Inertialess Flow Research Articles

An Investigation of a Squeeze Film Between Two Plane Annuli

An analysis of a laminar squeeze flow of an incompressible Newtonian fluid between parallel plane annuli is presented. The local and convective inertia of the flow are considered in the investigation. The solution is obtained as a power series in a single nondimensional parameter (squeeze Reynolds number) S, for small values of S. Expressions for the pressure and load capacity are given and are compared with those based on the assumption of inertialess flow. As applications, the constant velocity squeezing state and the case when the walls perform reciprocating harmonic oscillations with finite amplitude are considered. The changes in load capacity with the changes in the parameters that govern the motion have been investigated.

Visco-resistive magnetic reconnection due to steady inertialess flows. Part 1. Exact analytical solutions

Magnetic reconnection at an X-type neutral point of a two-dimensional magnetic field is studied in an incompressible viscous resistive fluid whose flow is assumed to be slow enough that its inertia is negligible. In both the ideal and resistive magnetohydrodynamic approximations current singularities appear at the X-point and along the separatrices. It is shown here analytically that the combined effect of viscosity and resistivity can resolve these singularities with the flow crossing the separatrices. A wide class of exact solutions describing the structure of the flow and current density distribution is found. The results suggest that reconnection may occur with localized distributions of strong current density restricted to finite regions.

Lubrication theory for a fibre suspension: Part 1: pressure-driven flow in a planar channel having slowly-varying cross-section

The steady inertialess flow of a semi-dilute fibre suspension through a planar channel having slowly varying cross-section is considered. For large enough fibre concentrations, flow separation occurs at the wall near the widest part of the channel, and regions of asymptotically stagnant fluid develop over a significant fraction of the channel. These stagnant zones contain very thin sub-layers which are difficult to resolve numerically. Lubrication theory breaks down when flow separation occurs. A new theory is developed which enables the stagnant regions to be removed from the numerical computation, with jump conditions that stitch together the flowing regions on either side. The analysis predicts a region of very weak reversed flow near the wall and a stagnant layer which, in the lubrication limit extends to the wall. The lubrication results are compared with a full numerical solution.

Fluid Inertia Effects on the Performance of Short and Long Squeeze Film Dampers Executing Periodic Vibration

A comprehensive study is made to examine effects of fluid inertia on pressure distribution, load capacity, wall shear stress differences (defined in Nomenclature), and velocity variation of flow in short and long Squeeze Film Dampers (SFDs). The SFD is assumed to execute a small excursion around an arbitrary static position. Exact solutions, in the form of a Fourier series, for fluid pressure and velocity are obtained for periodic motions of the SFD journal. An example of a horizontal motion, with various static positions in bearing clearance, is studied in detail for both short and long bearing configurations. It is found that the existence of fluid inertia generally increases the peak pressure value, and hence the load capacity. Wall shear stress differences and velocity distribution are also altered by the presence of the fluid inertia compared with inertialess flow, but the parabolic shape of the velocity may be maintained. Insight on how the fluid inertia effect is internally related to the viscous effects is also gained from this study.

A unified treatment of the direct contact melting processes in several geometric cases

This paper presents a theoretical study of direct contact melting processes for three geometries: the flat plate, the cylindrical wall and the spherical wall. Governing equations that include both fluid inertia terms and heat convection terms are solved analytically and numerically. It is shown that transport for the three geometries is essentially governed by one common equation system, and that the results for all three cases can be expressed as one basic solution for inertialess flow and as a perturbation solution for the full equations with inertia. Using a similarity transformation, the equations for inertialess flow are reduced to a system of ordinary differential equations in one independent variable. It is shown that there exists an exact solution to this system, and it can be obtained without using any approximate method or numerical technique. Comparison of the numerical and perturbation solutions with inertia to the exact solution results for inertialess flow indicates that the effect of the inertia terms is so small that they can be neglected over almost the entire practical range of Prandtl number.

