On the viscosity of magnetic fluid with low and moderate solid fraction

The design of a pressurized capillary rheometer operating at prescribed temperature is described to measure the viscosity of magnetic fluids (MFs) containing Fe3O4 magnetic nanoparticles (MNPs). The equipment constant of the rheometer was obtained using liquids with predetermined viscosities. Experimentally measured viscosities were used to evaluate different equations for suspension viscosities. Deviation of measured suspension viscosities from the Einstein equation was found to be basically due to the influence of spatial distribution and aggregation of Fe3O4 MNPs. By taking account of the coating layer on MNPs and the aggregation of MNPs in MFs, a modified Einstein equation was proposed to fit the experimental data. Moreover, the influence of external magnetic field on viscosity was also taken into account. Viscosities thus predicted are in good agreement with experimental data. Temperature effect on suspension viscosity was shown experimentally to be due to the shear-thinning behavior of the MFs.     

Bifurcation analysis for thermal convection in a rotating porous layer

The linear and weakly nonlinear thermal convection in a rotating porous layer is investigated by constructing a simplified model involving a system of fifth-order nonlinear ordinary differential equations. The flow in the porous medium is described by Lap wood-Brinkman-extended Darcy model with fluid viscosity different from effective viscosity. Conditions for the occurrence of possible bifurcations are obtained. It is established that Hopf bifurcation is possible only at a lower value of the Rayleigh number than that of simple bifurcation. In contrast to the non-rotating case, it is found that the ratio of viscosities as well as the Darcy number plays a dual role on the steady onset and some important observations are made on the stability characteristics of the system. The results obtained from weakly nonlinear theory reveal that, the steady bifurcating solution may be either sub-critical or supercritical depending on the choice of physical parameters. Heat transfer is calculated in terms of Nusselt number.     

On the viscosity of magnetic fluid with low and moderate solid fraction

The design of a pressurized capillary rheometer operating at prescribed temperature is described to measure the viscosity of magnetic fluids （MFs） containing Fe3O4 magnetic nanoparticles （MNPs）. The equipment constant of the rheometer was obtained using liquids with predetermined viscosities. Experimentally measured viscosities were used to evaluate different equations for suspension viscosities. Deviation of measured suspension viscosities from the Einstein equation was found to be basically due to the influence of spatial distribution and aggregation of Fe3O4 MNPs. By taking account of the coating layer on MNPs and the aggregation of MNPs in MFs, a modified Einstein equation was proposed to fit the experimental data. Moreover, the influence of external magnetic field on viscosity was also taken into account. Viscosities thus predicted are in good agreement with experimental data. Temperature effect on suspension viscosity was shown experimentally to be due to the shear-thinning behavior of the MFs.     

Weakly nonlinear instability of the flow of two immiscible liquids with different viscosities in a pipe

The weakly nonlinear stability due to an axisymmetric disturbance of two immiscible liquids with different viscosities in a pipe is worked out. The most important conclusions drawn from this work are the independence of the Reynolds number in the stability criterion, the existence of stable waves with sawtooth profile when the more viscous fluid is located at the core and it occupies most of the pipe, and the destabilizing effect of the surface tension, effect that can be overtaken by the stabilizing effect due to the difference in viscosity of the two fluids, when the viscosity of the ring fluid is much smaller than the viscosity of the core fluid and the thickness of the ring flow is smaller than the thickness of the core flow.     

A differential constitutive equation for polymer melts

A modification of the Giesekus constitutive equation is derived by incorporating (approximately, via the Peterlin approximation) the finite extensibility of polymer molecules into dumbbell kinetic theory along with the anisotropic hydrodynamic drag suggested by Giesekus. The constitutive equation that is obtained retains much of the simplicity of Giesekus' constitutive equation, but it involves terms that are cubic in the stress as well as those that are quadratic. It is shown that the constitutive equation quantitatively describes the steady elongational viscosity of the IUPAC polymer melt A (including the strain softening of the melt), but it cannot describe the elongational and shear viscosities simultaneously. It is also shown that the constitutive equation satisfies the Lodge-Meissner relation for shear strains less than unity.     

The onset of convection in a nanofluid saturated porous layer using Darcy model with cross diffusion

Linear and nonlinear stability analysis for the onset of convection in a horizontal layer of a porous medium saturated by a nanofluid is studied. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The modified Darcy equation that includes the time derivative term is used to model the momentum equation. In conjunction with the Brownian motion, the nanoparticle fraction becomes stratified, hence the viscosity and the conductivity are stratified. The nanofluid is assumed to be diluted and this enables the porous medium to be treated as a weakly heterogeneous medium with variation, in the vertical direction, of conductivity and viscosity. The critical Rayleigh number, wave number for stationary and oscillatory mode and frequency of oscillations are obtained analytically using linear theory and the non-linear analysis is made with minimal representation of the truncated Fourier series analysis involving only two terms. The effect of various parameters on the stationary and oscillatory convection is shown pictorially. We also study the effect of time on transient Nusselt number and Sherwood number which is found to be oscillatory when time is small. However, when time becomes very large both the transient Nusselt value and Sherwood value approaches to their steady state values.     

