Normal Stress Balance Research Articles

Pressure Corrections for the Potential Flow Analysis of Kelvin-Helmholtz Instability

The present paper deals with the study of viscous contribution to the pressure for the viscous potential flow analysis of Kelvin-Helmholtz instability of two viscous fluids. Viscosity enters through normal stress balance in the viscous potential flow theory and tangential stresses for two fluids are not continuous at the interface. Here we have considered viscous pressure in the normal stress balance along with the irrotational pressure and it is assumed that the addition of this viscous pressure will resolve the discontinuity between the tangential stresses and the tangential velocities at the interface. The viscous pressure is derived by mechanical energy equation and this pressure correction applied to compute the growth rate of Kelvin-Helmholtz instability. A dispersion relation is obtained and a stability criterion is given in the terms of critical value of relative velocity. It has been observed that the inclusion of irrotational shearing stresses stabilizes the system.

Viscous Potential Flow Analysis of Rayleigh-Taylor Instability of Cylindrical Interface

The present paper deals with the study of Rayleigh-Taylor instability at the cylindrical interface using viscous potential flow theory. In the inviscid potential flow theory, the viscous term in Navier-Stokes equation vanishes as viscosity is zero. In viscous potential flow, the viscous term in Navier-Stokes equation vanishes as vorticity is zero but viscosity is not zero. Viscosity enters through normal stress balance in viscous potential flow theory and tangential stresses are not considered. A dispersion relation is derived and stability is discussed in terms of various parameters such as Ohnesorge number, density ratio etc. A condition for neutral stability is obtained and it is given in terms of critical value of the wave number. It is observed that the Ohnesorge number has destabilizing effect while inner fluid fraction has stabilizing effect on the stability of the system.

The electrohydrodynamics of superimposed fluids subjected to a nonuniform transverse electric field

This study aims to investigate electrohydrodynamics of two superimposed fluids that are confined between a pair of two-dimensional flat plates and are exposed to a sinusoidal electric field in zero gravity. The goal is to identify the parameters that affect the flow structure and interface deformation using a simple closed form solution. The governing electrohydrodynamic equations are solved analytically for Newtonian and immiscible fluids in the framework of leaky-dielectric theory and in the limit of small electric field and fluid inertia. A detailed analysis of the electric and flow fields is presented and it is shown that the electric field induces sinusoidal electrical stresses at the interface, which lead to periodic convection cells. The parameters affecting the sense of flow circulation and strength are investigated and it is shown that the former depends on the relative magnitude of the electric permittivity and conductivity ratios while the latter is controlled by the relative thicknesses of the fluid layers and the ratio of the electric conductivities and viscosities of the fluids. The maximum flow strength is achieved at a relative thickness that is set by the competition between the electric and hydrodynamic effects. For small deformation, the distortion of the interface is examined using a normal stress balance at the interface, and it is shown that the degree of interface deformation scales with the square of the amplitude of the electric potential nonuniformity, while its wavenumber is twice that of the imposed potential nonuniformity. Furthermore, a zero-deformation curve is found, which delineates the region in the permittivity-conductivity space according to the sense of interface deformation. The results show that for certain ranges of fluid layer thicknesses and permittivity ratios, the interface will remain flat, despite the action of the nonuiform field.

Viscous potential flow analysis of Kelvin-Helmholtz instability of cylindrical interface

A linear analysis of Kelvin-Helmholtz instability of cylindrical interface is carried out using viscous potential flow theory. In the inviscid potential flow theory, the viscous term in Navier-Stokes equation vanishes as viscosity is zero. In viscous potential flow, the viscous term in Navier-Stokes equation vanishes as vorticity is zero but viscosity is not zero. Viscosity enters through normal stress balance in viscous potential flow theory and tangential stresses are not considered. Both asymmetric and axisymmetric disturbances are considered. A dispersion relation has been obtained and stability criteria are given in the terms of the relative velocity. A comparison between inviscid potential flow and viscous potential flow has been made. It has been observed that Reynolds number and inner fluid fraction both have destabilizing effect on the stability of the system.

