Normal Stress Balance Research Articles

Study on Capillary Instability With Heat and Mass Transfer Through Porous Media: Effect of Irrotational Viscous Pressure

In this paper, we investigate the effect of irrotational, viscous pressure on capillary instability of the interface between two viscous, incompressible, and thermally conducting fluids in a fully saturated porous medium when the phases are enclosed between two horizontal cylindrical surfaces coaxial with the interface and when there is mass and heat transfer across the interface. The analysis extends our earlier work in which the capillary instability of two viscous and thermally conducting fluids in a fully saturated porous medium 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. Here, we use another irrotational theory in which the discontinuities in the irrotational tangential velocity and shear stress are eliminated in the global energy balance by taking viscous contributions to the irrotational pressure. We use the Darcy's model, and a quadratic dispersion relation is obtained. It is observed that heat and mass transfer has a stabilizing effect on the stability of the system and this effect enhances in the presence of irrotational viscous pressure.

STUDY ON ELECTROHYDRODYNAMIC CAPILLARY INSTABILITY OF VISCOELASTIC FLUIDS WITH RADIAL ELECTRIC FIELD

We study the linear analysis of electrohydrodynamic capillary instability of the interface between a viscous fluid and viscoelastic fluid of Maxwell type, when the phases are enclosed between two horizontal cylindrical surfaces coaxial with the interface, and when fluids are subjected to the radial electric field. Here, we use an irrotational theory known as viscous potential flow (VPF) theory in which viscosity enters through normal stress balance but shearing stresses are assumed to be zero. A quadratic dispersion relation that accounts for the growth of axisymmetric waves is obtained and stability criterion is given in terms of a critical value of wave number as well as electric field. It is observed that the radial electric field has dual effect on the stability of the system.

Free surface due to a flow driven by a rotating disk inside a vertical cylindrical tank: Axisymmetric configuration

The flow driven by a rotating disk at the bottom of an open fixed cylindrical cavity is studied numerically and experimentally. The steady axisymmetric Navier-Stokes equations projected onto a curvilinear coordinate system are solved by a Newton-Raphson algorithm. The free surface shape is computed by an iterative process in order to satisfy a zero normal stress balance at the interface. In previous studies, regarding the free surface deflection, there is a significant disagreement between a first-order approximation [M. Piva and E. Meiburg, “Steady axisymmetric flow in an open cylindrical container with a partially rotating bottom wall,” Phys. Fluids 17, 063603 (2005)] and a full numerical simulation [R. Bouffanais and D. Lo Jacono, “Unsteady transitional swirling flow in the presence of a moving free surface,” Phys. Fluids 21, 064107 (2009)]. For a small deflection, the first-order approximation matches with our numerical simulation and for a large deflection a good agreement is found with experimental measurements.

Evaporative capillary instability for flow in porous media under the influence of axial electric field

We study the linear analysis of electrohydrodynamic capillary instability of the interface between two viscous, incompressible and electrically conducting fluids in a fully saturated porous medium, when the phases are enclosed between two horizontal cylindrical surfaces coaxial with the interface and, when there is mass and heat transfer across the interface. The fluids are subjected to a constant electric field in the axial direction. Here, we use an irrotational theory in which 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. A quadratic dispersion relation that accounts for the growth of axisymmetric waves is obtained and stability criterion is given in terms of a critical value of wave number as well as electric field. It is observed that heat transfer has stabilizing effect on the stability of the considered system while medium porosity destabilizes the interface. The axial electric field has dual effect on the stability analysis.

Weakly nonlinear analysis on radially growing interface with the effect of viscous normal stress

Two-phase Hele-Shaw problem is studied as a model for a two-dimensional radially growing interface. Instead of the Young–Laplace equation, which has been employed in almost all previous studies concerning Hele-Shaw problem, we apply the normal stress balance and derive the new mode coupling equation for perturbation of the interface. In the present research, weakly nonlinear analysis is carried out and time evolution of the interfaces is numerically calculated. These numerical results suggest that our model reflects more exactly the nonlinear features of a radially growing interface, rather than the previous ones based on the Young–Laplace equation.

