Normal Viscous Stresses Research Articles

A thermomechanical model of the formation of a rift valley

A numerical model describing the thermomechanical state of the “cold” upper mantle near a mid-oceanic ridge (MOR) spreading at a moderate rate is constructed in the approximation of the boundary layer theory. The condition of rift valley formation leads to a constraint on the temperature and shows what temperature distribution corresponds to the “cold” upper mantle. Taking into account the dependence of mantle rheology on the pressure, temperature, and viscous stresses, the model distributions of the pressure and normal viscous stresses at the base of the lithosphere result in a bend of the heterogeneous lithosphere near the MOR, producing a seafloor topography typical of a rift valley with a depth of a few hundred meters and a spreading rate of ∼2.5 cm/yr, characteristic of the Atlantic Ocean. The model width of the rift valley (∼10–15 km) agrees with observations fairly well. The model is consistent with the typical heat-flow values observed in the spreading zone.

Stretching of a confined ferrofluid: Influence of viscous stresses and magnetic field

An analytical investigation is presented for the stretch flow of a viscous Newtonian ferrofluid highly confined between parallel plates. We focus on the development of interfacial instabilities when the upper plate is lifted at a described rate, under the action of an applied magnetic field. We derive the mode-coupling differential equation for the interface perturbation amplitudes and study both linear and nonlinear flow regimes. In contrast to the great majority of works in stretch flow we take into account stresses originated from velocity gradients normal to the ferrofluid interface. The impact of such normal stresses is accounted for through a modified Young-Laplace pressure jump interfacial boundary condition, which also includes the contribution from magnetic normal traction. We study how the stability properties of the interface and the shape of the emerging patterns respond to the combined action of normal stresses and magnetic field, both in the presence and absence of surface tension. We show that the inclusion of normal viscous stresses introduces a pertinent dependence on the initial aspect ratio, indicating that the number of fingers formed would be overestimated if such stresses are not taken into account. At early linear stages it is found that such stresses regularize the system, acting as an effective interfacial tension. At weakly nonlinear stages we verified that normal stresses reduce finger competition, which can be completely suppressed with the assistance of an azimuthal magnetic field. We have also found that the magnetic normal traction introduces a purely nonlinear contribution to the problem, revealing the key role played by the magnetic susceptibility in the control of finger competition.

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.

Corner flow in free liquid films

A lubrication-flow model for a free film in a corner is presented. The model, written in the hyperbolic coordinate system ξ = x 2 −y 2, η = 2xy, applies to films that are thin in the η-direction. The lubrication approximation yields two coupled evolution equations for the film thickness and the velocity field which, to lowest order, describes plug flow in the hyperbolic coordinates. A free film in a corner evolving under surface tension and gravity is investigated. The rate of thinning of a free film is compared to that of a film evolving over a solid substrate. Viscous shear and normal stresses are both captured in the model and are computed for the entire flow domain. It is shown that normal stress dominates over shear stress in the far field, while shear stress dominates close to the corner.

Non-self-similar, linear dynamics during pinch-off of a hollow annular jet

Based on an experimental and computational study of the breakup of a drop (jet) of small viscosity in an ambient fluid of large viscosity, Doshi et al. [Science 302, 1185 (2003)] have shown that the breakup of a drop (jet) of zero viscosity in a very viscous ambient fluid gives rise to an unexpected, nonuniversal form of singularity. Doshi et al. conjectured that the nonuniversal dynamics result from the fact that stresses exerted by the inner fluid are negligible. To verify this conjecture and overcome computational difficulties associated with simulating systems in which the disparity between the viscosities of the inner and the outer fluids is large, the breakup of an annular jet whose core is a gas of negligible viscosity is analyzed. Calculations show that as the jet’s minimum radius hmin→0, both core- and shell-side pressures remain bounded while surface tension pressure, which diverges as 1/hmin, is balanced by viscous normal stress exerted by the shell fluid. Simulations show that interfacial points move radially inward with the same velocity. Fourier decomposition of interface shapes confirms that the dynamics are linear. As hmin→0, the axial length scale remains finite, its value varying with imposed initial and boundary conditions. Thus, the breakup is not self-similar and the final breakup profile is nonuniversal.