Viscous contribution to the pressure for electroviscous potential flow analysis of capillary instability

Viscous contribution for the Viscous Potential Flow (VPF) analysis of capillary instability with axial electric field at the interface of two viscous fluids has been analysed. In VPF, viscosity enters through normal stress balance and the effects of shearing stresses are completely ignored. To include the effect of shearing stresses, viscous pressure is included in normal stress balance along with irrotational pressure and it is assumed that the viscous contribution to the irrotational pressure will resolve the discrepancy between the tangential stresses at the interface. Stability criterion is given in terms of critical value of electric field. Various graphs have been drawn to show the effect of the parameters, such as electric field, inner fluid fraction, Ohnesorge number, conductivity ratio and permittivity ratio.

Mechanical model for fiber-laden membranes

An integrated mechanical model for fiber-laden membranes is presented and representative predictions of relevance to cellulose ordering and orientation in the plant cell wall are presented. The model describes nematic liquid crystalline self-assembly of rigid fibers on an arbitrarily curved fluid membrane. The mechanics of the fluid membrane is described by the Helfrich bending-torsion model, the fiber self-assembly is described by the 2D Landau-de Gennes quadrupolar Q-tensor order parameter model, and the fiber-membrane interactions (inspired by an extension of the 2D Maier-Saupe model to curved surfaces) include competing curvo-philic (curvature-seeking) and curvo-phobic (curvature-avoiding) effects. Analysis of the free energy reveals three fiber orientation regimes: (a) along the major curvature, (b) along the minor curvature, (c) away from the principal curvatures, according to the competing curvo-philic and curvo-phobic interactions. The derived shape equation (normal stress balance) now includes curvature-nematic ordering contributions, with both bending and torsion renormalizations. Integration of the shape and nematic order equations gives a complete model whose solution describes the coupled membrane shape/fiber order state. Applications to cylindrical membranes, relevant to the plant cell wall, shows how growth decreases the fiber order parameter and moves the fibers’ director from the axial direction towards the azimuthal orientation, eventually leading to a state of stress predicted by pure membranes. The ubiquitous 54.7° cellulose fibril orientation with respect to the long axis in a cylindrical plant cell wall is shown to be predicted by the preset model when the ratio of curvo-phobic and curvo-philic interactions is in the range of the cylinder radius.

Spherical bubble collapse in viscoelastic fluids

The collapse of a spherical bubble in an infinite expanse of viscoelastic fluid is considered. For a range of viscoelastic models, the problem is formulated in terms of a generalized Bernoulli equation for a velocity potential, under the assumptions of incompressibility and irrotationality. The boundary element method is used to determine the velocity potential and viscoelastic effects are incorporated into the model through the normal stress balance across the surface of the bubble. In the case of the Maxwell constitutive equation, the model predicts phenomena such as the damped oscillation of the bubble radius in time, the almost elastic oscillations in the large Deborah number limit and the rebound limit at large values of the Deborah number. A rebound condition in terms of ReDe is derived theoretically for the Maxwell model by solving the Rayleigh–Plesset equation. A range of other viscoelastic models such as the Jeffreys model, the Rouse model and the Doi-Edwards model are amenable to solution using the same technique. Increasing the solvent viscosity in the Jeffreys model is shown to lead to increasingly damped oscillations of the bubble radius.

A Quadratic Spline based Interface (QUASI) reconstruction algorithm for accurate tracking of two-phase flows