Inertial effects in electromagnetically driven liquid-metal flows with a periodically wavy boundary

We treat a liquid-metal flow in a gap between two solid perfect conductors with different voltages and with an externally applied, transverse magnetic field. One conductor has a plane surface and the other has a periodically wavy surface. There is a free surface and an electrically insulating, inviscid gas on each side of the conductor with the plane surface. Most of the electric current lines between the conductors are parallel to the magnetic field lines, but, near each corner where a free surface and the plane conductor meet, current lines fringe across the magnetic field lines. This fringing creates an electromagnetic body force which drives a flow parallel to the conductor corner. The inertialess flow for large values of the interaction parameter consists of flow over the tops of the hills of the wavy surface and a recirculation which is trapped between each pair of adjacent hills. Inertial effects convect the transverse parts of these recirculations to the crests of the hills and to the bottoms of the valleys, and this convection results in more circulations in planes which are perpendicular to the magnetic field. The flow is very different from the fully developed flow for two plane perfect conductors, and inertial effects significantly alter the periodic flow. This problem models the flow in a liquid-metal sliding electrical contact with a brush insert, and the periodically wavy boundary models the surface of the brush.

A finite volume approach for calculation of viscoelastic flow through an abrupt axisymmetric contraction

This paper reports the first convergent numerical algorithm for the steady inertialess flow of an Upper-Convected Maxwell (UCM) fluid through a four-to-one abrupt axisymmetric contraction. The Finite Volume Method (FVM) is adopted along with a stream function-vorticity approach in the Elastic Viscous Split Stress (EVSS) form with a first-order upwind approximation applied to the convective terms of the stress constitutive equation. A staggered volume discretization of the flow variables eliminates the stress singularity from the computational domain without loosing any flow physics. The volume integrals of the governing equations over the flow domain result in a system of nonlinear algebraic equations that are solved iteratively by a semi-implicit line-to-line method with a pseudo-trasient term added to the stress constitutive equation. Computations of an UCM fluid using this method are carried out to a much higher value of the Deborah number ( De) than previous numerical simulations using the Finite Element Method (FEM). The solutions are found to be smooth, stable, and convergent with the finger stress tensor remaining posiitive-definite throughout the flow domain. Calculations are not performed above De = 6.25 because of the decreasing pseudo-time step constraint at higher elasticity. The finite volume algorithm approximates better solutions upon mesh refinement, demonstrates the smoothness and the mathematical well-posedness of this problem, and predicts a Newtonian-like flow structure near the singularity for an UCM fluid. Using this new method, the inertialess flow of an UCM fluid through 4:1 abrupt axisymmetric contraction, for the first time, produces larger corner vortices at higher values of De.

Non-axisymmetric modes in viscoelastic taylor-couette flow

In this work, the linear stability analysis of the viscoelastic Taylor-Couette flow against non-axisymmetric disturbances is investigated. A pseudospectrally generated, generalized algebraic eigenvalue problem is constructed from the linearized set of the three-dimensional governing equations around the steady-state azimuthal solution. Numerical evaluation of the critical eigenvalues shows that for an upper-convected Maxwell model and for the specific set of geometric and kinematic parameters examined in this work, the azimuthal Couette (base) flow becomes unstable against non-axisymmetric time periodic disturbances before it does so for axisymmetric ones, provided the elasticity number ϵ ( De/ Re) is larger than some non-zero but small value (ϵ⪢ 0.01). In addition, as ϵ increases, different families of eigensolutions become responsible for the onset of instability. In particular, the azimuthal wavenumber of the critical eigensolution has been found to change from 1 to 2 to 3 and then back to 2 as ϵ increases from 0.01 to infinity (inertialess flow). In an analogous fashion to the axisymmetric viscoelastic Taylor-Couette flow, two possible patterns of time-dependent solutions (limit cycles) can emerge after the onset of instability: ribbons and spirals, corresponding to azimuthal and traveling waves, respectively. These patterns are dictated solely by the symmetry of the primary flow and have already been observed in conjunction with experiments involving Newtonian fluids but with the two cylinders counter-rotatng instead of co-rotating as considered here. Inclusion of a non-zero solvent viscosity (Oldroyd-B model) has been found to affect the results quantitatively but not qualitatively. These theoretical predictions are of particular importance for the interpretation of the experimental data obtained in a Taylor-Couette flow using highly elastic viscoelastic fluids.

