On the shock–vortex interaction in Schardin's problem

In this study we revisit Schardin's problem by investigating experimentally shock waves diffracting over a finite wedge and interacting with the tip vortices in a complicated manner. Holographic interferometry and shadowgraphy were used in a shock tube for a shock Mach number . Numerical simulations were carried out to obtain complementary flow data. The experimental results show that diverging acoustic waves are generated due to the interaction between shock waves and vortexlets along the slip layer. By means of the computational results obtained for short time intervals, and the corresponding optical images, analysis of the shock-vortex interactions became possible for extended time periods. Received 18 May 1998 / Accepted 4 March 1999     

Creeping motion of a sphere in tubes filled with Herschel–Bulkley fluids

Previous numerical simulations for the flow of Bingham plastics past a sphere contained in cylindrical tubes of different diameter ratios are extended to Herschel–Bulkley fluids with the purpose of comparing them with experiments. The emphasis is on determining the extent and shape of yielded/unyielded regions along with the drag coefficient as a function of the pertinent dimensionless groups. Good overall agreement is obtained between the numerical results and the experimental studies.     

Studies on the breather solutions for the $$\mathbf{(2+1)}$$ -dimensional variable-coefficient Kadomtsev–Petviashvili equation in fluids and plasmas

Nonlinear Dynamics - In this paper, we study the $$(2 + 1)$$ -dimensional variable-coefficient Kadomtsev–Petviashvili equation, which has certain applications in fluids and plasmas. Via the...     

Using the Euler–Lagrange variational principle to obtain flow relations for generalized Newtonian fluids

The Euler–Lagrange variational principle is used to obtain analytical and numerical flow relations in cylindrical tubes. The method is based on minimizing the total stress in theflow duct using the fluid constitutive relation between stress and rate of strain. Newtonian and non-Newtonian fluid models, which include power law, Bingham, Herschel–Bulkley, Carreau, and Cross, are used for demonstration.     

Effect of trigonometric sine,square and triangular wave-type time-periodic gravity-aligned oscillations on Rayleigh–Bénard convection in Newtonian liquids and Newtonian nanoliquids

The influence of trigonometric sine, square and triangular wave-types of time-periodic gravity-aligned oscillations on Rayleigh–Bénard convection in Newtonian liquids and in Newtonian nanoliquids is studied in the paper using the generalized Buongiorno two-phase model. The five-mode Lorenz model is derived under the assumptions of Boussinesq approximation, small-scale convective motion and some slip mechanisms. Using the method of multiscales, the Lorenz model is transformed to a Ginzburg–Landau equation the solution of which helps in quantifying the heat transport through the Nusselt number. Enhancement of heat transport in Newtonian liquids due to the presence of nanoparticles/nanotubes is clearly explained. The study reveals that all the three wave types of gravity modulation delay the onset of convection and thereby to a diminishment of heat transport. It is also found that in the case of trigonometric sine type of gravity modulation heat transport is intermediate to that of the cases of triangular and square types. The paper is the first such work that attempts to theoretically explain the effect of three different wave-types of gravity modulation on onset of convection and heat transport in the presence/absence of nanoparticles/nanotubes. Comparing the heat transport by the single-phase and by the generalized two-phase models, the conclusion is that the single-phase model under-predicts heat transport in nanoliquids irrespective of the type of gravity modulation being effected on the system. The results of the present study reiterate the findings of related experimental and numerical studies.

     

Numerical simulation of shock–vortex interaction in Schardin’s problem

Shock diffraction over a two-dimensional wedge and subsequent shock–vortex interaction have been numerically simulated using the AUSM $+$ + scheme. After the passage of the incident shock over the wedge, the generated tip vortex interacts with a reflected shock. The resulting shock pattern has been captured well. It matches the existing experimental and numerical results reported in the literature. We solve the Navier–Stokes equations using high accuracy schemes and extend the existing results by focussing on the Kelvin–Helmholtz instability generated vortices which follow a spiral path to the vortex core and on their way interact with shock waves embedded within the vortex. Vortex detection algorithms have been used to visualize the spiral structure of the initial vortex and its final breakdown into a turbulent state. Plotting the dilatation field we notice a new source of diverging acoustic waves and a lambda shock at the wedge tip.     

Soliton dynamics and interaction in the Bose–Einstein condensates with harmonic trapping potential and time-varying interatomic interaction

Investigated in this paper is the quasi-one-dimensional Gross–Pitaevskii equation, which describes the dynamics of the Bose–Einstein condensates with the harmonic trapping potential and time-varying interatomic interaction. Via the Horita method and symbolic computation, analytic bright N-soliton solution is obtained. One-, two- and three-soliton solutions are analyzed graphically. Based on the limit analysis on the one- and two-soliton solutions, the modulation on the speed of the matter-wave bright solitons is realized. Via the parameters, the interaction between the matter-wave solitons are adjustable. Furthermore, an approach to construct the interference between the matter-wave solitons has been proposed. Finally, investigation on the three-soliton solution verifies our conclusions drawn from the one and two solitons. Our conclusions might be useful in the fields of the control on the matter-wave solitons, atom lasers, and atomic accelerators.     

