Instability of two streaming conducting and dielectric bounded fluids in porous medium under time-varying electric field

In this paper, we consider the instability of the interface between two superposed streaming conducting and dielectric fluids of finite depths through porous medium in a vertical electric field varying periodically with time. A damped Mathieu equation with complex coefficients is obtained. The method of multiple scales is used to obtain an approximate solution of this equation, and then to analyze the stability criteria of the system. We distinguish between the non-resonance case, and the resonance case, respectively. It is found, in the first case, that both the porosity of porous medium, and the kinematic viscosities have stabilizing effects, and the medium permeability has a destabilizing effect on the system. While in the second case, it is found that each of the frequency of the electric field, and the fluid velocities, as well as the medium permeability, has a stabilizing effect, and decreases the value of the resonance point, while each of the porosity of the porous medium, and the kinematic viscosities has a destabilizing effect, and increases the value of the resonance point. In the absence of both streaming velocities and porous medium, we obtain the canonical form of the Mathieu equation. It is found that the fluid depth and the surface tension have a destabilizing effect on the system. This instability sets in for any value of the fluid depth, and by increasing the depth, the instability holds for higher values of the electric potential; while the surface tension has no effect on the instability region for small wavenumber values. Finally, the case of a steady electric field in the presence of a porous medium is also investigated, and the stability conditions show that each of the fluid depths and the porosity of the porous medium ɛ has a destabilizing effect, while the fluid velocities have stabilizing effect. The stability conditions for two limiting cases of interest, the case of purely fluids), and the case of absence of streaming, are also obtained and discussed in detail.     

Stability criteria of an immersed annulus swirling jet in a zero gravity environment

The present work deals with the stability properties of an immersed annulus swirling jet in a zero gravity environment. The considered system is composed of two streaming coaxial fluid cylinders, embedded in a third streaming fluid, where the intermediate one (annulus) has a uniform swirl speed. The fluids are immiscible, inviscid, and incompressible. The linear stability criteria of the model are discussed analytically and stability diagrams are obtained. We conclude that the radii ratio, the Weber number and the swirl number play a significant role in determining the dynamics of the developing interfacial patterns.     

Linear instability of the electrified free interface between two cylindrical shells of viscoelastic fluids through porous media

In this paper, we have discussed the linear stability analysis of the electrified surface separating two coaxial Oldroyd-B fluid layers confined between two impermeable rigid cylinders in the presence of both interfacial insoluble surfactant and surface charge through porous media. The case of long waves interfacial stability has been studied. The dispersion relation is solved numerically and hence the effects of various parameters are illustrated graphically. Our results reveal that the influence of the physicochemical parameter β is to shrink the instability region of the surface and reduce the growth rate of the unstable normal modes. Such important effects of the surfactant on the shape of interfacial structures are more sensitive to the variation of the β corresponding to non-Newtonian fluids-model compared with the Newtonian fluids model. In the case of long wave limit, it is demonstrated that increasing β, has a dual role in-fluence (de-stabilizing effects) depending on the viscosity of the core fluid. It has a destabilizing effect at the large values of the core fluid viscosity coefficient, while this role is exchanged to a regularly stabilizing influence at small values of such coefficient.     

Electroviscous potential flow in nonlinear analysis of capillary instability

We study the nonlinear stability of electrohydrodynamic of a cylindrical interface separating two conducting fluids of circular cross section in the absence of gravity using electroviscous potential flow analysis. The analysis leads to an explicit nonlinear dispersion relation in which the effects of surface tension, viscosity and electricity on the normal stress are not neglected, but the effect of shear stresses is neglected. Formulas for the growth rates and neutral stability curve are given in general. In the nonlinear theory, it is shown that the evolution of the amplitude is governed by a Ginzburg–Landau equation. When the viscosities are neglected, the cubic nonlinear Schrödinger equation is obtained. Further, it is shown that, near the marginal state, a nonlinear diffusion equation is obtained in the presence of viscosities. The various stability criteria are discussed both analytically and numerically and stability diagrams are obtained. It is also shown that, the viscosity has effect on the nonlinear stability criterion of the system, contrary to previous belief.     

Magnetohydrodynamic stability of heterogeneous dissipative conducting liquids

Summary This discussion which is restricted to the flow of heterogeneous, incompressible, inviscid and conducting liquids between two nonrotating coaxial cylinders is divided into three parts. In the first part the problem in question is investigated with an applied radial magnetic field. A sufficient condition for stability is found. In particular, if the steady flow velocity is uniform it is shown that the flow is always stable.In part two the stability of homogeneous viscous conducting fluids through two fixed coaxial cylinders with an applied radial magnetic field has been discussed. A sufficient condition for stability is derived and an upper bound for the amplification factor is given.In part three the problem in question is treated with an applied axial magnetic field. A sufficient condition for stability is given and some particular cases are investigated.     