The hydrodynamic stability of channel flow with compliant boundaries

An asymptotic theory is developed for the hydrodynamic stability of an incompressible fluid flowing in a channel in which one wall is rigid and the other is compliant. We exploit the multideck structure of the flow to investigate theoretically the development of disturbances to the flow in the limit of large Reynolds numbers. A simple spring-plate model is used to describe the motion of the compliant wall, and this study considers the effect of the various wall parameters, such as tension, inertia, and damping, on the stability properties. An amplitude equation for a modulated wavetrain is derived and the properties of this equation are studied for a number of cases including linear and nonlinear theory. It is shown that in general the effect of viscoelastic damping is destabilizing. In particular, for large damping, the analysis points to a fast travelling wave, short-scale instability, which may be related to a flutter instability observed in some experiments. This work also demonstrates that the conclusions obtained by previous investigators in which the effect of tension, inertia, and other parameters is neglected, may be misleading. Finally it is shown that a set of compliant-wall parameters exists for which the Haberman type of critical layer analysis leads to stable equilibrium amplitudes, in contrast to many other stability problems where such equilibrium amplitudes are unstable.P.S. is grateful to the University of Zimbabwe for financial support. J.S.B.G. is grateful to the E.P.S.R.C. for the computing resources acquired under Grants GR/H58568-C88 and GR/H 83683 used in this research.     

Magnetohydrodynamics instability of interfacial waves between two immiscible incompressible cylindrical fluids

The problem of nonlinear instability of interfacial waves between two immiscible conducting cylindrical fluids of a weak Oldroyd 3-constant kind is studied. The system is assumed to be influenced by an axial magnetic field, where the effect of surface tension is taken into account. The analysis, based on the method of multiple scale in both space and time, includes the linear as well as the nonlinear effects. This scheme leads to imposing of two levels of the solvability conditions, which are used to construct like-nonlinear Schr6dinger equations （1-NLS） with complex coefficients. These equations generally describe the competition between nonlinearity and dispersion. The stability criteria are theoret- ically discussed and thereby stability diagrams are obtained for different sets of physical parameters. Proceeding to the nonlinear step of the problem, the results show the appearance of dual role of some physical parameters. Moreover, these effects depend on the wave kind, short or long, except for the ordinary viscosity parameter. The effect of the field on the system stability depends on the existence of viscosity and differs in the linear case of the problem from the nonlinear one. There is an obvious difference between the effect of the three Oldroyd constants on the system stability. New instability regions in the parameter space, which appear due to nonlinear effects, are shown.     

Constitutive equation of co-rotational derivative type for anisotropic-viscoelastic fluid

A constitutive equation theory of Oldroyd fluid B type, i.e. the co-rotational derivative type, is developed for the anisotropic-viscoelastic fluid of liquid crystalline (LC) polymer. Analyzing the influence of the orientational motion on the material behavior and neglecting the influence, the constitutive equation is applied to a simple case for the hydrodynamic motion when the orientational contribution is neglected in it and the anisotropic relaxation, retardation times and anisotropic viscosities are introduced to describe the macroscopic behavior of the anisotropic LC polymer fluid. Using the equation for the shear flow of LC polymer fluid, the analytical expressions of the apparent viscosity and the normal stress differences are given which are in a good agreement with the experimental results of Baek et al. For the fiber spinning flow of the fluid, the analytical expression of the extensional viscosity is given. The project supported by the National Natural Science Foundation of China (19832050 and 10372100)     

The stability of two stratified non-newtonian liquids in couette flow

Consideration is given to the stability of the interface between two Oldroyd liquids with shear-dependent viscosities, flowing in distinct layers while undergoing plane Couette flow. Results are presented as regions of stability in the plane determined by the logarithms of the viscosity and depth ratios. The work of previous authors for two Newtonian, power-law and constant-viscosity Oldroyd liquids is revealingly presented in a similar fashion. It is found that the dependence of the viscosities on shear-rate can drastically affect the regions of interfacial stability in a way over and above that due to just a change in the effective viscosity ratio. It is also found that for the Oldroyd liquids this viscosity variation affects the stability when it is present in the less-viscous layer.     

