A note on the damping and oscillations of a fluid drop moving in another fluid

The oscillations of a drop moving in another fluid medium have been studied at low values of Reynolds number and Weber number by taking into consideration the shape of the drop and the viscosities of the two phases in addition to the interfacial tension. The deformation of the drop modifies the Lamb's expression for frequency by including a correction term while the viscous effects split the frequency into a pair of frequencies—one lower and the other higher than Lamb's. The lower frequency mode has ample experimental support while the higher frequency mode has also been observed. The two modes almost merge with Lamb's frequency for the asymptotic cases of a drop in free space or a bubble in a dense viscous fluid but the splitting becomes large when the two fluids have similar properties. Instead of oscillations, aperiodic damping modes are found to occur in drops with sizes smaller than a critical size ($\sim\hat{\rho}\hat{\nu}^2/T $). With the help of these calculations, many of the available experimental results are analyzed and discussed.

Stability of the flow of a fluid through a flexible tube at high Reynolds number

The stability of the Hagen-Poiseuille flow of a Newtonian fluid in a tube of radius R surrounded by an incompressible viscoelastic medium of radius R < r < HR is analysed in the high Reynolds number regime. The dimensionless numbers that affect the fluid flow are the Reynolds number Re = (ρVR / η), the ratio of the viscosities of the wall and fluid ηr = (ηs/η), the ratio of radii H and the dimensionless velocity Γ = (ρV2/G)1/2. Here ρ is the density of the fluid, G is the coefficient of elasticity of the wall and Vis the maximum fluid velocity at the centre of the tube. In the high Reynolds number regime, an asymptotic expansion in the small parameter ε = (1/Re) is employed. In the leading approximation, the viscous effects are neglected and there is a balance between the inertial stresses in the fluid and the elastic stresses in the medium. There are multiple solutions for the leading-order growth rate do), all of which are imaginary, indicating that the fluctuations are neutrally stable, since there is no viscous dissipation of energy or transfer of energy from the mean flow to the fluctruations due to the Reynolds strees. There is an O(ε1/2) correction to the growth rate, s(1), due to the presence of a wall layer of thickness ε1/2R where the viscous stresses are O(ε1/2) smaller than the inertial stresses. An energy balance analysis indicates that the transfer of energy from the mean flow to the fluctuations due to the Reynolds stress in the wall layer is exactly cancelled by an opposite transfer of equal magnitude due to the deformation work done at the interface, and there is no net transfer from the mean flow to the fluctuations. Consequently, the fluctuations are stabilized by the viscous dissipation in the wall layer, and the real part of s(1) is negative. However, there are certain values of Γ and wavenumber k where s(l) = 0. At these points, the wail layer amplitude becomes zero because the tangential velocity boundary condition is identically satisfied by the inviscid flow solution. The real part of the O(ε) correction to the growth rate s(2) turns out to be negative at these points, indicating a small stabilizing effect due to the dissipation in the bulk of the fluid and the wall material. It is found that the minimum value of s(2) increases [is proportional to] (H − 1)−2 for (H − 1) [double less-than sign] 1 (thickness of wall much less than the tube radius), and decreases [is proportional to] (H−4 for H [dbl greater-than sign] 1. The damping rate for the inviscid modes is smaller than that for the viscous wall and centre modes in a rigid tube, which have been determined previously using a singular perturbation analysis. Therefore, these are the most unstable modes in the flow through a flexible tube.