Compatible pseudospectral approximations for incompressible flow in an undulating tube

The flow of an Oldroyd-B fluid through an undulating tube is considered. The effect of elasticity and inertia on flow resistance is investigated numerically using a time-splitting technique. This technique can be used to solve both steady and transient viscoelastic flows. A pseudospectral method based on mixed Fourier–Chebyshev expansions is used to represent the flow variables in space. An approximation space for pressure is constructed which is compatible with that for the velocity. This is achieved by removing the spurious modes using a singular value decomposition. A projection method ensures that mass is conserved identically at the collocation points. Numerical results are presented in such a way as to highlight the inertial and elastic effects. To this end two sets of results are given: the first, inertialess viscoelastic flow; the second, flow at nonzero Reynolds number holding the Weissenberg number constant. The two cases are shown to have quite opposite effects upon the flow rate and resistance.

On the numerical stability of mixed finite-element methods for viscoelastic flows governed by differential constitutive equations

Mixed finite-element methods for computation of viscoelastic flows governed by differential constitutive equations vary by the polynomial approximations used for the velocity, pressure, and stress fields, and by the weighted residual methods used to discretize the momentum, continuity, and constitutive equations. This paper focuses on computation of the linear stability of the planar Couette flow as a test of the numerical stability for solution of the upper-convected Maxwell model. Previous theoretical results prove this inertialess flow to be always stable, but that accurate calculation is difficult at high De because eigenvalues with fine spatial structure and high temporal frequency approach neutral stability. Computations with the much used biquadratic finite-element approximations for velocity and deviatoric stress and bilinear interpolation for pressure demonstrate numerical instability beyond a critical value of De for either the explicitly elliptic momentum equation (EEME) or elastic-viscous split-stress (EVSS) formulations, applying Galerkin's method for solution of the momentum and continuity equations, and using streamline upwind Petrov-Galerkin (SUPG) method for solution of the hyperbolic constitutive equation. The disturbance that causes the instability is concentrated near the stationary streamline of the base flow. The removal of this instability in a slightly modified form of the EEME formulation suggests that the instability results from coupling the approximations to the variables. A new mixed finite-element method, EVSS-G, is presented that includes smooth interpolation of the velocity gradients in the constitutive equation that is compatible with bilinear interpolation of the stress field. This formulation is tested with SUPG, streamline upwinding (SU), and Galerkin least squares (GLS) discretization of the constitutive equation. The EVSS-G/SUPG and EVSS-G/SU do not have the numerical instability described above; linear stability calculations for planar Couette flow are stable to values of De in excess of 50 and converge with mesh and time step. Calculations for the steady-state flow and its linear stability for a sphere falling in a tube demonstrate the appearance of linear instability to a time-periodic instability simultaneously with the apparent loss of existence of the steady-state solution. The instability appears as finely structured secondary cells that move from the front to the back of the sphere.