Three-dimensional magnetohydrodynamic Kelvin–Helmholtz instability of cylindrical flow with permeable boundaries

We study the linear magnetohydrodynamic Kelvin–Helmholtz instability of the interface between two viscous, incompressible, and electrically conducting fluids. The phases are enclosed between two coaxial cylindrical porous layers with the interface through which mass and heat transfer takes place. The fluids are subjected to a constant magnetic field parallel to the streaming direction, and the suction/injection velocities for the fluids at the permeable boundaries are also taken into account. Here, we use an irrotational theory in which 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. We consider both asymmetric and axisymmetric disturbances in our analysis. A quadratic dispersion relation is deduced and stability criterion is given in terms of a critical value of relative velocity, as well as, magnetic field. It has been observed that in the case of permeable boundaries, heat and mass transfer phenomena play a dual in the stability analysis. The flow through porous medium is more stable than the pure flow.

Viscous potential flow analysis of magnetohydrodynamic Rayleigh–Taylor instability with heat and mass transfer

The effect of tangential magnetic field on the linear analysis of Rayleigh–Taylor instability of two viscous, incompressible and electrically conducting fluids is studied when there is heat and mass transfer across the interface. We use an irrotational theory known as viscous potential flow theory; in which viscosity enters through normal stress balance but shearing stresses are assumed to be zero. A quadratic dispersion relation that accounts for the growth of disturbance waves is obtained and stability criterion is given in terms of a critical value of wave number as well as applied magnetic field. We observe that heat and mass transfer and magnetic field both stabilize the interface while vapour thickness has destabilizing effect.

Study on Electrohydrodynamic Capillary Instability with Heat and Mass Transfer

The effect of an axial electric field on the capillary instability of a cylindrical interface in the presence of heat and mass transfer has been investigated using viscous potential flow theory. 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 the viscous potential flow theory and tangential stresses are not considered. A dispersion relation that accounts for the growth of axisymmetric waves is derived and stability is discussed theoretically as well as numerically. Stability criterion is given by critical value of applied electric field as well as critical wave number. Various graphs have been drawn to show the effect of various physical parameters such as electric field, heat transfer capillary number, conductivity ratio, permittivity ratio on the stability of the system. It has been observed that the axial electric field and heat and mass transfer both have stabilizing effect on the stability of the system. Keywords—Capillary instability, Viscous potential flow, Heat and mass transfer, Axial electric field.

Viscous Corrections for the Viscous Potential Flow Analysis of Rayleigh–Taylor Instability With Heat and Mass Transfer

Viscous corrections for the viscous potential flow analysis of Rayleigh–Taylor instability of two viscous fluids when there is heat and mass transfer across the interface have been considered. Both fluids are taken as incompressible and viscous with different kinematic viscosities. In viscous potential flow theory, viscosity enters through a normal stress balance and the effects of shearing stresses are completely neglected. We include the viscous pressure in the normal stress balance along with irrotational pressure and it is assumed that this viscous pressure will resolve the discontinuity of the tangential stresses at the interface of the two fluids. It has been observed that heat and mass transfer has a stabilizing effect on the stability of the system. It has been shown that the irrotational viscous flow with viscous corrections gives rise to exactly the same dispersion relation as the dissipation method in which no pressure term is required and the viscous effect is accounted for by evaluating viscous dissipation using irrotational flow. It has been observed that the inclusion of irrotational shearing stresses has a stabilizing effect on the stability of the system.

Study on hydro-magnetic capillary instability with mass transfer through porous media

The linear analysis of hydro-magnetic capillary instability of a cylindrical interface between two viscous and magnetic fluids in a fully saturated porous medium has been carried out, when the fluids are subjected to a constant axial magnetic field and, when there is heat and mass transfer across the interface. The viscous potential flow theory has been used for the investigation. Viscosity enters through normal stress balance in the viscous potential flow theory and tangential stresses are not considered. A dispersion relation that accounts for the growth of axisymmetric waves is derived and stability is discussed theoretically as well as numerically. Stability criterion is given by critical value of applied magnetic field as well as critical wave number. Various graphs have been drawn to show the effect of various physical parameters such as magnetic field strength, heat transfer capillary number, vapour fraction, permeability and porosity on the stability of the system. It has been observed that heat transfer and magnetic field both have stabilizing effect while porosity has destabilizing effect on the stability of the system.

Pressure corrections for viscous potential flow analysis of radial fingering in Hele-Shaw cell

Pressure corrections for the viscous potential flow analysis of radial fingering in two-phase gas/liquid flow in a Hele-Shaw cell were determined. In viscous potential flow theory, viscosity enters through the normal stress balance and the effect of shearing stresses is completely neglected. We have included viscous pressure in the normal stress balance, along with irrotational pressure, and it is assumed that this viscous pressure will resolve the discontinuity of the shearing stresses at the interface of two fluids. Both fluids are taken as viscous and incompressible with different mobilities. A dispersion relation was obtained, and stability is discussed both theoretically as well as numerically. It was observed that the effect of irrotational shearing stresses stabilizes the system.

