measures the relative importance of gravitational to surface tension force is moderate, G≈0.5, and the Ohnesorge number Oh that measures the relative importance of viscous to surface tension force is low, Oh≈0.1, have shown that the dynamics changes from (a) simple dripping, i.e., period-1 dripping with or without satellites, to (b) complex dripping, where the system exhibits period doubling bifurcations and hysteresis, to (c) jetting, as the Weber number We that measures the relative importance of inertial to surface tension force increases. New experiments and computations reveal that lowering the Bond number to G≈0.3 while holding Oh fixed results in profound simplification of the behavior of the faucet. At the lower value of G, the faucet exhibits simply period-1 dripping, period-2 dripping, and jetting as We increases. Experimental and computational bifurcation diagrams when G≈0.3 and Oh≈0.1 that depict the variation of drop length or volume at breakup with We are reported and shown to agree well with each other. The range of We over which the faucet exhibits complex dripping when G≈0.3 is shown by both experiment and computation to shrink as Oh increases. Computations are also used to develop a comprehensive phase diagram when G≈0.3 that shows transitions between simple dripping and complex dripping, and those between dripping and jetting in (We,Oh) space. Similar to the case of G≈0.5, dripping faucets of high viscosity (Oh) liquids are shown to transition directly from simple dripping to jetting without exhibiting complex dripping when G≈0.3. When G≈0.3, computed values of We that signal transition from dripping to jetting are further shown to accord well with estimates obtained from scaling analyses. By contrast, new computations in which the Bond number is increased to G≈1, while Oh is held fixed at Oh≈0.1, reveal that the faucet’s response becomes quite complex for large G. In such situations, the computations predict theoretical occurrence of (a) rare period-3 dripping and period-3 intermittence, which have previously been surmised solely by the use of ad hoc spring-mass models of dripping, and (b) chaotic attractors. Therefore, by combining insights from earlier studies and the detailed response of dripping which has been obtained here by varying (i) Oh as 0.01⩽Oh⩽2, a range that is typical of most practical applications, (ii) We from virtually zero to a value just exceeding that at which the system transitions from dripping to jetting, and (iii) G from a small value to a value approaching that beyond which controlled formation of drops is prohibited, this paper provides a comprehensive understanding of the effect of the governing parameters on the nonlinear dynamics of dripping.

Dynamics and breakup of a contracting liquid filament

Contraction of a filament of an incompressible Newtonian liquid in a passive ambient fluid is studied computationally to provide insights into the dynamics of satellite drops created during drop formation. This free boundary problem, which is composed of the Navier–Stokes system and the associated initial and boundary conditions that govern the evolution in time of the filament shape and the velocity and pressure fields within it, is solved by the method of lines incorporating the finite element method for spatial discretization. The finite element algorithm developed here utilizes an adaptive elliptic mesh generation technique that is capable of tracking the dynamics of the filament up to the incipience of pinch-off without the use of remeshing. The correctness of the algorithm is verified by demonstrating that its predictions accord with (a) previously published results of Basaran (1992) on the analysis of finite-amplitude oscillations of viscous drops, (b) simulations of the dynamics of contracting filaments carried out with the well-benchmarked algorithm of Wilkes et al. (1999), and (c) scaling laws governing interface rupture and transitions that can occur from one scaling law to another as pinch-off is approached. In dimensionless form, just two parameters govern the problem: the dimensionless half-length which measures the relative importance of viscous force to capillary force. Regions of the parameter space are identified where filaments (a) contract to a sphere without breaking into multiple droplets, (b) break via the so-called endpinching mechanism where daughter drops pinch-off from the ends of the main filament, and (c) break after undergoing a series of complex oscillations. Predictions made with the new algorithm are also compared to those made with a model based on the slender-jet approximation. A region of the parameter space is found where the slender-jet approximation fares poorly, and its cause is elucidated by examination of the vorticity dynamics and flow fields within contracting filaments.