Unsteady MUD rotating flow over a rotating sphere near the equator

The unsteady rotating flow of a laminar incompressible viscous electrically conducting fluid over a rotating sphere in the vicinity of the equator has been studied. The fluid and the body rotate either in the same direction or in opposite directions. The effects of surface suction and magnetic field have been included in the analysis. There is an initial steady state that is perturbed by a sudden change in the rotational velocity of the sphere, and this causes unsteadiness in the flow field. The nonlinear coupled parabolic partial differential equations governing the boundary-layer flow have been solved numerically by using an implicit finite-difference scheme. For large suction or magnetic field, analytical solutions have also been obtained. The magnitude of the radial, meridional and rotational velocity components is found to be higher when the fluid and the body rotate in opposite directions than when they rotate in the same direction. The surface shear stresses in the meridional and rotational directions change sign when the ratio of the angular velocities of the sphere and the fluid lambda greater than or equal to lambda(0). The final (new) steady state is reached rather quickly which implies that the spin-up time is small. The magnetic field and surface suction reduce the meridional shear stress, but increase the surface shear stress in the rotational direction.

The generalized Onsager model for the secondary flow in a high-speed rotating cylinder

The generalizations of the Onsager model for the radial boundary layer and the Carrier-Maslen model for the end-cap axial boundary layer in a high-speed rotating cylinder are formulated for studying the secondary gas flow due to wall heating and due to insertion of mass, momentum and energy into the cylinder. The generalizations have wider applicability than the original Onsager and Carrier-Maslen models, because they are not restricted to the limit A >> 1, though they are restricted to the limit R e >> 1 and a high-aspect-ratio cylinder whose length/diameter ratio is large. Here, the stratification parameter A = root m Omega(2)R(2)/2k(B)T). This parameter A is the ratio of the peripheral speed, Omega R, to the most probable molecular speed, root 2k(B)T/m, the Reynolds number Re = rho w Omega R(2)/mu, where m is the molecular mass, Omega and R are the rotational speed and radius of the cylinder, k(B) is the Boltzmann constant, T is the gas temperature, rho(w) is the gas density at wall, and mu is the gas viscosity. In the case of wall forcing, analytical solutions are obtained for the sixth-order generalized Onsager equations for the master potential, and for the fourth-order generalized Carrier-Maslen equation for the velocity potential. For the case of mass/momentum/energy insertion into the flow, the separation-of-variables procedure is used, and the appropriate homogeneous boundary conditions are specified so that the linear operators in the axial and radial directions are self-adjoint. The discrete eigenvalues and eigenfunctions of the linear operators (sixth-order and second-order in the radial and axial directions for the Onsager equation, and fourth-order and second-order in the axial and radial directions for the Carrier-Maslen equation) are determined. These solutions are compared with direct simulation Monte Carlo (DSMC) simulations. The comparison reveals that the boundary conditions in the simulations and analysis have to be matched with care. The commonly used `diffuse reflection' boundary conditions at solid walls in DSMC simulations result in a non-zero slip velocity as well as a `temperature slip' (gas temperature at the wall is different from wall temperature). These have to be incorporated in the analysis in order to make quantitative predictions. In the case of mass/momentum/energy sources within the flow, it is necessary to ensure that the homogeneous boundary conditions are accurately satisfied in the simulations. When these precautions are taken, there is excellent agreement between analysis and simulations, to within 10 %, even when the stratification parameter is as low as 0.707, the Reynolds number is as low as 100 and the aspect ratio (length/diameter) of the cylinder is as low as 2, and the secondary flow velocity is as high as 0.2 times the maximum base flow velocity. The predictions of the generalized models are also significantly better than those of the original Onsager and Carrier-Maslen models, which are restricted to thin boundary layers in the limit of high stratification parameter.

Removal of unwanted fluid

This work is concerned with the removal of unwanted fluid through the source-sink pair. The source consists of fluid issuing out of a nozzle in the form of a jet and the sink is a pipe that is kept some distance from the source pipe. Of concern is the percentage of source fluid sucked through the sink. The experiments have been carried in a large glass water tank. The source nozzle diameter is 6Â mm and the sink pipe diameter is either 10 or 20Â mm. The horizontal and vertical separations and angles between these source and sink pipes are adjustable. The flow was visualized using KMnO 4 dye, planer laser induced fluorescence and particle streak photographs. To obtain the effectiveness (that is percentage of source fluid entering the sink pipe), titration method is used. The velocity profiles with and without the sink were obtained using particle image velocimetry. The sink flow rate to obtain a certain effectiveness increase dramatically with lateral separation. The sink diameter and the angle between source and the sink axes don't influence effectiveness as much as the lateral separation.

