Formation of a jet in case of non-stationary flow of a perfect fluid from a slit

The problem of flow of a perfect fluid from a slit separating two parallel planes is given a mathematical formulation for the case of a submerged jet, as well as for the case of a flow with a free boundary. Various types of flow are classified. The results of numerical solutions are compared with experimental data. The phenomenon of reversal of the vortex sheet which occurs when the flow rate through the slit is reduced, is discussed. The self-modelling problem is solving using the method of matching the asymptotic expansions. A cumulative effect is discovered, namely that the rate of penetration of a narrow central part of the self-modelling jet exceeds the rate of flow of the main part of the jet by one order of magnitude.     

Nonaxisymmetric Free Convection Flow over a Rotating Disk in a Viscoelastic fluid with Magnetic Field

The non axisymmetric motion produced by a buoyancy-induced secondary flow of a viscoelastic fluid over an infinite rotating disk in a verticalplane with a magnetic field applied normal to the disk has been studied.The governing Navier Stokes equations and the energy equation admit a self similar solution. The system of ordinary differential equations has been solved numerically using Runge-Kutta Gill subroutine.The turning moment for the viscoelastic fluid is found to be less than that of the Newtonian fluid but the turning moment is increased due to the magnetic parameter. The resultant force due to the buoyancy-induced secondary flow increases with the magnetic parameter but reduces as the viscoelastic parameter increases. The quantity of fluid, which is pumped outwards due to the centrifuging action of the disk, for the viscoelastic fluid is more than that of the Newtonian fluid. The buoyancy-induced secondary flow boundary layer is much thicker than the primary boundary layer thickness. The thermal boundary layer due to the primary flow increases with the magnetic parameter decreases as the viscoelastic parameter increases. The heat transfer increases with the viscoelastic parameter but decreases as the magnetic parameter increases. The effect of the viscoelastic parameter is more pronounced on the secondary flow than on the primary flow.     

Numerical investigation into a liquid displacing a gas in thin porous layers

To examine the filling process in a lithium-ion battery, a numerical model to characterize the displacing flow of a liquid in air-filled pores of thin heterogeneous porous materials is elaborated. The investigation is based on the volume-averaged Navier-Stokes equations for small Reynolds numbers, using a volume-of-fluid method to cover the multiphase flow. The flow is investigated with respect to the wall effect and to capillary action within the porous matrix. On the one hand, model experiments with similar particle-size distributions as in the battery layers are conducted to extract the porosity as function of the wall distance. On the other hand, experiments with the three different porous layers of the battery are performed to receive mean values for the most important properties related to the two-phase flow. Results for the displacement flow in parts of the battery are presented and discussed, showing a considerable influence of the modeled effects onto the flow characteristics. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Peristaltic motion of a magnetohydrodynamic micropolar fluid in a tube

This study is concerned with the magnetohydrodynamic flow of a micropolar fluid in a circular cylindrical tube. The equations governing the flow are modeled using the assumptions of long wavelength and low Reynolds number. It is found that the governing equations are coupled partial differential equations for the flow velocity and the microrotation. The finite difference scheme is used to integrate the equations and the results are graphically presented and discussed. Special emphasis is given to the effects of micropolar fluid parameters, tube wall peristaltic amplitude and magnetic parameter on the transverse profiles of velocity and microrotation as well as pumping characteristics and trapping phenomena.     

Environmental dispersion in a tidal flow through a depth-dominated wetland

Presented in this paper is a theoretical analysis for longitudinal evolution of mean concentration of an environmental emission into a tidal wetland flow. The velocity distribution of the periodic flow through the wetland is derived, with that for a fully developed steady wetland flow included as a special case. The zero-th, first and the growth of the second order moments of the concentration are rigorously obtained by applying Aris’s method of concentration moments to derive the environmental dispersivity. The necessary time needed for the environmental dispersivity to attain a steady oscillating status is analyzed. The effects of some characteristic parameters, especially one representing the impact of vegetation in the wetland on both velocity profile and environmental dispersivity, and another identifying the effect of flow oscillation on the environmental dispersivity, are illustrated in detail. To reflect the dispersion enhancement by the flow oscillation, a typical example is given to characterize the critical length and duration of the contaminant cloud.     