The theoretical analysis of the anticipation lattice models for traffic flow

Based on the anticipation lattice hydrodynamic models, which are described by the partial differential equations, the continuum version of the model is investigated through a reductive perturbation method. The linear stability theory is used to discuss the stability of the continuum model. The Korteweg–de Vries (for short, KdV) equation near the neutral stability line and the modified Korteweg–de Vries (for short, mKdV) equation near the critical point are obtained by using the nonlinear analysis method. And the corresponding solutions for the traffic density waves are derived, respectively. The results display that the anticipation factor has an important influence on traffic flow. From the simulation, it is shown that the traffic jam is suppressed efficiently with taking into account the anticipation effect, and the analytical result is consonant with the simulation one.     

On the nonlinear interfacial instability of rotating core-annual flow

The interfacial stability of rotating core-annular flows is investigated. The linear and nonlinear effects are considered for the case when the annular region is very thin. Both asymptotic and numerical methods are used to solve the flow in the core and film regions which are coupled by a difference in viscosity and density. The long-time behavior of the fluid-fluid interface is determined by deriving its nonlinear evolution in the form of a modified Kuramoto-Sivashinsky equation. We obtain a generalization of this equation to three dimensions. The flows considered are applicable to a wide array of physical problems where liquid films are used to lubricate higher- or lower-viscosity core fluids, for which a concentric arrangement is desired. Linearized solutions show that the effects of density and viscosity stratification are crucial to the stability of the interface. Rotation generally destabilizes nonaxisymmetric disturbances to the interface, whereas the centripetal forces tend to stabilize flows in which the film contains the heavier fluid. Nonlinear effects allow finite-amplitude helically traveling waves to exist when the fluids have different viscosities.This research was partially supported by the National Aeronautics and Space Administration under NASA Contract No. NAS1-18605 while the second author was in residence at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23665. This work was also supported by the Science and Engineering Research Council.     

Viscosity effect on the flow patterns in T-type micromixers

Flow patterns and mixing of liquids with different viscosities in T-type micromixers are numerically investigated on the Reynolds number range from 1 to 250. The viscosity ratio of the mixing media varied from 1 to 2; its effect on the flow structure and the mixing is studied. The dependences of the mixing efficiency and the pressure difference in the channel on the viscosity ratio and the Reynolds number are obtained. It is shown that the viscosity ratio has a considerable effect on the flow structure before and after transition from the symmetric to the asymmetric flow pattern. The self-similar behavior of the asymmetric flow pattern is established.     

Elongational viscosity by fiber spinning

Isothermal melt, fiber-spinning was recently analyzed by means of a nonlinear, integral, constitutive equation that incorporates shear history effects, spectrum of relaxation times, shear-thinning, and extension thinning or thickening when either the drawing force or the draw ratio is specified. The predictions agreed with experimental data on spinning of polystyrene, low-density polyethylene, and polypropylene melts. The predicted apparent elongational viscosity along the threadline (which, as shown in this work, must be identical to that measured experimentally by fiber spinning type of elongational rheometers) is compared with the true elongational viscosity predicted by the same constitutive equation under well-defined experimental conditions of constant extension rate, independent of any strain history. It is concluded that the apparent elongational viscosity, as measured by fiber-spinning, approaches the true elongational viscosity at low Weissenberg numbers (defined as the product of the liquid's relaxation time multiplied by the extension rate). At moderate Weissenberg numbers, the two viscosities may differ by an order of magnitude and their difference grows even larger at high Weissenberg numbers.     

Weakly nonlinear EHD stability of slightly viscous jet

A weakly nonlinear approach is utilized here to study the electrohydrodynamic (EHD) instability of an incompressible viscous liquid jet stressed by an axial electric field. The linear motion equations is solved in the light of nonlinear boundary conditions. The viscosity is assumed to be small. The study takes into account both the shear and radial components of the stresses at the interface. In the linear theory, we discuss the breakup phenomena of liquid jets. Also, it is found that, the electrical shearing stresses have no effect at the linear marginal state, while the linear cutoff wavenumber depends on the electrical shearing stresses. A nonlinear perturbation method is introduced. This method can be described our problem precisely. The nonlinear stability is compared with the linear stability condition in the weak viscosity case. It is found that, the weak viscosity has effect on the nonlinear stability condition, in contrast with the linear analysis, whereas the nonlinear cutoff wavenumber doesn't depend on the weak viscosity in both the linear and nonlinear theory.     