Unsteady Rotating Flow Over an Impulsively Rotating Infinite Disk with Axial Magnetic Field and Suction

The unsteady rotating flow of an incompressible laminar viscous electrically conducting fluid over an impulsively rotated infinite disk in the presence of magnetic field and suction is investigated. We have considered the situation where there is a steady state initially (i.e., at t = 0, the fluid is rotating with constant angular velocity over a stationary disk). Then at t > 0, the disk is suddenly rotated with a constant angular velocity either in the same direction or in opposite direction to that of the fluid rotation which causes unsteadiness in the flow field. The effect of the impulsive motion is found to be more pronounced on the tangential shear stress than on the radial shear stress. When the disk and the fluid rotate in the same direction, the tangential shear stress at the surface changes sign in a small time interval immediately after the start of the impulsive motion.

Removal of unwanted fluid

This work is concerned with the removal of unwanted fluid through the source-sink pair. The source consists of fluid issuing out of a nozzle in the form of a jet and the sink is a pipe that is kept some distance from the source pipe. Of concern is the percentage of source fluid sucked through the sink. The experiments have been carried in a large glass water tank. The source nozzle diameter is 6 mm and the sink pipe diameter is either 10 or 20 mm. The horizontal and vertical separations and angles between these source and sink pipes are adjustable. The flow was visualized using KMnO4 dye, planer laser induced fluorescence and particle streak photographs. To obtain the effectiveness (that is percentage of source fluid entering the sink pipe), titration method is used. The velocity profiles with and without the sink were obtained using particle image velocimetry. The sink flow rate to obtain a certain effectiveness increase dramatically with lateral separation. The sink diameter and the angle between source and the sink axes don't influence effectiveness as much as the lateral separation.

Effect of hinged leaflets on vortex pair generation

We experimentally study the effect of having hinged leaflets at the jet exit on the formation of a two-dimensional counter-rotating vortex pair. A piston-cylinder mechanism is used to generate a starting jet from a high-aspect-ratio channel into a quiescent medium. For a rigid exit, with no leaflets at the channel exit, the measurements at a central plane show that the trailing jet in the present case is never detached from the vortex pair, and keeps feeding into the latter, unlike in the axisymmetric case. Passive flexibility is introduced in the form of rigid leaflets or flaps that are hinged at the exit of the channel, with the flaps initially parallel to the channel walls. The experimental arrangement closely approximates the limiting case of a free-to-rotate rigid flap with negligible structural stiffness, damping and flap inertia, as these limiting structural properties permit the largest flap openings. Using this arrangement, we start the flow and measure the flap kinematics and the vorticity fields for different flap lengths and piston velocity programs. The typical motion of the flaps involves a rapid opening and a subsequent more gradual return to its initial position, both of which occur when the piston is still moving. The initial opening of the flaps can be attributed to an excess pressure that develops in the channel when the flow starts, due to the acceleration that has to be imparted to the fluid slug between the flaps. In the case with flaps, two additional pairs of vortices are formed because of the motion of the flaps, leading to the ejection of a total of up to three vortex pairs from the hinged exit. The flaps' length (L-f) is found to significantly affect flap motions when plotted using the conventional time scale L/d, where L is the piston stroke and d is the channel width. However, with a newly defined time scale based on the flap length (L/L-f), we find a good collapse of all the measured flap motions irrespective of flap length and piston velocity for an impulsively started piston motion. The maximum opening angle in all these impulsive velocity program cases, irrespective of the flap length, is found to be close to 15 degrees. Even though the flap kinematics collapses well with L/L-f, there are differences in the distribution of the ejected vorticity even for the same L/L-f. Such a redistribution of vorticity can lead to important changes in the overall properties of the flow, and it gives us a better understanding of the importance of exit flexibility in such flows.