Study on the subgrade deformation under high-speed train loading and water–soil interaction

It is important to study the subgrade characteristics of high-speed railways in consideration of the water–soil coupling dynamic problem,especially when high-speed trains operate in rainy regions.This study develops a nonlinear water–soil interaction dynamic model of slab track coupling with subgrade under high-speed train loading based on vehicle–track coupling dynamics.By using this model,the basic dynamic characteristics,including water–soil interaction and without water induced by the high-speed train loading,are studied.The main factors-the permeability coefficien and the porosity-influencin the subgrade deformation are investigated.The developed model can characterize the soil dynamic behaviour more realistically,especially when considering the influenc of water-rich soil.     

The axisymmetric contraction–expansion: the role of extensional rheology on vortex growth dynamics and the enhanced pressure drop

The flow of a polystyrene Boger fluid through axisymmetric contraction–expansions having various contraction ratios (2≤β≤8) and varying degrees of re-entrant corner curvatures are studied experimentally over a large range of Deborah numbers. The ideal elastic fluid is dilute, monodisperse and well characterized in both shear and transient uniaxial extension. A large enhanced pressure drop above that of a Newtonian fluid is observed independent of contraction ratio and re-entrant corner curvature. Streak images, laser Doppler velocimetry (LDV) and digital particle image velocimetry (DPIV) are used to investigate the flow kinematics upstream of the contraction plane. LDV is used to measure velocity fluctuation in the mean flow field and to characterize a global elastic flow instability which occurs at large Deborah numbers. For a contraction ratio of β=2, a steady elastic lip vortex is observed while for contraction ratios of 4≤β≤8, no lip vortex is observed and a corner vortex is seen. Rounding the re-entrant corner leads to shifts in the onset of the flow transitions at larger Deborah numbers, but does not qualitatively change the overall structure of the flow field. We describe a simple rescaling of the deformation rate which incorporates the effects of lip curvature and allows measurements of vortex size, enhanced pressure drop and critical Deborah number for the onset of elastic instability to be collapsed onto master curves. Transient extensional rheology measurements are utilized to explain the significant differences in vortex growth pathways (i.e. elastic corner vortex versus lip vortex growth) observed between the polystyrene Boger fluids used in this research and polyisobutylene and polyacrylamide Boger fluids used in previous contraction flow experiments. We show that the role of contraction ratio on vortex growth dynamics can be rationalized by considering the dimensionless ratio of the elastic normal stress difference in steady shear flow to those in transient uniaxial extension. It appears that the differences in this normal stress ratio for different fluids at a given Deborah number arise from variations in solvent quality or excluded volume effects.     

Correction to: Effect of trigonometric sine,square and triangular wavetype time-periodic gravity-aligned oscillations on Rayleigh–Bénard convection in Newtonian liquids and Newtonian Nanoliquids

Meccanica - The original article has been updated due to typesetting mistakes made in Section 2.2 Mathematical formulation, Table 1 and in Equation (51).     

Some basic questions on mechanosensing in cell–substrate interaction

Cells constantly probe their surrounding microenvironment by pushing and pulling on the extracellular matrix (ECM). While it is widely accepted that cell induced traction forces at the cell–matrix interface play essential roles in cell signaling, cell migration and tissue morphogenesis, a number of puzzling questions remain with respect to mechanosensing in cell–substrate interactions. Here we show that these open questions can be addressed by modeling the cell–substrate system as a pre-strained elastic disk attached to an elastic substrate via molecular bonds at the interface. Based on this model, we establish analytical and numerical solutions for the displacement and stress fields in both cell and substrate, as well as traction forces at the cell–substrate interface. We show that the cell traction generally increases with distance away from the cell center and that the traction-distance relationship changes from linear on soft substrates to exponential on stiff substrates. These results indicate that cell adhesion and migration behaviors can be regulated by cell shape and substrate stiffness. Our analysis also reveals that the cell traction increases linearly with substrate stiffness on soft substrates but then levels off to a constant value on stiff substrates. This biphasic behavior in the dependence of cell traction on substrate stiffness immediately sheds light on an existing debate on whether cells sense mechanical force or deformation when interacting with their surroundings. Finally, it is shown that the cell induced deformation field decays exponentially with distance away from the cell. The characteristic length of this decay is comparable to the cell size and provides a quantitative measure of how far cells feel into the ECM.     

Partitioned solution to fluid–structure interaction problem in application to free-surface flows

In this work we discuss a way to compute the impact of free-surface flow on nonlinear structures. The approach chosen relies on a partitioned strategy that allows us to solve the strongly coupled fluid–structure interaction problem. It is then possible to re-use the existing and validated strategy for each sub-problem. The structure is formulated in a Lagrangian way and solved by the finite element method. The free-surface flow approach considers a Volume-Of-Fluid (VOF) strategy formulated in an Arbitrary Lagrangian–Eulerian (ALE) framework, and the finite volume is used to discrete and solve this problem. The software coupling is ensured in an efficient way using the Communication Template Library (CTL). Numerical examples presented herein concern the 2D validation case but also 3D problems with a large number of equations to be solved.     