Instability analysis of nonlinear surface waves in a circular cylindrical container subjected to a vertical excitation

Singular perturbation theory of two-time scale expansions was developed both in inviscid and weak viscous fluids to investigate the motion of single surface standing wave in a liquid-filled circular cylindrical vessel, which is subject to a vertical periodical oscillation. Firstly, it is assumed that the fluid in the circular cylindrical vessel is inviscid, incompressible and the motion is irrotational, a nonlinear evolution equation of slowly varying complex amplitude, which incorporates cubic nonlinear term, external excitation and the influence of surface tension, was derived from solvability condition of high-order approximation. It shows that when forced frequency is low, the effect of surface tension on mode selection of surface wave is not important. However, when forced frequency is high, the influence of surface tension is significant, and can not be neglected. This proved that the surface tension has the function, which causes free surface returning to equilibrium location. Theoretical results much close to experimental results when the surface tension is considered. In fact, the damping will appear in actual physical system due to dissipation of viscosity of fluid. Based upon weakly viscous fluids assumption, the fluid field was divided into an outer potential flow region and an inner boundary layer region. A linear amplitude equation of slowly varying complex amplitude, which incorporates damping term and external excitation, was derived from linearized Navier–Stokes equation. The analytical expression of damping coefficient was determined and the relation between damping and other related parameters (such as viscosity, forced amplitude and depth of fluid) was presented. The nonlinear amplitude equation and a dispersion, which had been derived from the inviscid fluid approximation, were modified by adding linear damping. It was found that the modified results much reasonably close to experimental results. Moreover, the influence both of the surface tension and the weak viscosity on the mode formation was described by comparing theoretical and experimental results. The results show that when the forcing frequency is low, the viscosity of the fluid is prominent for the mode selection. However, when the forcing frequency is high, the surface tension of the fluid is prominent. Finally, instability of the surface wave is analyzed and properties of the solutions of the modified amplitude equation are determined together with phase-plane trajectories. A necessary condition of forming stable surface wave is obtained and unstable regions are illustrated.     

Adaptive boundary control of the unforced generalized Korteweg–de Vries–Burgers equation

This paper deals with the adaptive control problem of the unforced generalized Korteweg?Cde Vries?CBurgers (GKdVB) equation when the spatial domain is [0,1]. Three adaptive control laws are designed for the GKdVB equation when either the kinematic viscosity ?? or the dynamic viscosity ?? is unknown, or when both viscosities ?? and ?? are unknowns. Using the Lyapunov theory, the L ²-global exponential stability of the solutions of this equation is shown for each of the proposed control laws. Also, numerical simulations based on the Finite Element method (FEM) are given to illustrate the analytical results.     

Investigation of the effect of the Coriolis force on a thin fluid film on a rotating disk

The effect of the Coriolis force on the evolution of a thin film of Newtonian fluid on a rotating disk is investigated. The thin-film approximation is made in which inertia terms in the Navier–Stokes equation are neglected. This requires that the thickness of the thin film be less than the thickness of the Ekman boundary layer in a rotating fluid of the same kinematic viscosity. A new first-order quasi-linear partial differential equation for the thickness of the thin film, which describes viscous, centrifugal and Coriolis-force effects, is derived. It extends an equation due to Emslie et al. [J. Appl. Phys. 29, 858 (1958)] which was obtained neglecting the Coriolis force. The problem is formulated as a Cauchy initial-value problem. As time increases the surface profile flattens and, if the initial profile is sufficiently negative, it develops a breaking wave. Numerical solutions of the new equation, obtained by integrating along its characteristic curves, are compared with analytical solutions of the equation of Emslie et al. to determine the effect of the Coriolis force on the surface flattening, the wave breaking and the streamlines when inertia terms are neglected.     

An extended two-lane traffic flow lattice model with driver’s delay time

In this paper, a new two-lane lattice model is presented by considering the effect of drivers’ delay in sensing relative flux. By means of the linear stability analysis, the effect of drivers’ delay time on the stability of two-lane traffic flow is examined and shown that with the drivers’ delay time increasing, the unstable areas expand accordingly on the phase diagram, which is also confirmed by direct computer simulations. Through nonlinear analysis method, the modified Korteweg–deVries equation near the critical point is obtained and solved to describe the traffic- jamming transitions in a two-lane system.     

Cross-flow-induced instability and nonlinear dynamics of cylinder arrays with consideration of initial axial load

The instability and nonlinear dynamics of planar motions of a cylinder array subjected to cross-flow have been studied via a five-mode discretization of the governing partial differential equation, focusing on the effect of initial axial load externally imposed on the cylinder. Theoretical results based on a stability analysis have indicated that, with increasing initial axial load and flow velocity, the system may lose stability either via flutter or via buckling. The boundaries of these two forms of instability are predicted analytically. To explore the post-instability dynamics of the system, a Runge–Kutta scheme is used to solve the nonlinear governing equation of motion. Three typical behaviors, including limit cycle motions of the system, are obtained. It is shown that, for relatively low flow velocity, with increasing initial axial load, just beyond the pitchfork bifurcation the cylinder would settle in a buckled equilibrium position; and for high flow velocity, however, this phenomenon only occurs when the initial axial load becomes sufficiently large.