A new Quadratic Spline based Interface (QUASI) reconstruction algorithm is presented which provides an accurate and continuous representation of the interface in a multiphase domain and facilitates the direct estimation of local interfacial curvature. The fluid interface in each of the mixed cells is represented by piecewise parabolic curves and an initial discontinuous PLIC approximation of the interface is progressively converted into a smooth quadratic spline made of these parabolic curves. The conversion is achieved by a sequence of predictor–corrector operations enforcing function ( C 0) and derivative ( C 1) continuity at the cell boundaries using simple analytical expressions for the continuity requirements. The efficacy and accuracy of the current algorithm has been demonstrated using standard test cases involving reconstruction of known static interface shapes and dynamically evolving interfaces in prescribed flow situations. These benchmark studies illustrate that the present algorithm performs excellently as compared to the other interface reconstruction methods available in literature. Quadratic rate of error reduction with respect to grid size has been observed in all the cases with curved interface shapes; only in situations where the interface geometry is primarily flat, the rate of convergence becomes linear with the mesh size. The flow algorithm implemented in the current work is designed to accurately balance the pressure gradients with the surface tension force at any location. As a consequence, it is able to minimize spurious flow currents arising from imperfect normal stress balance at the interface. This has been demonstrated through the standard test problem of an inviscid droplet placed in a quiescent medium. Finally, the direct curvature estimation ability of the current algorithm is illustrated through the coupled multiphase flow problem of a deformable air bubble rising through a column of water.

Deformation of a viscoelastic drop in planar extensional flows of a Newtonian fluid

This communication presents experimental data for steady and transient deformations of a purely elastic drop (PIB/PB Boger fluid) suspended in a Newtonian fluid (PDMS) undergoing a planar extensional flow produced by a four-roll mill. The viscoelastic effects alter the steady drop shape from being ellipsoidal to drop shapes with more blunt ends. The results reflect a balance between the direct tensile stress contribution of the viscoelastic fluid to the normal stress balance, and modifications of the viscous (i.e. Newtonian) stress and pressure due to viscoelastic changes in the flow. This is qualitatively consistent with the conclusion from recent simulation results [S. Ramaswamy, L.G. Leal, The deformation of a viscoelastic drop subjected to steady uniaxial extensional flow of a Newtonian fluid, Journal of Non-Newtonian Fluid Mechanics 85 (2–3) (1999) 127–163]. There is no overshoot of the drop deformation upon start-up of the flow, and a relative insensitivity to the Deborah number. However, when the external flow is turned off, there is a broad spectrum of relaxation times for the drop to return to a spherical shape, between the characteristic capillary timescale for a Newtonian fluid, aμ/ σ, to the longer polymer relaxation time, τ R .

Viscous potential flow analysis of capillary instability with heat and mass transfer

We carry out the linear viscous-irrotational analysis of capillary instability with heat transfer and phase change. We consider the cylindrical interface shared by two viscous incompressible fluids enclosed by two concentric cylinders. In viscous potential flow, viscosity enters the model through the balance of normal stresses at the interface. We write the dispersion relation from the stability analysis for axisymmetric disturbances in terms of a set of dimensionless numbers that arise in this phase change problem. For the film boiling condition, plots depicting the effect of some of these parameters on the maximum growth rate for unstable perturbations and critical wavenumber for marginal stability are presented and interpreted. Viscous effects of a purely irrotational motion in the presence of heat and mass transfer can stabilize an otherwise unstable gas–liquid interface.

Numerical and analytical studies of non-linear gravity capillary waves in fluid layers under normal electric fields

Non-linear gravity-capillary waves travelling at constant speed are considered in the presence of a normal electric field. The fluid, which is assumed to be inviscid, irrotational and a perfect dielectric, is bounded below by a solid plate electrode held at constant voltage, and the region above the free surface is a hydrodynamically passive perfect dielectric, e.g. air. A second parallel flat plate electrode is placed laterally far away and drives a uniform normal electric field there. Electrohydrodynamic coupling occurs at the free surface through the Maxwell stresses which act to modify the normal stress balance and consequently the Bernoulli equation boundary condition there. Three harmonic problems in deforming domains need to be solved, one for the hydrodynamics and one each for the electrostatics above and below the free surface, respectively. We derive and implement an accurate boundary integral method to compute travelling waves of arbitrary wavelength and amplitude. In addition, we consider a long-wave non-linear model of the full problem and compare solutions with the direct simulations. In both cases, we establish the existence of multiple families of solutions extending the classical theory of gravity-capillary waves. An asymptotic theory is also developed to construct periodic waves with ripples. These are used in comparisons with the numerical calculations and the agreement is very good.