Evolution Equations for the Surface Concentration of an Insoluble Surfactant; Applications to the Stability of an Elongating Thread and a Stretched Interface

The general form of the convection–diffusion equation governing the evolution of the surface concentration of an insoluble surfactant over an evolving interface is reviewed and discussed for three-dimensional, axisymmetric, and two-dimensional configurations. The linearized form of the evolution equation is then derived around cylindrical and planar shapes in a framework that is suitable for carrying out a linear stability analysis for axisymmetric or two-dimensional perturbations. Particular attention is paid to the cases of quiescent unperturbed fluids, unidirectional shear flow, and elongational flow. By way of application, the linearized transport equations are combined with Stokes-flow hydrodynamics to investigate the stability of an elongating cylindrical viscous thread suspended in an ambient viscous fluid or in a vacuum, and the stability of a two-dimensional interface separating two semi-infinite fluids and stretched under the action of an orthogonal stagnation-point flow. The results illustrate the possibly important role of the surfactant on the linear growth of periodic waves on the cylindrical interface, and reveal that the surfactant has no effect on the stability of the planar interface.     

Numerical simulations of Boger fluids through different contraction configurations for the development of a measuring system for extensional viscosity

This paper reports the flow behaviour of Newtonian and Boger fluids through various axisymmetric contraction configurations by means of numerical predictions. A principal aim has been to evaluate the geometrical design choice of the hyperbolic contraction flow. The FENE-CR model has been used to reflect the behaviour of Boger fluids, with constant shear viscosity, finite (yet large) extensional viscosity and less than quadratic first normal stress difference. Numerical calculations have been performed on six different contraction configurations to evaluate an optimized geometry for measuring extensional viscosity in uniaxial extensional flow. The influence of a sharp or rounded recess-corner on the nozzle has also been investigated. Few commercial measuring systems are currently available for measurement of the extensional rheology of medium-viscosity fluids, such as foods and other biological systems. In this context, a technique based on the hyperbolic contraction flow would be a suitable alternative. The pressure drop, the velocity field, the first normal stress difference and the strain rate across the geometry have each been evaluated for Newtonian and Boger fluids. This numerical study has shown that the hyperbolic configuration is superior to the other geometry choices in achieving a constant extension rate. In this hyperbolic configuration, no vortices are formed, the measuring range is broader and the strain rate is constant throughout the geometric domain, unlike in the alternative configurations tested. The difference between sharp and rounded recess-corner configurations proved to be negligible and a rise in excess pressure drop (epd) for increasing deformation rates has been observed.     

MHD flows of a second grade fluid between two side walls perpendicular to a plate through a porous medium

Exact analytical solutions for magnetohydrodynamic (MHD) flows of an incompressible second grade fluid in a porous medium are developed. The modified Darcy's law for second grade fluid has been used in the flow modelling. The Hall effect is taken into account. The exact solutions for the unsteady flow induced by the time-dependent motion of a plane wall between two side walls perpendicular to the plane has been constructed by means of Fourier sine transforms. The similar solutions for a Newtonian fluid, performing the same motion, appear as limiting cases of the solutions obtained here. The influence of various parameters of interest on the velocity and shear stress at the bottom wall has been shown and discussed through several graphs. A comparison between a Newtonian and a second grade fluids is also made.     

MHD free-convective flow of micropolar and Newtonian fluids through porous medium in a vertical channel

We have studied the fully-developed free-convective flow of an electrically conducting fluid in a vertical channel occupied by porous medium under the influence of transverse magnetic field. The internal prefecture of the channel is divided into two regions; one region filled with micropolar fluid and the other region with a Newtonian fluid or both the regions filled by Newtonian fluids. Analytical solutions of the governing equations of fluid flow are found to be in excellent agreement with analytical prediction. Analytical results for the details of the velocity, micro-rotation velocity and temperature fields are shown through graphs for various values of physical parameters. It is noticed that Newtonian fluids prop up the linear velocity of the fluid in contrast to micropolar fluid. Also the skin friction coefficient at both the walls is derived and its numerical values are offered through tables.     