Finite element analysis of steady viscoelastic flow around a sphere in a tube: calculations with constant viscosity models

Finite element analysis is used to compute the inertialess flow of a viscoelastic fluid around a sphere falling in a cylindrical tube. Calculations are reported for three differential constitutive models with constant viscosities: the Upper-Convected Maxwell model (UCM), the Oldroyd-B model (OLDB) and the dumbbell model of Chilcott and Rallison (CR). Calculations are based on the Explicitly Elliptic Momentum Equation (EEME) and the Elastic Viscous Split Stress (EVSS) methods for calculations with models without and with a Newtonian solvent contribution, respectively. The calculations converged with mesh refinement for all three models for values of De below 1.6. Calculations with the UCM and OLDB models are limited below De = 2 by loss of convergence of the Newton iteration. Mesh refinement does not seem to alleviate these limits. Calculations with the CR model and moderate values of the maximum extensibility of the dumbbell L converge to higher values of De; however, computations for large values of L require fine meshes in the stagnation region behind the sphere.

A purely elastic instability in Taylor–Couette flow

A non-inertial (zero Taylor number) viscoelastic instability is discovered for Taylor–Couette flow of dilute polymer solutions. A linear stability analysis of the inertialess flow of an Oldroyd-B fluid (using both approximate Galerkin analysis and numerical solution of the relevant small-gap eigenvalue problem) show the growth of an overstable (oscillating) mode when the Deborah number exceeds f(S) e−½, where e is the ratio of the gap to the inner cylinder radius, and f(S) is a function of the ratio of solvent to polymer contributions to the solution viscosity. Experiments with a solution of 1000 p.p.m. high-molecular-weight polyisobutylene in a viscous solvent show an onset of secondary toroidal cells when the Deborah number De reaches 20, for e of 0.14, and a Taylor number of 10−6, in excellent agreement with the theoretical value of 21. The critical De was observed to increase as e decreases, in agreement with the theory. At long times after onset of the instability, the cells become small in wavelength compared to those that occur in the inertial instability, again in agreement with our linear analysis. For this fluid, a similar instability occurs in cone-and-plate flow, as reported earlier. The driving force for these instabilities is the interaction between a velocity fluctuation and the first normal stress difference in the base state. Instabilities of the kind that we report here are likely to occur in many rotational shearing flows of viscoelastic fluids.

Cortical shell-liquid core model for passive flow of liquid-like spherical cells into micropipets

Many nonadherent cells exist as spheres in suspension and when sucked into pipets, deform continuously like liquids within the fixed surface area limitation of a plasma membrane envelope. After release, these cells eventually recover their spherical form. Consequently, pipet aspiration test provides a useful method to assay the apparent viscosity of such cells. For this purpose, we have analyzed the inertialess flow of a liquid-like model cell into a tube at constant suction pressure. The cell is modeled as a uniform liquid core encapsulated by a distinct cortical shell. The method of analysis employs a variational approach that minimizes errors in boundary conditions defined by the equations of motion for the cortical shell where the trial functions are exact solutions for the flow field inside the liquid core. For the particular case of an anisotropic liquid cortex with persistent tension, we have determined universal predictions for flow rate scaled by the ratio of excess pressure (above the threshold established by the cortical tension) and core viscosity which is the reciprocal of the dynamic resistance to entry. The results depend on pipet to cell size ratio and a parameter that characterizes the ratio of viscous flow resistance in the cortex to that inside the cytoplasmic core. The rate of entry increases markedly as the pipet size approaches the outer segment diameter of the cell. Viscous dissipation in the cortex strongly influences the entry flow resistance for small tube sizes but has little effect for large tubes. This indicates that with sufficient experimental resolution, measurement of cell entry flow with different-size pipets could establish both the cortex to cell dissipation ratio as well as the apparent viscosity of the cytoplasmic core.

Parallel running of a modular simulation scheme

The idea of modularity in simulation is discussed in relation to parallel processing. A mathematical model for parallel simulation is presented. In this model a possibly nonlinear set of consistency conditions is satisfied in between time steps. Two examples are given which are fundamental to HVAC simulation: (1) heat flow in networks of heat paths and thermal capacitors, and (2) inertialess flow of fluid in networks of pipes and other components. These examples present in abstract from many of the difficulties encountered in thermal simulation of buildings with heating and ventilating plants. The question of convergence of networks of parallel processes is discussed, convergence is proved for the two examples, and a general theorem which implies unconditional convergence is developed. This theorem applies in many cases in which the processes represent physical components, and each process resets some variables to attain local steady state with its neighbors, so the sum of the local distances from steady state is at least not increased by any process.