The generalized Onsager model for a binary gas mixture

The Onsager model for the secondary flow field in a high-speed rotating cylinder is extended to incorporate the difference in mass of the two species in a binary gas mixture. The base flow is an isothermal solid-body rotation in which there is a balance between the radial pressure gradient and the centrifugal force density for each species. Explicit expressions for the radial variation of the pressure, mass/mole fractions, and from these the radial variation of the viscosity, thermal conductivity and diffusion coefficient, are derived, and these are used in the computation of the secondary flow. For the secondary flow, the mass, momentum and energy equations in axisymmetric coordinates are expanded in an asymptotic series in a parameter epsilon = (Delta m/m(av)), where Delta m is the difference in the molecular masses of the two species, and the average molecular mass m(av) is defined as m(av) = (rho(w1)m(1) + rho(w2)m(2))/rho(w), where rho(w1) and rho(w2) are the mass densities of the two species at the wall, and rho(w) = rho(w1) + rho(w2). The equation for the master potential and the boundary conditions are derived correct to O(epsilon(2)). The leading-order equation for the master potential contains a self-adjoint sixth-order operator in the radial direction, which is different from the generalized Onsager model (Pradhan & Kumaran, J. Fluid Mech., vol. 686, 2011, pp. 109-159), since the species mass difference is included in the computation of the density, viscosity and thermal conductivity in the base state. This is solved, subject to boundary conditions, to obtain the leading approximation for the secondary flow, followed by a solution of the diffusion equation for the leading correction to the species mole fractions. The O(epsilon) and O(epsilon(2)) equations contain inhomogeneous terms that depend on the lower-order solutions, and these are solved in a hierarchical manner to obtain the O(epsilon) and O(epsilon(2)) corrections to the master potential. A similar hierarchical procedure is used for the Carrier-Maslen model for the end-cap secondary flow. The results of the Onsager hierarchy, up to O(epsilon(2)), are compared with the results of direct simulation Monte Carlo simulations for a binary hard-sphere gas mixture for secondary flow due to a wall temperature gradient, inflow/outflow of gas along the axis, as well as mass and momentum sources in the flow. There is excellent agreement between the solutions for the secondary flow correct to O(epsilon(2)) and the simulations, to within 15 %, even at a Reynolds number as low as 100, and length/diameter ratio as low as 2, for a low stratification parameter A of 0.707, and when the secondary flow velocity is as high as 0.2 times the maximum base flow velocity, and the ratio 2 Delta m/(m(1) + m(2)) is as high as 0.5. Here, the Reynolds number Re = rho(w)Omega R-2/mu, the stratification parameter A = root m Omega R-2(2)/(2k(B)T), R and Omega are the cylinder radius and angular velocity, m is the molecular mass, rho(w) is the wall density, mu is the viscosity and T is the temperature. The leading-order solutions do capture the qualitative trends, but are not in quantitative agreement.

Axially homogeneous, zero mean flow buoyancy-driven turbulence in a vertical pipe