Wall material and roughness effects on transmittable shear stresses of magnetorheological fluids in plate–plate magnetorheometry

A direct comparison of plate–plate magnetorheometry results for nonmagnetic (titanium/brass) and ferromagnetic plates is presented, using a modified Anton Paar magnetocell MRD180/1T. Necessary corrections to derive the true flux density in the magnetorheological fluid (MRF) from the online Hall probe reading and to account for the gap opening effect caused by normal forces on shear stress and flux density are addressed. The measured shear stress versus magnetic flux density characteristics agree in the low flux density regime <0.1 T but yield distinctly higher transmittable shear stresses for ferromagnetic plates at elevated flux densities (49% increase at 1 T for 90% by weight carbonyl iron powder (CIP) and 84% for 85% by weight CIP). Remarkably, the normal force, if corrected for its magnetostatic part, remains independent of the type of plates up to about 0.6 T. We address the role of normal forces, of magnetic interactions between CIP and wall, as well as the role of wall roughness in a solid body friction model. A systematic variation of wall properties and materials was achieved by introducing both a modular rotor and stator, which ease the variation of the walls in contact to the MRF. The transmittable shear stress of nonmagnetic plates (e.g., brass) may be increased up to the level of ferromagnetic disks by a higher wall roughness or by grooves. No shear stress increase is obtained for grooves in ferromagnetic plates, which is explained by the different local flux density modulation at the grooves for ferromagnetic compared to nonmagnetic plates. Finally, we address the effect of ferromagnetic and nonmagnetic coatings on brass and steel disks, and show that, e.g., a layer of CIP on brass efficiently increases the transmittable shear stress.     

Flow characteristics and mixing performance of electrokinetically driven non-Newtonian fluid in contraction–expansion microchannel

A numerical investigation is performed into the flow characteristics and mixing performance of electrokinetically driven non-Newtonian fluid in a contraction–expansion microchannel. In the study, the rheological behavior of the fluid is characterized using a power-law model. The results show that the volumetric flow rate reduces as the flow behavior index increases, and thus an improved mixing performance is obtained. Furthermore, it is shown that for all considered values of the flow behavior index, the mixing performance can be enhanced by increasing the ratio of the main channel width to the contraction channel width, extending the length of the contraction channel, assigning a smaller value to the nondimensional Debye–Hückel parameter, and applying an appropriate electric field strength. Finally, it is shown that although the mixing efficiency reduces with a reducing flow behavior index, an acceptable mixing performance can still be obtained given an appropriate specification of the flow conditions and geometry parameters.     

The flow pattern map of a two-phase non-Newtonian liquid–gas flow in the vertical pipe

A map for the determination of flow pattern for two-phase flow of gas and non-Newtonian liquid in the vertical pipe has been presented. Our own experimental data confirm applicability of such a map.     

Effect of flow nonparallelism on instability of the Taylor-Görtler waves in supersonic axisymmetric jets

Within the framework of the linear theory of hydrodynamic stability, the characteristics of the Taylor-Görtler waves are numerically simulated at the initial section of a supersonic axisymmetric jet taking into account the effects of flow nonparallelism and expansion. The special features of the streamwise dynamics of the growth rates of various wave components for turbulent. weakly nonisobaric, and laminar jets are studied. It is shown that the growth rates depend strongly on the quantity on which their determination is based, the position of the point where it is measured, and the flow regime. Some experimental results are discussed, and a method for determining the growth rates is proposed.     

Experiments on the wavy instability in Rayleigh–Bénard–Poiseuille convection: Spatial and temporal features in the linear and nonlinear regime

An experimental study of a Rayleigh–Bénard–Poiseuille air flow in a rectangular channel is presented. The aim of the paper is to characterize a secondary instability, referred to as wavy instability and known to be a convective instability, with the objective to identify the best conditions for an optimal homogenization of heat transfers in the system. A periodic mechanical forcing is introduced at channel inlet and the spatial and temporal evolution of the temperature fluctuations are analyzed, depending on the Rayleigh and Reynolds numbers, the forcing frequency and the forcing amplitude. As the saturation state is a priori the best situation to homogenize the transfers, the objective is to expand the saturation area and to generate a maximum saturation amplitude value by conducting experiments at high Rayleigh numbers. It is shown that the change in the Rayleigh number value influences the saturation length but does not act on the saturation magnitude while the change in the Reynolds number value causes antagonist effects on the saturation parameters. The key parameter acting on the saturation amplitude is the forcing frequency. The most efficient forcing configuration is to introduce the external perturbation into the fully developed region of the longitudinal rolls and to apply a specific low forcing frequency associated with a moderate forcing magnitude.