A computational model of a transcritical R744 ejector based on a homogeneous real fluid approach

A mathematical model of the compressible transonic single- and two-phase flow of a real fluid is discussed in this paper. The model was originally developed to simulate a refrigerant flow through a heat pump ejector. In the proposed approach, a temperature-based energy equation is replaced with an enthalpy-based formulation, in which the specific enthalpy, instead of the temperature, is an independent variable. A thermodynamic and mechanical equilibrium between gaseous and liquid phases is assumed for the two-phase flow. Consequently, real fluid properties, such as the density, the dynamic viscosity and the diffusion coefficient, are defined as functions of the pressure and the specific enthalpy. The energy equation formulation is implemented in commercial CFD software using subroutines that were developed in-house. The formulations was tested extensively for a single-phase flow of the R141b refrigerant, and for a two-phase flow of the R744 fluid (carbon dioxide) that occurred in a 3-D model of the ejector motive nozzle. In the model validation procedure, a satisfactory comparison between the experimental and computational results of the primary and secondary mass flow rates was obtained for both flow regimes. In addition, in the case of the R744 flow, the pressure distribution along the centre line of the ejector was accurately predicted as well. Furthermore, the results also shows that geometry modelling and measurement accuracy play an important in the final numerical results. As a result of the reasonable computational times, this method can be effectively used for the design of ejectors and also in geometric optimisation computations.     

Taylor dispersion in a packed tube

Presented in this paper is a theoretical analysis for longitudinal scalar spread of mean concentration under a fully developed flow in a tube packed with porous media. A general form of momentum equation for superficial flow in porous media is introduced as a combination of the Navier–Stokes equation and Darcy’s law plus a superficial dispersion term due to phase discontinuity between the fluid flow and solid frame. The analytical solution presented for the fully developed superficial flow includes that for the Poiseulle flow in an evacuated tube as a limiting case. As an extension of Taylor’s classical work on dispersion of soluble matter in solvent flowing slowly through an evacuated tube, a one-dimensional dispersion equation valid for overall environmental assessment of contaminant is rigorously derived by cross-sectionally averaging the superficial mass equation and introducing a closure relation for a new unknown out of the averaging procedure, and corresponding Taylor dispersivity determined is shown to be a generalization of Taylor’s well-known result for the Poiseulle flow.     

A sphere in a second degree polynomial creeping flow parallel to a wall

Comprehensive results are provided for the creeping flow arounda spherical particle in a viscous fluid close to a plane wall,when the external velocity is parallel to the wall and variesas a second degree polynomial in the coordinates. By linearityof Stokes equations, the solution is a sum of flows for typicalunperturbed flows: a pure shear flow, a ‘modulated shearflow’, for which the rate of shear varies linearly inthe direction normal to the wall, and a quadratic flow. Solutionsconsidered here use the bipolar coordinates technique. Theycomplement the accurate results of Chaoui and Feuillebois (2003)for the pure shear flow. The solution of Goren and O'Neill (1971)for the quadratic flow is reconsidered and a new analyticalsolution is derived for the ambient modulated shear flow. Theperturbed flow fields for these two cases are presented in detailand discussed. Results for the force and torque friction factorsare provided with a 5 x 10^–17 accuracy as a reference.For the quadratic flow, there is a force and a torque on a fixedsphere. A minimum value of the torque is found for a gap ofabout 0·18a, where a is the sphere radius. This minimumis interpreted in term of the corresponding flow structure.For the modulated shear flow, there is only a torque. The freemotion of a sphere in an ambient quadratic flow is also determined.     

Modeling of flow in a very small surface separation

When a fluid flows in a very small surface separation, the very thin boundary layer physically adhering to the solid surface will participate in the flow, while between the two boundary layers is a continuum fluid flow. An analysis is here presented for this multiscale flow. The continuum fluid is treated as Newtonian. The physical adsorbed boundary layer is treated as non-continuum across the layer thickness. The interfacial slippage can occur on the adsorbed layer-solid surface interface, while it is absent on the adsorbed layer-fluid interface. Three flow equations are derived respectively for the two adsorbed layers and the intermediate continuum fluid. They together govern the multiscale flow in such a small surface separation.     

固壁近旁Stokes流中粘性液滴的运动和变形

发展了一种模拟固壁近旁轴对称Stokes流中粘性液滴的运动和变形及直接计算固壁上应力的边界积分方法．用此方法对不同的液滴-固壁初始相对间距、粘度比、表面张力和浮力联合参数以及环境流动参数情况进行了数值实验．数值结果显示,由于环境流动和浮力的作用,随着时间的推进,液滴在轴向压缩,在径向拉伸．当环境流动的作用弱于浮力作用时,随着时间的推移,液滴上升并向上弯,固壁上由液滴运动所引起的应力不断减小．当环境流动的作用强于浮力作用时,随着时间的推移,液滴变得越来越扁．在这种情形,当大初始间距时,壁面上的应力随液滴的演变而增大;当小初始间距时,由环境流动、浮力及壁面对流动的较强作用的联合影响,此应力随液滴的演变而减小．由于液滴运动所引起的壁面应力的有效作用仅限于对称轴附近的一个小范围内,且此范围随液滴与固壁的初始间距增大而增大．应力的大小随初始间距增大而大为减小．表面张力对液滴变形有阻止作用．液滴粘性会减小液滴的变形和位置迁移．     