We report an experimental study of a new type of turbulent flow that is driven purely by buoyancy. The flow is due to an unstable density difference, created using brine and water, across the ends of a long (length/diameter = 9) vertical pipe. The Schmidt number Sc is 670, and the Rayleigh number (Ra) based on the density gradient and diameter is about 10(8). Under these conditions the convection is turbulent, and the time-averaged velocity at any point is `zero'. The Reynolds number based on the Taylor microscale, Re-lambda, is about 65. The pipe is long enough for there to be an axially homogeneous region, with a linear density gradient, about 6-7 diameters long in the midlength of the pipe. In the absence of a mean flow and, therefore, mean shear, turbulence is sustained just by buoyancy. The flow can be thus considered to be an axially homogeneous turbulent natural convection driven by a constant (unstable) density gradient. We characterize the flow using flow visualization and particle image velocimetry (PIV). Measurements show that the mean velocities and the Reynolds shear stresses are zero across the cross-section; the root mean squared (r.m.s.) of the vertical velocity is larger than those of the lateral velocities (by about one and half times at the pipe axis). We identify some features of the turbulent flow using velocity correlation maps and the probability density functions of velocities and velocity differences. The flow away from the wall, affected mainly by buoyancy, consists of vertically moving fluid masses continually colliding and interacting, while the flow near the wall appears similar to that in wall-bound shear-free turbulence. The turbulence is anisotropic, with the anisotropy increasing to large values as the wall is approached. A mixing length model with the diameter of the pipe as the length scale predicts well the scalings for velocity fluctuations and the flux. This model implies that the Nusselt number would scale as (RaSc1/2)-Sc-1/2, and the Reynolds number would scale as (RaSc-1/2)-Sc-1/2. The velocity and the flux measurements appear to be consistent with the Ra-1/2 scaling, although it must be pointed out that the Rayleigh number range was less than 10. The Schmidt number was not varied to check the Sc scaling. The fluxes and the Reynolds numbers obtained in the present configuration are Much higher compared to what would be obtained in Rayleigh-Benard (R-B) convection for similar density differences.

Effects of the filling process on the evolution of the mushy zone and macrosegregation in alloy casting

A numerical model of the entire casting process starting from the mould filling stage to complete solidification is presented. The model takes into consideration any phase change taking place during the filling process. A volume of fluid method is used for tracking the metal–air interface during filling and an enthalpy based macro-scale solidification model is used for the phase change process. The model is demonstrated for the case of filling and solidification of Pb–15 wt%Sn alloy in a side-cooled two-dimensional rectangular cavity, and the resulting evolution of a mushy region and macrosegregation are studied. The effects of process parameters related to filling, namely degree of melt superheat and filling velocity on macrosegregation in the cavity, are also investigated. Results show significant differences in the progress of the mushy zone and macrosegregation pattern between this analysis and conventional analysis without the filling effect.

Purification and functional characterization of type II DNA topoisomerase from rat testis and comparison with topoisomerase II from liver

A number of studies in yeast have shown that DNA topoisomerase TI is essential for chromosome condensation and disjunction during mitosis at the metaphase/anaphase transition and meiosis I. Accordingly, kinetic and mechanistic studies have implied a role for topoisomerase rr in chromosome disjunction. As a step toward understanding the nature and role of topoisomerase II in a mammalian germline in vivo, we have purified topoisomerase II from rat testis to homogeneity and ascertained several of its catalytic activities in conjunction with that of the purified enzyme from liver. The purified enzymes appeared to be monomers under denaturing conditions; however, they differed in their relative molecular mass. Topoisomerase II from testis and liver have apparent molecular masses of 150 +/- 10 kDa and 160 +/- 10 kDa, respectively. The native molecular mass of testis topoisomerase II as assayed by immunoblot analysis of cell-foe extracts, prepared in the presence of SDS and a number of protease inhibitors, corroborated with the size of the purified enzyme. Both enzymes are able to promote decatenation and relax supercoiled DNA substrates in an ATP and Mg2+-dependent manner. However, quantitative comparison of catalytic properties of topoisomerase II from testis with that of the enzyme from liver displayed significant differences in their efficiencies. Optimal pH values for testis enzyme are 6.5 to 8.5 while they are 6 to 7.5 for the liver enzyme. Intriguingly, the relaxation activity of liver topoisomerase II was inhibited by potassium glutamate at 1 M, whereas testis enzyme required about half its concentration. These findings argue that topoisomerase II from rat testis is structurally distinct from that of its somatic form and the functional differences between the two enzymes parallels with the physiological environment that is unique to these two tissues.