Sheet formation during drop deformation and breakup in polyethylene/polycarbonate systems sheared between parallel plates

A polycarbonate drop was sheared inside a polyethylene matrix in a transparent rotating parallel plate device at 220°C and low shear rates. A flat sheet was formed during the initial shearing of the drop. The drop then developed into either a thin thread or a sheet with a thin cylindrical tip. Sheet formation was found to occur at a critical strain or time. A stress ratio (Sr) between the matrix breakup stress, made up of the matrix normal stress and viscous stress, and the drop restoring stress, made up of the drop normal stress and the interfacial stress, is used to characterize the sheet formation during the drop deformation and breakup process. It was found that the viscosity ratio (ηr), stress ratio (Sr) and Deborah number (De) of the system could be used to predict the drop deformation and breakup.

Dynamic free-surface deformations in axisymmetric liquid bridges

Thermocapillary convection in a differentially heated liquid bridge is investigated in a two-dimensional numerical study. The deformable free surface is obtained as a solution of the coupled transport equations at fixed unit Prandtl number and aspect ratio. Only steady convection can be realized in this axisymmetric computation with either non-deformable or deformable surfaces. Dynamic free-surface deformations do not induce transitions to oscillatory convection even at large Reynolds numbers. Free surfaces are convex near the cold wall due to large surface pressures at the stagnation-point. Near the hot wall they change from concave to convex with increasing Re due to large normal viscous stresses. Large Capillary numbers cause additional surface ripples there while having little effect on the interior. Heat loss from the free surface has dramatic influence on the dynamics and must be included in theoretical models of thermocapillary instabilities.

Numerical Simulation of the Buoyancy-Driven Bouncing of a 2-D Bubble at a Horizontal Wall

The rise of a buoyant bubble and its interaction with a target horizontal wall is simulated with a 2-D numerical code based on the Boundary Element Method (BEM). Developed from a viscous potential flow approximation, the BEM takes into account only the part of the energy dissipation related to the normal viscous stresses. Hence, a simple analytical model based on lubrication approximation is coupled to the BEM in order to compute the drainage of the interstitial liquid film filling the gap between the bubble and the near wall. In this way the bubble–wall interaction is fully computed: the approach stage, the bubble deformation stage and, depending on the values of the Reynolds number and the Weber number, the rebound and the bubble oscillations. From computation of both the bubble interface motion and the liquid velocity field, a physical analysis in terms of energy budget is proposed. Though, in the present study, the bubble under consideration is basically supposed to be a 2-D gaseous cylinder, a comparison between our numerical results and the experiments of Tsao and Koch (1997) enlightens interestingly the physics of bouncing.

Expansion of a compressible gas bubble in Stokes flow

The flow-induced deformation of an inviscid bubble occupied by a compressible gas and suspended in an ambient viscous liquid is considered at low Reynolds numbers with particular reference to the pressure developing inside the bubble. Ambient fluid motion alters the bubble pressure with respect to that established in the quiescent state, and requires the bubble to expand or contract according to an assumed equation of state. When changes in the bubble volume are prohibited by a global constraint on the total volume of the flow, the ambient pressure is modified while the bubble pressure remains constant during the deformation. A numerical method is developed for evaluating the pressure inside a two-dimensional bubble in an ambient Stokes flow on the basis of the normal component of the interfacial force balance involving the capillary pressure, the normal viscous stress, and the pressure at the free surface on the side of the liquid; the last is computed by evaluating a strongly singular integral. Dynamical simulations of bubble deformation are performed using the boundary integral method properly implemented to remove the multiplicity of solutions due to the a priori unknown rate of expansion, and three particular problems are discussed in detail: the shrinkage of a bubble at a specified rate, the deformation of a bubble subject to simple shear flow, and the deformation of a bubble subject to a purely elongational flow. In the case of shrinkage, it is found that the surface tension plays a critical role in determining the behaviour of the bubble pressure near the critical time when the bubble disappears. In the case of shear or elongational flow, it is found that the bubble contracts during an initial period of deformation from the circular shape, and then it expands to obtain a stationary shape whose area is higher than that assumed in the quiescent state. Expansion may destabilize the bubble by raising the capillary number above the critical threshold under which stationary shapes can be found.