Experimental studies of interfacial instabilities in multilayer pressure-driven flow of polymeric melts

The interfacial deformation and stability of two-(A-B) as well as three-layer symmetric (A-B-A) and asymmetric (A-B-C) pressure-driven flow of viscoelastic fluids has been investigated. Flow visualization in conjunction with digital image processing has been used to observe and measure the rate of encapsulation and interfacial stability/instability of the flow. Specifically, the encapsulation behavior as well as stability/instability of the interface and the corresponding growth or decay rate of disturbances as a function of various important parameters, namely, number of layers and their arrangement, layer depth ratio, viscosity and elasticity ratio as well as disturbance frequency, have been investigated. Based on these experiments, we have shown that the encapsulation phenomena occurs irrespective of the stability/instability of the interface and in cases when both encapsulation and instability occur simultaneously their coupling leads to highly complex and three-dimensional interfacial wave patterns. Moreover, it has been shown that the simple notion that less viscous fluids encapsulate more viscous fluids is incorrect and depending on the wetting properties of the fluid as well as their first and second normal stresses the reverse could occur. Additionally, in two- and three-layer flows it has been shown that by placing a thin, less viscous layer adjacent to the wall longwave disturbances can be stabilized while short and intermediate wavelength disturbances are stabilized when the more elastic fluid is the majority component. Furthermore, in three-layer flows it has been demonstrated that in the linear instability regime no dynamic interaction between the two interfaces is possible for short and intermediate wavenumber disturbances. However, in the nonlinear stability regime dynamic interactions between interfaces have been observed in this range of disturbance wavenumbers leading to highly chaotic flows. Finally, in the parameter space of this study no subcritical bifurcations were observed while supercritical bifurcations resulting in waves with a pointed front and a gradual tail were observed.     

Rayleigh-Taylor instability of the interface between conducting and nonconducting fluids in an alternating magnetic field

The influence of an alternating magnetic field on the Rayleigh-Taylor instability of a conducting fluid has been investigated [1, 2] in the limiting cases of long- and short-wave (compared with the skin-layer thickness) perturbations. Garnier [3] has reported a new mechanism of instability of the interface between conducting and nonconducting fluids which differs not only from the classical Rayleigh-Taylor instability but also from parametric excitation of perturbations. From the instability criterion obtained in [3] without restriction on the spatial scale of the perturbation a paradoxical result follows: An increase in the frequency of the field leads to instability, whereas the practical results of metal casting in an electromagnetic crystallizing tank [4] indicate the opposite effect. In the present paper, it is shown that a plane-polarized high-frequency field effectively stabilizes part of the spectrum of three-dimensional perturbations of an interface but does not completely suppress the Rayleigh-Taylor instability mechanism. The instability generated by the self-field has the nature of parametric resonance.     

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.     

Multilayer film casting of modified Giesekus fluids Part 2. Linear stability analysis

A linear stability analysis of the multilayer film casting of polymeric fluids has been conducted. A modified Giesekus model was used to characterize the rheological behaviors of the fluids. The critical draw ratio at the onset of draw resonance was found to depend on the elongational and shear viscosities of the fluids. Extensional-thickening has a stabilizing effect, whereas shear-thinning and extensional-thinning have destabilizing effects. The critical draw ratios for bilayer films of various thickness fractions are bounded by those for single layer films of the two fluids. When the two fluids have a comparable elongational viscosity, the critical draw ratio at a given Deborah number varies linearly with thickness fraction. When one fluid has a much larger elongational viscosity, it dominates the flow and the critical draw ratios at most thickness fractions remain close to its critical draw ratio as a single layer film. When the dominating fluid exhibits extensional-thickening, a film with a certain thickness fraction has more than one critical draw ratio at a given Deborah number and may not exhibit draw resonance within some range of the Deborah number.     

Instability of the interface in two-layer flows with large viscosity contrast at small Reynolds numbers

The Kelvin–Helmholtz instability is believed to be the dominant instability mechanism for free shear flows at large Reynolds numbers. At small Reynolds numbers, a new instability mode is identified when the temporal instability of parallel viscous two fluid mixing layers is extended to current-fluid mud systems by considering a composite error function velocity profile. The new mode is caused by the large viscosity difference between the two fluids. This interfacial mode exists when the fluid mud boundary layer is sufficiently thin. Its performance is different from that of the Kelvin–Helmholtz mode. This mode has not yet been reported for interface instability problems with large viscosity contrasts.These results are essential for further stability analysis of flows relevant to the breaking up of this type of interface.     