Vortex Generation in a Liquid Metal Duct Flow near a Magnetic Dipole

The generation of vortical structures by a strong magnetic dipole field in a liquid metal duct flow is studied by means of three-dimensional direct numerical simulations. The dipole is considered as the paradigm for a magnetic obstacle which will deviate the streamlines due to Lorentz forces which act on the fluid elements. Our model uses the quasi-static approximation applicable in the limit of small magnetic Reynolds numbers. The analysis covers the stationary flow regime at smaller flow Reynolds numbers Re as well as the fully time-dependent regimes at higher values with a turbulent flow in the wake of the magnetic obstacle. We present a systematic study of these two basic flow regimes on Re and the Hartmann number Ha, a measure of the strength of the magnetic dipole field. Furthermore, three orientations of the dipole are compared, the streamwise, spanwise and wall-normal ones. The most efficient generation of turbulence at a fixed distance above the duct occurs for the spanwise orientation in which we can observe the formation of Hartmann layers at the top plate. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hydromagnetic instability of a power-law liquid film flowing down a vertical cylinder using numerical approximation approach techniques

The long-wave perturbation method is employed to investigate the hydromagnetic stability of a thin electrically-conductive power-law liquid film flowing down the external surface of a vertical cylinder in a magnetic field. The validity of the numerical results is improved through the introduction of the flow index and the magnetic force into the governing equation. In contrast to most previous studies presented in the literature, the solution scheme employed in this study is based on a numerical approximation approach rather than an analytical method. The normal mode approach is used to analyze the stability of the film flow. The modeling results reveal that the stability of the film flow system is weakened as the radius of the cylinder is reduced. However, the flow stability can be enhanced by increasing the intensity of the magnetic field and the flow index, respectively. In general, the optimum conditions can be found through the use of a system to alter stability of the film flow by controlling the applied magnetic field.     

Analytic solution for MHD Transient rotating flow of a second grade fluid in a porous space

This article looks into the unsteady rotating magnetohydrodynamic (MHD) flow of an incompressible second grade fluid in a porous half space. The flow is induced by a suddenly moved plate in its own plane. Both the fluid and plate rotate in unison with the same angular velocity. Analytic solution of the governing flow problem is obtained by using Fourier sine transform. Based on the modified Darcy's law, expression for velocity is obtained. The influence of pertinent parameters on the flow is delineated and appropriate conclusions are drawn. Several existing solutions of Newtonian fluid have been also deduced as limiting cases.     

Flow distribution and environmental dispersivity in a tidal wetland channel of rectangular cross-section

Presented in this paper is a theoretical analysis on flow distribution and environmental dispersivity for a tidal wetland channel of rectangular cross-section. The analytical solution of velocity distribution for the tidal wetland flow is obtained and illustrated with a limiting case covering the known solution for a steady wetland flow. By use of Aris’s method of concentration moments, the environmental dispersivity for a pulsed contaminant emission into the tidal wetland flow is rigorously derived and characterized in terms of dimensionless parameters. The solution is shown to be a generalization of the environmental dispersivity for the corresponding steady wetland flow, taking into account the combined action of periodic oscillation and cross-sectional variation of superficial flow as well as the difference between superficial mass dispersivities in the vertical and lateral directions. For a long time evolution of the contaminant cloud, the environmental dispersivity may approach a stable stage of oscillation with a period equal to the period of the superficial flow. The evolution of environmental dispersivity at the initial stage for the tidal wetland flow is shown not monotonous as it does in the case of the steady wetland flow. It is also found that the period of superficial flow has no impact on the necessary time for the environmental dispersivity to attain the stable stage.     

Flow in a circular tube with a centred strip

The flow in a circular tube with a centred strip or vaned coreis studied by domain decomposition and matching. It is foundthat even a very small core has considerable effect on the velocityand the flow rate. Asymptotic solutions for the elliptic tubewith a centred strip and the circular tube with a circular coresuggest an inverse logarithmic singularity as the core approacheszero.     