Numerical simulation of the dynamics of acoustically levitated drops

Numerical simulation of the dynamics of acoustically levitated drops is accomplished by solving the Helmholtz equation governing the acoustic scattering problem and the Laplace equation governing the motion of an inviscid and incompressible drop. The boundary element method with quadratic elements is used to solve the surface integral forms of both the Helmholtz equation and the Laplace equation simultaneously. Boundary conditions on the drop surface render a system of first-order differential equations which is integrated with respect to time by the use of fourth-order Runge-Kutta integration to study the drop translational frequency, shape oscillation frequency, and the static equilibrium shape. A spherical shape approximation method is used to obtain an analytic estimate of the minimum trapping pressure (MTP) and the translational frequency of the drop. In particular, the translational frequency is shown to agree with the analytic result obtained from the spherical shape approximation for a small acoustic number or a small wave number. The shape oscillation frequencies are found to increase initially with the increasing acoustic number, D [D=(a0Prms2)/(ρoc2σ), Prms is the root-mean-squared sound intensity, a0 is the equivalent radius of drop, ρo is the density of host medium, c is the sound speed in host medium, σ is the surface tension of the drop liquid] up to a critical value and then decreases with the increasing acoustic number due to the nonlinear nature of the acoustic radiation pressure. The equilibrium shape of the drop is computed by incorporating the normal viscous stress terms into the dynamical system intentionally to get rid of the transients. The equilibrium shapes are found to agree with the existing result for small acoustic numbers but differ for large acoustic numbers.

Extrudate Swelling: Physics, Models, and Computations

Viscous, viscoelastic, or elastic normal stresses are superimposed to pressure within flowing fluids. These stresses act normal to the boundaries of the flow that may deform depending on their modulus or viscosity. At absolutely rigid boundaries of infinite modulus of elasticity any boundary deformation and therefore any fluid expansion or swelling is surpressed (eg, flow in rigid pipes, annuli, channels). Elastic boundaries (eg, flow in veins and arteries, flow by membranes, around inflating/deflating balloons) deform under the action of normal stresses, allowing expansion or swelling of fluid. The same mechanism prevails in lubrication, where pressure and superimposed normal viscoelastic stresses keep surfaces in relative motion apart, with simultaneous increase in load capacity. Viscous boundaries (eg, liquid jet in air or in immiscible liquid, slow extrusion of viscoelastic liquids from dies, expanding/collapsing air-bubbles or liquid-droplets) are displaced by flowing adjacent immiscible fluids, allowing swelling or imposing contraction depending on relative rheological characteristics. Thus, the kind of swelling examined here is independent of density, ie, incompressible, and is due to the action of normal stresses against the boundary that is imposed either by adjacent deformable obstacles or else by surface tension. The resulting swelling is dynamic (ie, it initiates, changes and ceases with the flow) and can be made permanent by solidification, crystallization or glassification. The most profound form of incompressible swelling is the extrude swelling that controls the ultimate shape of extruded parts. Incompressible swelling is enhanced by the ability of macromolecules to deform and recover (eg, viscoelastic) and by the design of flow conduits to impose sharp transitions of deformation modes (eg, singular exit flows). The same swelling is reduced by the ability of molecules (or fibers in fiber-suspensions) to align with the flow streamines, as well as any tendency of solid-like structure formulation (eg, viscoplastic).

Effects of Viscous Normal Stresses in Thin Draining Films

The hydrodynamics of film drainage is normally treated using lubrication theory. Bird et al. suggested an alternative approach. Here, we compare these two approaches and find that lubrication theory neglects an edge effect. When the surfaces of the film are mobile, there is an additional normal force. Though typically negligible, in some cases this normal force may have an effect.

Temperature Distribution and Thermal Degradation of the Lubricant in EHL Line Contact Conjunctions

Temperatures of the lubricant in the inlet zone, the central contact zone, and the outlet zone of EHL line contact conjunctions were simultaneously calculated for a complete range of sliding conditions by using the new thermal circular non-Newtonian EHL model. The complete range of sliding conditions includes pure rolling, normal sliding, simple sliding, and opposite sliding. The result reveals the strong effect of sliding condition on the magnitude and distribution of the lubricant temperature. It also shows the diffusion time effect on the lubricant temperature, which builds up near the slower moving solid surface in the inlet zone. The effect of viscous normal stress on the temperature distribution was also investigated. Results indicate that the effect is not significant except for applications at very high entraining velocities. In addition, thermal degradation of the lubricant was studied. Results show that it may become a problem for applications at high speeds and high sliding conditions.

Wave- and Wind-Driven Flow in Water of Finite Depth

Abstract The authors first derive both Coriolis-induced and viscosity-induced stresses for arbitrary water depth and arbitrary wave direction. Opportunity is taken here to succinctly and rigorously derive the Longuet-Higgins virtual tangential stress due to wave motion. It is shown that the virtual stress is a projection on the surface slope of two viscous normal stresses acting on the vertical and horizontal planes. Then a simple Eulerian model is presented for the steady flow driven by waves and by waves and winds This simple Eulerian model demonstrates that the wave forcing can he easily incorporated with other conventional forcing, rather than resorting to a complicated and lengthy perturbation analysis of the Lagrangian equations of motion. A further focus is given to the wave-driven flow when the various limits of the wave-driven steady flow are discussed. The wave-driven steady flow given by the model yields a unified formula between Ursell and Hasselmann's inviscid but rotational theory and the Lo...