An integral constitutive law for viscoelastic fluids based on the concept of evolving natural configurations: stability analysis

A general conceptual framework has recently been developed within the context of large deformations for the study of materials (solids and fluids) that either undergo microstructural changes or for which stress states are related to the presence of several relaxation mechanisms. The corner stone of this new approach is that all materials exhibit an infinity of stress-free natural configurations that are evolving in a thermodynamically admissible process. In this paper, that is exploratory in nature, we study the behavior of a non-linear integral viscoelastic constitutive equation (CE) for a fluid assumed to posses an infinity of evolving natural configurations. The CE stability pattern with respect to small perturbations about the rest state is also addressed.     

Magnetic particle microrheometric dynamics in Newtonian fluids: numerical simulations on the early motion and beyond

Rotating magnetic particle microrheometry has been a promising technique in measuring material properties in limited-sample high-viscosity fluids. Experimental limitations in the early motion require further theoretical exploration. In this work, the rotation of a ferromagnetic particle is considered under the influence of an external uniform magnetic field in an infinite highly viscous Newtonian fluid. The motion is restricted at the very low Reynolds number limit. Early-time analytical approximations are utilised to initiate numerical calculations in an attempt to describe the azimuthal velocity dependency on scaled time and radius. The equation of motion is solved by implementing a Crank–Nicholson finite-difference scheme, while the driving time-dependent boundary condition is discretised according to a Lax–Wendroff scheme. Stability and convergence criteria for the PDE are also discussed. It is demonstrated that the step function form of the applied magnetic field does not cause finite displacement other than that expected from Newtonian fluid flow for the typical magnetic field magnitude ranges encountered in micro-rheometric studies. The numerical solution is compared against analytical values available for particle ‘zero-total-torque’ condition and it was found to be second-order accurate in time and radial dimension.     

MHD flow and heat transfer of micropolar fluid between two porous disks

A numerical study is carried out for the axisymmetric steady laminar incompressible flow of an electrically conducting micropolar fluid between two infinite parallel porous disks with the constant uniform injection through the surface of the disks. The fluid is subjected to an external transverse magnetic field. The governing nonlinear equations of motion are transformed into a dimensionless form through von Karman’s similarity transformation. An algorithm based on a finite difference scheme is used to solve the reduced coupled ordinary differential equations under associated boundary conditions. The effects of the Reynolds number, the magnetic parameter, the micropolar parameter, and the Prandtl number on the flow velocity and temperature distributions are discussed. The results agree well with those of the previously published work for special cases. The investigation predicts that the heat transfer rate at the surfaces of the disks increases with the increases in the Reynolds number, the magnetic parameter, and the Prandtl number. The shear stresses decrease with the increase in the injection while increase with the increase in the applied magnetic field. The shear stress factor is lower for micropolar fluids than for Newtonian fluids, which may be beneficial in the flow and thermal control in the polymeric processing.     

Study of heat transport by stationary magneto-convection in a Newtonian liquid under temperature or gravity modulation using Ginzburg–Landau model

The present paper deals with a weak non-linear stability problem of magneto-convection in an electrically conducting Newtonian fluid, confined between two horizontal surfaces, under a constant vertical magnetic field, and subjected to an imposed time-periodic boundary temperature (ITBT) or gravity modulation (ITGM). In the case of ITBT, the temperature gradient between the walls of the fluid layer consists of a steady part and a time-dependent oscillatory part. The temperature of both walls is modulated in this case. In the problem involving ITGM, the gravity field has two parts: a constant part and an externally imposed time periodic part, which can be realized by oscillating the fluid layer. The disturbance is expanded in terms of power series of amplitude of convection, which is assumed to be small. Using Ginzburg–Landau equation, the effect of modulations on heat transport is analyzed. Effect of various parameters on the heat transport is also discussed.     

On the stability of natural convection in a porous vertical slab saturated with an Oldroyd-B fluid

The stability of the conduction regime of natural convection in a porous vertical slab saturated with an Oldroyd-B fluid has been studied. A modified Darcy’s law is utilized to describe the flow in a porous medium. The eigenvalue problem is solved using Chebyshev collocation method and the critical Darcy–Rayleigh number with respect to the wave number is extracted for different values of physical parameters. Despite the basic state being the same for Newtonian and Oldroyd-B fluids, it is observed that the basic flow is unstable for viscoelastic fluids—a result of contrast compared to Newtonian as well as for power-law fluids. It is found that the viscoelasticity parameters exhibit both stabilizing and destabilizing influence on the system. Increase in the value of strain retardation parameter \(\Lambda _2 \) portrays stabilizing influence on the system while increasing stress relaxation parameter \(\Lambda _1\) displays an opposite trend. Also, the effect of increasing ratio of heat capacities is to delay the onset of instability. The results for Maxwell fluid obtained as a particular case from the present study indicate that the system is more unstable compared to Oldroyd-B fluid.