Application of the k-ε Turbulence Model to the Squeeze Film Damper

The one-dimensional squeeze film damper is modeled for high speed flow by using the two-equation (k-ε) turbulent transport model. The assumption is made that the fluid flow at each local region of the squeeze film damper has similar behavior to inertialess flow in a channel at comparable Reynolds number. Using the k-ε model, the inertialess channel flow case is solved. Based on this result, correlations are obtained for the mean velocity, inertia and viscous terms of the integrated momentum equation for the squeeze film damper. It is found that turbulence increases the magnitude of the fluid pressure and the tangential force, while fluid inertia causes a shift on the pressure creating a significant radial force. In applications, turbulence may be a beneficial effect, increasing the principal damping force; while inertia may be detrimental increasing the cross-coupling forces.

Quasisteady pulsed flow of a thixotropic fluid in a cylindrical pipe

An approximate solution is obtained to the problem of inertialess periodic flow of fluid with a variable structure in a pipe of circular cross section. A study is made of the effect of the parameters which define the kinetics of the variations in the structure and the fluctuations in the pressure gradient on the effective viscosity and the other mean hydrodynamic characteristics. A comparison is made between the solutions to the problems of the flow of a thixotropic and a nonlinearly viscous fluid. The results are discussed in connection with their application to the circulation of the blood.

Asymptotic MHD flows in channels

A numerical and experimental study is made of the possibility of the onset of inertialess flow regimes in a conducting fluid in channels of complex geometry under the influence of a strong external field. The region of existence is determined for the inertialess regime from the Hartmann numbers and the MHD interaction parameter and asymptotic estimates are obtained for the drag coefficients.

Fluid Inertia Effects in a Squeeze Film Between Two Plane Annuli

An analysis is presented for the laminar squeezing flow of a Newtonian in-compressible fluid between parallel plane annuli. Both local and convective inertia of the flow are considered in the analysis. Power series expansions in terms of pertinent flow parameters are used to obtain a solution of the equations of motion. Expressions for the pressure and load capacity are presented, and compared with those based on the assumption of inertialess flow.

Coaxial-disk flow of an Oldroyd-B fluid: exact solution and stability

This paper reports an exact solution for the coaxial disk flow of an Oldroyd-B fluid. The flow is approximately generated by the parallel-plate viscometer. Asymptotic and numerical solutions are reported showing that there is a critical Weissenberg number based on the angular velocity and the Maxwellian relaxation time, above which the flow is unstable. A linearized stability analysis for the basic inertialess flow confirms this numerical instability and yields the critical Weissenberg number.

Birefringence and pressure drop for the orifice flow of a polymer solution

By means of laser illumination, we have measured the birefringence and tractracer-particle trajectories in the converging flow of a concentrated poly-acrylamide solution just upstream of a small circular orifice. For different orifice diameters D and mean discharge velocities V we obtain a close correlation between the birefringence, based on D, and the strain-rate parameter V/D for values of V/D up to 50s −1. However, the local values of axial extension rate, as deduced approximately from tracer streakline photographs, appear much smaller in magnitude than V/D. Our birefringence measurements also reveal what appear to be flow instabilities of the type observed in previous experiments on converging flow of polymer solutions. As an especially surprising and interesting result, we find that measured pressure drops across various orifices do not conform to predictions based on general dimensional arguments for inertialess viscoelastic flow from infinite reservoirs. This leads to the tentative conclusion that a filamentary- core structure associated with viscoelastic flow may generally give rise to pronounced hydrodynamic interactions with the upstream flow-chamber walls, even for nominally large contraction (chamber/orifice diameter) ratios.