Viscous correction for the viscous potential flow analysis of capillary instability with heat and mass transfer

The linear analysis of capillary instability of the cylindrical interface between two viscous, incompressible fluids confined in a concentric annulus in the presence of heat and mass transfer is carried out. The viscous correction for the viscous potential flow theory is used for the investigation. Both fluids are taken as incompressible and viscous with different kinematic viscosities. To study the effect of shearing stresses, viscous pressure in the normal stress balance is included along with irrotational pressure. It is observed that heat and mass transfer has a stabilizing effect on the stability of the system and this effect enhances in the presence of irrotational shearing stresses. A comparison between viscous potential flow (VPF) solution and viscous correction for the viscous potential flow (VCVPF) is made and it is found that the VCVPF solution is more stable than the VPF solution.

PRESSURE CORRECTIONS FOR THE VISCOELASTIC POTENTIAL FLOW ANALYSIS OF ELECTROHYDRODYNAMIC CAPILLARY INSTABILITY

Pressure corrections for the viscoelastic potential flow analysis of capillary instability in the presence of axial electric field has been carried out. In viscoelastic potential flow theory, viscosity enters through normal stress balance and the effects of shearing stresses are completely neglected. We include the viscous pressure in the normal stress balance along with irrotational pressure and it is assumed that this viscous pressure will resolve the discontinuity of the tangential stresses at the interface for two fluids. A dispersion relation is derived for the case of axially imposed electric field and stability is discussed in terms of various parameters such as electric field, Deborah number, Ohnesorge number, permittivity ratio and conductivity ratio etc. Stability criterion is given in terms of critical value of wave number as well as critical value of applied electric field. The system is unstable when electric field is less than the critical value of electric field, otherwise it is stable. It has been found that irrotational shearing stresses have stabilizing effect on the stability of the system.

Viscous corrections for the viscous potential flow analysis of magnetohydrodynamic Kelvin-Helmholtz instability with heat and mass transfer

Viscous corrections for the viscous potential flow analysis of Kelvin-Helmholtz instability at the interface of two viscous fluids in the presence of a tangential magnetic field has been carried out when there is heat and mass transfer across the interface. Both fluids are taken as incompressible, viscous and magnetic with different kinematic viscosities and different magnetic permeabilities. In the viscous potential flow theory, viscosity enters through the normal stress balance whereas the effect of shearing stresses is completely neglected. We include the viscous pressure in the normal stress balance along with the irrotational pressure and it is assumed that this viscous pressure will resolve the discontinuity of the tangential stresses at the interface for two fluids. A dispersion relation has been derived and stability is discussed theoretically as well as numerically. A stability criterion is given in terms of a critical value of relative velocity as well as a critical value of applied magnetic field. It has been observed that the tangential magnetic field and the vapor fraction both have stabilizing effect on the stability of the system while heat and mass transfer destabilize the interface. Also, it has been found that the effect of irrotational shearing stresses stabilizes the system.

Instability of pressure driven viscous fluid streams in a microchannel under a normal electric field

This paper investigates analytically and experimentally electrohydrodynamic instability of the interface between two viscous fluids with different electrical properties under constant flow rates in a microchannel. In the three-dimensional analytical model, the two-layer system is subjected to an electric field normal to the interface between the two fluids. There is no assumption on the magnitude of the ratio of fluid to electric time scales, and thus the linear Poisson–Boltzmann equation are solved using separation of variable method for densities of bulk charge and surface charge. The electric field and fluid dynamics are coupled only at the interface through the tangential and normal interfacial stress balance equations. In the experiments, two immiscible fluids, aqueous NaHCO3 (the high electrical mobility fluid) and silicone oil (polydimethylsiloxane, the low electrical mobility fluid) are pumped into a microchannel made in polymethyl methacrylate) (PMMA) substrate. The normal electric field is added using a high voltage power supply. The results showed that the external electric field and increasing width of microchannel destabilize the interface between the immiscible fluids. At the same time, the viscosity of the high electrical mobility fluid and flow rates of fluids has a stabilizing effect. The experimental results and the analytical results show a reasonable agreement.