On the evolution of two-dimensional patterns in an inviscid liquid sheet

We perform an analysis of the pattern formation for a moving sheet of inviscid fluid. The sheet, which is assumed to have an infinite horizontal extent, moves at some prescribed velocity into a passive surrounding gas. The sheet’s thickness is assumed much smaller than the horizontal scale of the fluid motion. By considering a system that is symmetric with respect to the horizontal planes, long scale asymptotics are used to reduce the full governing equations in three dimensions to a set of three coupled nonlinear partial differential equations for the horizontal components of the velocity field and the height of the interface profile. The interfacial conditions consisting of the kinematic and normal stress balance are incorporated into these evolution equations. Investigations are carried out as function of the sole dimensionless parameter, namely the Weber number. A small amplitude stability analysis around the planar gas–liquid interface reveals that wave patterns in the form of traveling plane waves occur subcritically, and are therefore unstable. The reduced evolution equations are solved numerically for fixed values of the Weber number. Since the reduced system of equations is homogeneous, the wave motion is generated by initial conditions. Five initial conditions have been imposed: one-dimensional rolls, two-dimensional squares, two-dimensional hexagons, two-dimensional ridges, and smooth peaks. The ensuing evolution of the liquid sheet’s shape and corresponding flow fields are described by illustrations of the changes in the sheet’s morphology with time.

Stokes flow response to a localised change in interfacial tension

The interfacial tension contribution to the normal stress balance leads to a rather complicated low-Reynolds number hydrodynamic response to the imposed surface tension variation. It is shown that the assumptions dictated by the model prevent the steady state of zero response from ever being attained and thereby demand that the forcing be switched off, after which the interface displacement decays algebraically in time with a small amplitude.

Purely irrotational theories of the effect of the viscosity on the decay of free gravity waves

A purely irrotational theory of the effect of viscosity on the decay of free gravity waves is derived and shown to be in excellent agreement with Lamb’s (1932) exact solution. The agreement is achieved for all waves numbers k excluding a small interval around a critical k=kc where progressive waves change to monotonic decay. Very detailed comparisons are made between the purely irrotational and exact theory. 1. Introduction Lamb (1932, §348, §349) performed an analysis of the effect of viscosity on free gravity waves. He computed the decay rate by a dissipation method using the irrotational flow only. He also constructed an exact solution for this problem, which satisfies both the normal and shear stress conditions at the interface. Joseph and Wang (2004) studied Lamb’s problem using the theory of viscous potential flow (VPF) and obtained a dispersion relation which gives rise to both the decay rate and wave-velocity. They also computed a viscous correction for the irrotational pressure and used this pressure correction in the normal stress balance to obtain another dispersion relation. This method is called a viscous correction of the viscous potential flow (VCVPF). Since VCVPF is an irrotational theory the shear stress cannot be made to vanish. However, the corrected pressure eliminates this uncompensated shear stress from the power of traction integral arising in an energy analysis of the irrotational flow. Here we find that the viscous pressure correction of the irrotational motion gives rise to a higher order irrotational correction to the irrotational velocity which is proportional to the viscosity and does not have a boundary layer structure. The corrected velocity depends strongly on viscosity and is not related to vorticity; the whole package is purely irrotational. The corrected irrotational flow gives rise to a dispersion relation which is in splendid agreement with Lamb’s exact solution, which has no explicit viscous pressure. The agreement with the exact solution holds for fluids even 10 4 times more viscous than water and for small and large wave numbers where the cutoff wave number kc marks the place where progressive waves give rise to monotonic decay. We find that VCVPF gives rise to the same decay rate as in Lamb’s exact solution and in his dissipation calculation when k kc. The effects of vorticity are sensible only in a small interval centered on the cutoff wave number. We will do a comprehensive comparison for the decay rate and wave-velocity given by Lamb’s exact solution and Joseph and Wang’s VPF and VCVPF theories. 2. Irrotational viscous corrections for the potential flow solution The gravity wave problem is governed by the continuity equation 0 u ,