Two-dimensional adiabatic Newtonian flow with temperature-dependent viscosity

We study the analytical and numerical stability of the two-dimensional adiabatic flow of an incompressible Newtonian liquid with temperature-dependent viscosity. We show that the uniform shearing is the asymptotic solution if the time becomes very large in a rate dependent on the temperature sensitivity and the referencial Eckert and Reynolds numbers. Numerical solutions for flows between parallel plates caused by steady boundary velocity are presented. The results indicate that the boundary velocity controls the thermomechanical process in complete agreement with the analytical behaviour of the flow.

A finite element model of interaction between viscous free surface waves and submerged cylinders

This paper describes the simulation of the flow of a viscous incompressible Newtonian liquid with a free surface. The Navier–Stokes equations are formulated using a streamline upwind Petrov–Galerkin scheme, and solved on a Q-tree-based finite element mesh that adapts to the moving free surface of the liquid. Special attention is given to fitting the mesh correctly to the free surface and solid wall boundaries. Fully non-linear free surface boundary conditions are implemented. Test cases include sloshing free surface motions in a rectangular tank and progressive waves over submerged cylinders.

Free Surface Flow and Film Thickness Induced by Blade Coating.

Viscous free surface flows with a blade of various inclination angles are studied using Finite Difference Method. Consideration is restricted to the steady two-dimensional Stokes flow of an incompressible Newtonian liquid carried out of a narrow gap by a moving plate. The flow upstream is represented by Taylor-Hamel solution and the flow downstream by Coyne-Elrod solution, whose unknown constant k is determined in this study. And so, the flow downstream except a small part in the vicinity of blade edge can be described analytically. Free surface meniscus, velocity profile and so on are calculated numerically. Investigation shows that the effect of blade angle on free surface flow is diminishingly small. To this peculiar result, the physical explanation is given by a qualitative estimation for the system of the governing equations. Consequently, uniform value of pressure far upstream, about which little attention has been paid heretofore, is revealed to be an important factor affecting film thickness. Adopting this quantity as one of the parameters makes it possible to predict the film thickness.

Non-continuum anomalies in the apparent viscosity experienced by a test sphere moving through an otherwise quiescent suspension

A comparison is made of the ‘‘apparent viscosity’’ (as defined by Stokes law) between two different cases of a test sphere moving slowly through an unbounded, otherwise quiescent, globally homogeneous, dilute suspension of identical, neutrally buoyant spherical particles dispersed in an incompressible Newtonian liquid. In case I the force on the test sphere is maintained constant for all time (and the torque-free sphere allowed to rotate) — corresponding to the so-called ‘‘falling ball’’ case — and its instantaneous velocity allowed to vary with proximity to each suspended sphere encountered during its trajectory; in case II the non-rotating test sphere is towed with a uniform (instantaneous) velocity through the suspension and the force experienced by it allowed to vary with proximity to each suspended sphere. Allowing for two-body hydrodynamic interactions between the ball and a suspended particle, the ensemble-average velocity of the test sphere is calculated in case I and ensemble-average force in case II, and Stokes law used to calculate the apparent viscosity of the suspension from the ensemble-averaged, linear force/velocity ratio obtained. In each case the ‘‘apparent suspension viscosity’’ coefficient attains, as expected, the limiting, continuum, Einstein value of 2.5 when the test sphere is much larger than the freely suspended particle. However, in the case of disparate relative sizes, the apparent viscosity is found to be significantly larger in case II than in case I. The difference arises from the locally inhomogeneous nature of the suspension and points up a fundamental non-continuum aspect of suspension behavior above and beyond the expected test/suspended-sphere size ratio ‘‘Knudsen’’ non-continuum effect.

Interfacial viscosity in viscous free surface flows. A sample case

We present a technique for introducing the surface viscosity terms into the numerical codes employed in the analysis of viscous free surface flows, namely, coating flows. Also, it qualitatively describes the effects produced by the surface viscosity on the interfacial dynamic behavior of an incompressible Newtonian liquid in a steady-state flow. The test problem chosen is the flow occurring in the slot coating process, particularly in the region where one end of a curved free surface (a meniscus) detaches from a stationary solid wall while the other end contacts a moving substrate