Numerical studies on columnar-to-equiaxed transition in directional solidification of binary alloys

A numerical study on columnar-to-equiaxed transition (CET) during directional solidification of binary alloys is presented using a macroscopic solidification model. The position of CET is predicted numerically using a critical cooling rate criterion reported in literature. The macroscopic solidification model takes into account movement of solid phase due to buoyancy, and drag effect on the moving solid phase because of fluid motion. The model is applied to simulate the solidification process for binary alloys (Sn-Pb) and to estimate solidification parameters such as position of the liquidus, velocity of the liquidus isotherm, temperature gradient ahead of the liquidus, and cooling rate at the liquidus. Solidification phenomena under two cooling configurations are studied: one without melt convection and the other involvin thermosolutal convection. The numerically predicted positions of CET compare well with those of experiments reported in literature. Melt convection results in higher cooling rate, higher liquidus isotherm velocities, and stimulation of occurrence of CET in comparison to the nonconvecting case. The movement of solid phase aids further the process of CET. With a fixed solid phase, the occurrence of CET based on the same critical cooling rate is delayed and it occurs at a greater distance from the chill.

On the origin of the inflectional instability of a laminar separation bubble

This is an experimental and theoretical Study of a laminar separation bubble and the associated linear stability mechanisms. Experiments were performed over a flat plate kept in a wind tunnel, with an imposed pressure gradient typical of an aerofoil that would involve a laminar separation bubble. The separation bubble was characterized by measurement of surface-pressure distribution and streamwise velocity using hot-wire anemometry. Single component hot-wire anemometry was also used for a detailed study of the transition dynamics. It was foundthat the so-called dead-air region in the front portion of the bubble corresponded to a region of small disturbance amplitudes, with the amplitude reaching a maximum value close to the reattachment point. An exponential growth rate of the disturbance was seen in the region upstream of the mean maximum height of the bubble, and this was indicative of a linear instability mechanism at work. An infinitesimal disturbance was impulsively introduced into the boundary layer upstream of separation location, and the wave packet was tracked (in an ensemble-averaged sense) while it was getting advected downstream. The disturbance was found to be convective in nature. Linear stability analyses (both the Orr-Sommerfeld and Rayleigh calculations) were performed for mean velocity profiles, starting from an attached adverse-pressure-gradient boundary layer all the way up to the front portion of the separation-bubble region (i.e. up to the end of the dead-air region in which linear evolution of the disturbance could be expected). The conclusion from the present work is that the primary instability mechanism in a separation bubble is inflectional in nature, and its origin can be traced back to upstream of the separation location. In other words, the inviscid inflectional instability of the separated shear layer should be logically seen as an extension of the instability of the upstream attached adverse-pressure-gradient boundary layer. This modifies the traditional view that pegs the origin of the instability in a separation bubble to the detached shear layer Outside the bubble, with its associated Kelvin-Helmholtz mechanism. We contendthat only when the separated shear layer has moved considerably away from the wall (and this happens near the maximum-height location of the mean bubble), a description by the Kelvin-Helmholtz instability paradigm, with its associated scaling principles, Could become relevant. We also propose a new scaling for the most amplified frequency for a wall-bounded shear layer in terms of the inflection-point height and the vorticity thickness and show it to be universal.

Finite volume TVD formulation of lattice Boltzmann simulation on unstructured mesh

A numerical scheme is presented for accurate simulation of fluid flow using the lattice Boltzmann equation (LBE) on unstructured mesh. A finite volume approach is adopted to discretize the LBE on a cell-centered, arbitrary shaped, triangular tessellation. The formulation includes a formal, second order discretization using a Total Variation Diminishing (TVD) scheme for the terms representing advection of the distribution function in physical space, due to microscopic particle motion. The advantage of the LBE approach is exploited by implementing the scheme in a new computer code to run on a parallel computing system. Performance of the new formulation is systematically investigated by simulating four benchmark flows of increasing complexity, namely (1) flow in a plane channel, (2) unsteady Couette flow, (3) flow caused by a moving lid over a 2D square cavity and (4) flow over a circular cylinder. For each of these flows, the present scheme is validated with the results from Navier-Stokes computations as well as lattice Boltzmann simulations on regular mesh. It is shown that the scheme is robust and accurate for the different test problems studied.