Microscale Heat and Mass Transport of Evaporating Thin Film of Binary Mixture

Analytical and computational studies are presented to examine the effect of binary mixture (pentane/decane) on the microscale heat and mass transport of an evaporating meniscus formed inside a two-dimensional slotted pore. Mass conservation in the liquid film is combined with the momentum equations, energy balance, and normal stress balance and then scaled yielding two constitutive equations: 1) a fourth-order, nonlinear, ordinary differential equation for thin-film profile [Eq. (27)] and 2) a first-order, linear, ordinary differential equation for concentration profile [Eq. (30)]. The numerical results showed that the magnitude of distillation-driven capillary stress due to the composition gradient of a binary mixture can be larger than the thermocapillary stress due to temperature gradient while they are acting in opposite direction. Henceforth, the proof-of-concept has been established in that the binary mixture could facilitate improvement of the evaporating thin-film stability. It was also shown that the resulting stress elongated the length of the evaporating thin-film region without degradation of heat transport effectiveness.

Stability of a liquid jet into incompressible gases and liquids: Part 2. Effects of the irrotational viscous pressure

In this paper we investigate the effects of an irrotational, viscous pressure on the stability of a liquid jet into gases and liquids. The analysis extends our earlier work (part 1) in which the stability of the viscous jet was studied assuming that the motion and pressure are irrotational and the viscosity enters through the jump in the viscous normal stress in the normal stress balance at the interface. The liquid jet is always unstable; at high Weber numbers the instability is dominated by capillary instability; at low W the instability is dominated by Kelvin–Helmholtz (KH) waves generated by pressures driven by the discontinuous velocity. In the irrotational analysis the viscosity is important but the effects of shear are neglected. In fact a discontinuous velocity is not compatible with the continuity of the tangential components of velocity and shear stress so that KH instability is not properly posed for exact study using the no-slip condition but some of the effects of viscosity can be ascertained using viscous potential flow. The theory is called viscous potential flow (VPF). Here we develop another irrotational theory in which the discontinuities in the irrotational tangential velocity and shear stress are eliminated in the global energy balance by selecting viscous contributions to the irrotational pressure. These pressures generate a hierarchy of potential flows in powers of the viscosity, but only the first one, linear in viscosity, in the irrotational viscous stress, is thought to have physical significance. The tangential velocity and shear stress in an irrotational study cannot be made continuous, but the effects of the discontinuous velocity and stress in the mechanical energy balance can be removed “in the mean.” This theory with the additional viscous pressure is called VCVPF, viscous correction of VPF. VCVPF is VPF with the additional pressures. The theory here cannot be compared with an exact solution, which would not allow the discontinuous velocity and stress. In other problems, like capillary instability, in which VCVPF can be compared with an exact solution, the agreements are uniformly excellent in the wave number when one of the fluids is gas and in good but not uniform, agreement when both fluids are liquids.

Viscous contributions to the pressure for potential flow analysis of capillary instability of two viscous fluids

Capillary instability of a liquid cylinder immersed in another liquid is analyzed using viscous potential flow. An effect of viscosity on the irrotational motion may be introduced by evaluating the viscous normal stress at the liquid–liquid interface on the irrotational motions. In a second approximation, the explicit effects of the discontinuity of the shear stress and tangential component of velocity which cannot be resolved pointwise in irrotational flows, can be removed in the mean from the power of traction integrals in the energy equation by the selection of two viscous corrections of the irrotational pressure. The actual resolution of these discontinuities presumably takes place in a boundary layer which is not computed or needed. We include the irrotational stress and pressure correction in the normal stress balance and compare the computed growth rates to the growth rates of the exact viscous flow solution. The agreement is excellent when one of the liquids is a gas; for two viscous liquids, the agreement is good to reasonable for the maximum growth rates but poor for long waves. Calculations show that good agreement is obtained when the vorticity is relatively small or the irrotational part is dominant in the exact viscous solution. We show that the irrotational viscous flow with pressure corrections gives rise to exactly the same dispersion relation as the dissipation method in which no pressure at all is required and the viscous effect is accounted for by evaluating the viscous dissipation using the irrotational flow.