The effect of viscoelasticity on the dynamics of two gas bubbles near a rigid boundary

Multi-bubble dynamics is investigated in the presence of a rigid boundary. A simplified configuration is considered in which two bubbles, not necessarily of the same size, are positioned along an axis of symmetry. The problem is formulated in terms of a generalized Bernoulli equation for a velocity potential. A boundary element method is adopted to determine the normal velocity at the bubble surface. Viscoelastic effects are incorporated into the model through the normal stress balance across the bubble surface. The model predicts that bubble dynamics, and jet formation, in particular, is strongly influenced by the initial bubble sizes, the initial stand-off distances and on the fluid parameters. Viscoelastic effects are shown to be responsible for some interesting dynamics. Although inter-bubble interaction is shown to play an important role, it is not always dominant for viscoelastic fluids.

VISCOUS CONTRIBUTIONS TO THE PRESSURE FOR THE POTENTIAL FLOW ANALYSIS OF MAGNETOHYDRODYNAMIC KELVIN–HELMHOLTZ INSTABILITY

The present paper deals with the study of viscous contributions to the pressure for the viscous potential flow analysis of Kelvin–Helmholtz instability with tangential magnetic field at the interface 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 magnetohydrodynamic Kelvin–Helmholtz instability. A dispersion relation is obtained and stability criterion is given in the terms of critical value of relative velocity. It has been observed that the inclusion of irrotational shear stresses have stabilizing effect on the stability of the system.

Pressure corrections for the potential flow analysis of Kelvin–Helmholtz instability with heat and mass transfer

Pressure corrections for the viscous potential flow analysis of Kelvin–Helmholtz instability at the interface of two viscous fluids have been carried out when there is heat and mass transfer across the interface. Both fluids are taken as incompressible and viscous with different kinematic viscosities. In viscous potential flow theory, viscosity enters through normal stress balance and effect of shearing stresses is completely neglected. We include the viscous pressure in the normal stress balance along with irrotational pressure and it is assumed that this viscous pressure will resolve the discontinuity of the tangential stresses at the interface for two fluids. It has been observed that heat and mass transfer has destabilizing effect on the stability of the system. A comparison between viscous potential flow (VPF) solution and viscous contribution to the pressure for potential flow (VCVPF) solution has been made and it is found that the effect of irrotational shearing stresses stabilizes the system.

Viscous potential flow analysis of capillary instability with radial electric field

A linear analysis of capillary instability with radial electric field is carried out using viscous potential flow theory. In viscous potential flow, the viscous term in the 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 the tangential stresses are not considered. The effect of gravity and free surface charges at the interface are neglected. A dispersion relation is derived for the case of radially imposed electric field and stability is discussed in terms of various parameters such as Ohnesorge number, permittivity ratio, etc. A condition for neutral stability is obtained and it is given in terms of critical value of electric field. It is observed that the radial electric field has dual effect on the stability of the system corresponding to the values of conductivities and permittivities of the fluids.

Viscous irrotational analysis of the deformation and break-up time of a bubble or drop in uniaxial straining flow

AbstractThe nonlinear deformation and break-up of a bubble or drop immersed in a uniaxial extensional flow of an incompressible viscous fluid is analysed by means of viscous potential flow. In this approximation, the flow field is irrotational and viscosity enters through the balance of normal stresses at the interface. The governing equations are solved numerically to track the motion of the interface by coupling a boundary-element method with a time-integration routine. When break-up occurs, the break-up time computed here is compared with results obtained elsewhere from numerical simulations of the Navier–Stokes equations (Revuelta, Rodríguez-Rodríguez & Martínez-BazánJ. Fluid Mech., vol. 551, 2006, p. 175), which thus keeps vorticity in the analysis, for several combinations of the relevant dimensionless parameters of the problem. For the bubble, for Weber numbers$3\leqslant \mathit{We}\leqslant 6$, predictions from viscous potential flow shows good agreement with the results from the Navier–Stokes equations for the bubble break-up time, whereas for larger$\mathit{We}$, the former underpredicts the results given by the latter. When viscosity is included, larger break-up times are predicted with respect to the inviscid case for the same$\mathit{We}$. For the drop, and considering moderate Reynolds numbers,$\mathit{Re}$, increasing the viscous effects of the irrotational motion produces large, elongated drops that take longer to break up in comparison with results for inviscid fluids. For larger$\mathit{Re}$, it comes as a surprise that break-up times smaller than the inviscid limit are obtained. Unfortunately, results from numerical analyses of the incompressible, unsteady Navier–Stokes equations for the case of a drop have not been presented in the literature, to the best of the authors’ knowledge; hence, comparison with the viscous irrotational analysis is not possible.