Finite‐element simulation of stratified multiphase flows

AbstractThe analysis of stratified multiphase flow fields is difficult because the position of the interface is unknown a priori and there is a discontinuity in the normal viscous stress and/or pressure at the interface. A finite‐element technique that uses double nodes along the interface has been implemented. The immiscible liquid displacement in a capillary tube was examined in detail. Fountain flow in the advancing fluid, reverse fountain flow in the receding fluid, and a recirculating flow region in the less viscous fluid were determined. In Newtonian flat film coextrusion the entry region, where the two immiscible fluids form an interface, and the characteristic swelling and bending of the extrudate at the die exit were studied. The computational results compare favorably with available experimental observations.

Analysis of lubricated planar stagnation die flow

Lubricated die flows with a low-viscosity liquid separating a viscous material from the solid walls have been proposed for extensional rheometers. Steady, two-dimensional flow through a lubricated planar stagnation die is analyzed here by Galerkin's method of weighted residuals with finite element basis functions on streamlined subdomains. For the more viscous liquid, the shear-thinning Carreau model is chosen. Alternatives for modelling the Newtonian lubricant are two-dimensional flow analysis, lubrication approximation, and Navier's slip boundary condition: all are examined. An approximate analytical solution suggested by the computations is also presented. Portraits are shown of lubricant layer thickness and pressure velocity, deformation rate, γ, and vorticity, ω, fields. From the last two, Giesekus's local flow classification, Φ =(γ − ω)(γ + ω), is evaluated. Both the approximate solution and the theoretical calculations are compared to available experimental data. The approximate analytical solution leads to a new relation between the extensional viscosity and the pressure profile in the die. Pure planar extension throughout the die is prevented by the interaction of the die shape with the normal viscous stresses at the free interface. The sign change of the normal viscous stresses as the flow switches from a diverging to a converging channel is the key feature. However, two regions of the flow are nearly extensional and can be analyzed with the analytic approximation. Implications for extensional rheometry and design of lubricated flows are discussed.

Visco-elasticity under large, uniform deformations of rubber-like polymers in the transition range

The relaxation moduli of crosslinked polymers as a function of the main elongation ratios have been calculated on the basis of a non-Gaussian sub-chains model. The effect of a mobility reduction of the sub-chains as a result of elongation has been examined on freely jointed chain models and those having a weak correlation in orientationally adjacent chain units. The movements are thought to be instant rotations by 180° of an adjacent chain unit pair. The effect of a chain rigidity increase in the case of the freely jointed chain model in fully compensated by a reduction in movement as regards the relaxation time, while the existence of a correlation reduces the latter. A reduction of the sub-chain movements due to elongation makes stronger the dependence of the relaxation functions only on the elongation ratio. The simple elongations, uniform expansion, pure shear at constant rate of natural deformations, and also pure shear at constant rate, were examined for constant relaxation times. The finite extensibility of the chains results in a typical increase of the viscous stress in the network and the appearance of normal viscous stresses in simple shear.

Newtonian and non-Newtonian liquids rotating adjacent to a stationary surface

The interaction of a rotating flow and a stationary surface is discussed for a second-order non-Newtonian liquid. Similarity solutions of the governing partial differential equations are obtained for the case of the outer flow in solid-body rotation. The results for the Newtonian case are compared with Bodewadt's series solution of this problem. The non-Newtonian solutions indicate that for certain values of the parameters characterizing the non-linear viscous response and normal stress effects a larger secondary flow is induced in the boundary layer than in the Newtonian case.

Experiments on hydrodynamically developing flow in rectangular ducts of arbitrary aspect ratio

A comprehensive experimental investigation of laminar flow development in rectangular ducts is performed employing an apparatus which permits study of any cross sectional aspect ratio. The primary concern of the research is the determination of the pressure field associated with the hydrodynamic development of the flow. Experimental results are presented in terms of both the local pressure defect and the incremental pressure drop due to flow development. Comparisons are made with the available predictions of analysis and with prior experimental information. Hydrodynamic development lengths are deduced and the threshold of significant effects of the viscous normal stress is identified.