A Hybrid Boundary/Finite Element Method for Simulating Viscous Flows and Shapes of Droplets in Electric Fields

A mixed boundary element and finite element numerical algorithm for the simultaneous prediction of the electric fields, viscous flow fields, thermal fields and surface deformation of electrically conducting droplets in an electrostatic field is described in this paper. The boundary element method is used for the computation of the electric potential distribution. This allows the boundary conditions at infinity to be directly incorporated into the boundary integral formulation, thereby obviating the need for discretization at infinity. The surface deformation is determined by solving the normal stress balance equation using the weighted residuals method. The fluid flow and thermal fields are calculated using the mixed finite element method. The computational algorithm for the simultaneous prediction of surface deformation and fluid flow involves two iterative loops, one for the electric field and surface deformation and the other for the surface tension driven viscous flows. The two loops are coupled through the droplet surface shapes for viscous fluid flow calculations and viscous stresses for updating the droplet shapes. Computing the surface deformation in a separate loop permits the freedom of applying different types of elements without complicating procedures for the internal flow and thermal calculations. Tests indicate that the quadratic, cubic spline and spectral boundary elements all give approximately the same accuracy for free surface calculations; however, the quadratic elements are preferred as they are easier to implement and also require less computing time. Linear elements, however, are less accurate. Numerical simulations are carried out for the simultaneous solution of free surface shapes and internal fluid flow and temperature distributions in droplets in electric fields under both microgravity and earthbound conditions. Results show that laser heating may induce a non-uniform temperature distribution in the droplets. This non-uniform thermal field results in a variation of surface tension along the surface of the droplet, which in turn produces a recirculating fluid flow in the droplet. The viscous stresses cause additional surface deformation by squeezing the surface areas above and below the equator plane.

Steady flow and interfacial shapes of a highly viscous dispersed phase

A perturbation theory for the steady flow of immiscible liquids is developed when the dispersed phase is much more viscous than the continuous phase, as is the case in emulsions of highly viscous bitumen in water and in water lubricated pipelines of heavy crude. The perturbation is nonsingular, but nonstandard; the partitioning of the boundary conditions at different orders is not conventional. At zero-th order the dispersed phase moves as a rigid solid with an as yet unknown, to-be-determined, pressure. The flow of the continuous phase at zero-th order is determined by a Dirichlet problem with prescribed velocities on a to-be-iterated interfacial boundary. The first order problem in the dispersed phase is determined from the solution of a Stokes flow problem driven by the previously determined shear strain on the as yet undetermined interfacial boundary. This Stokes problem determines the unknown, to-be-determined, lowest order pressure distribution. At this point we have enough information to test the balance of normal stresses at lowest order; by iterating the interface shapes we may now complete the description of the lowest order problems. The perturbation sequence in powers of the viscosity ratio has a similar structure at every order and all the problems may be solved sequentially with the caveat that interface shape must be determined iteratively in each perturbation loop. A perturbation solution for the wavy interfacial shapes on core-annular flows of very viscous oil is presented and the results are compared with experiments and a simpler approximation in which the core moves as rigid, but deformable body with no secondary motions. The perturbation theory gives rise to an accurate description of the bamboo waves observed in experiments when the holdup ratio measured in the experiments is assumed in the theoretical calculation. The perturbation solution and the rigid body approximation are in a relatively good agreement with errors of the order 10% in the flow curves and wave shapes; the error is associated with the neglect of the secondary motion in the rigid-deformable core approximation.

General slow viscous flows in a two-fluid system

A general solution of the creeping flow equations suitable for a flow that is bounded by a nondeforming planar interface is presented. New compact representations for the velocity and pressure fields are given in terms of two scalar functions which describe arbitrary Stokes flow. A general reflection theorem is derived for a fluid-fluid interface problem containing Lorentz reflection formula as a particular case. The theorem allows a better interpretation of the image system for various singularities in the presence of a planar interface. The general solution is further used to describe the first-order approximation of the deformed interface by performing normal stress balance. It is found that the normal stress imbalance and the interface displacement are independent of the viscosity ratio of two fluids (!) and only depend on the location of initial singularity.