Laminar magnetohydrodynamic duct flow in the presence of a magnetic dipole

We study magnetohydrodynamic flow of a liquid metal in a straight duct. The magnetic field is produced by an exterior magnetic dipole. This basic configuration is of fundamental interest for Lorentz force velocimetry (LFV), where the Lorentz force opposing the relative motion of conducting medium and magnetic field is measured to determine the flow velocity. The Lorentz force acts in equal strength but opposite direction on the flow as well as on the dipole. We are interested in the dependence of the velocity on the flow rate and on strength of the magnetic field as well as on geometric parameters such as distance and position of the dipole relative to the duct. To this end, we perform numerical simulations with an accurate finite-difference method in the limit of small magnetic Reynolds number, whereby the induced magnetic field is assumed to be small compared with the external applied field. The hydrodynamic Reynolds number is also assumed to be small so that the flow remains laminar. The simulations allow us to quantify the magnetic obstacle effect as a potential complication for local flow measurement with LFV. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analysis of blood flow through a modelled artery with an aneurysm

The intention of the present work is to carry out a systematic analysis of flow features in a tube, modelled as artery, having a local aneurysm in presence of haematocrit. The arterial model is treated to be axi-symmetric and rigid. The blood, flowing through the modelled artery, is treated to be Newtonian and non-homogeneous. For a thorough quantitative analysis of the flow characteristics such as wall pressure, flow velocity, wall shear stress, the unsteady incompressible Navier-Stokes equations in cylindrical polar co-ordinates under the laminar flow conditions are solved by using the finite-difference method. Finally, the numerical illustrations presented in this paper provide an effective measure to estimate the combined influence of haematocrit and aneurysm on flow characteristics. It is found that the magnitude of wall shear stress and also the length of separation increase with increasing values of the haematocrit parameter. The length of flow separation increases but the peak value of wall shear stress decreases with the increasing length of aneurysm. The peak value of wall shear stress as well as the length of separation increases with the increasing height of the aneurysm.     

The critical cross-section of a vortex

Summary This paper discusses the problem of critical-flow cross-sections in vortex flows. It is shown that there are two different types of vortex flows, A-type and B-type vortices (say). An A-type vortex approaches its critical flow state as its cross-sectional area increases and departs from the critical state as the cross-sectional area is decreased. This property is associated with the particular dependence of total pressure and circulation on the stream function, and it holds for both subcritical and supercritical A-type vortices. On the other hand, both subcritical and supercritical B-type vortices approach their critical flow states as their cross-sectional areas are decreased and depart from their critical states for increasing cross-sectional area. As was shown by Benjamin, setting the first variation of the flow force with respect to the stream function equal to zero leads to Euler's equation of motion. The second variation also vanishes if the corresponding flow state is critical. In this case the sign of the third variation decides whether the flow is an A-type or a B-type vortex. Within the framework of inviscid-fluid flow theory the type of a vortex is preserved unless vortex breakdown occurs. Making use of the knowledge that vortex flows are controlled by two different types of critical-flow cross-sections a variety of vortex flow phenomena are investigated, including the two types of inlet vortices that are observed upstream of jet engines, the behavior of vortex valves, the flow characteristics of liquid-fuel atomizers and the bath tub vortex.     

Flow Past a Swept Wing with a Compliant Surface: Stabilizing the Attachment-Line Boundary Layer

Many aquatic species such as dolphins and whales have fins, which can be modeled as swept wings. Some of these fins, such as the dorsal fin of a dolphin, are semi-rigid and therefore can be modeled as a rigid swept wing with a compliant surface. An understanding of the hydrodynamics of the flow past swept compliant surfaces is of great interest for understanding potential drag reduction mechanisms, especially since swept wings are widely used in hydrodynamic and aerodynamic design. In this paper, the flow past a swept wing with a compliant surface is modeled by an attachment-line boundary layer flow, which is an exact similarity solution of the Navier–Stokes equations, flowing past a compliant surface modeled as an elastic plate. The hydrodynamic stability of the coupled problem is studied using a new numerical framework based on exterior algebra. The basic instability of the attachment line boundary layer on a rigid surface is a traveling wave instability that propagates along the attachment line, and numerical results show that the compliance results in a substantial reduction in the instability region. Moreover, the results show that, although the flow-field is three-dimensional, the qualitative nature of the instability suppression is very similar to the qualitative reduction of instability of the two-dimensional Tollmien–Schlichting modes in the classical boundary-layer flow past a compliant surface.     

Numerical computation of pulsatile flow through a locally constricted channel

This paper deals with the numerical solution of a pulsatile laminar flow through a locally constricted channel. A finite difference technique has been employed to solve the governing equations. The effects of the flow parameters such as Reynolds number, flow pulsation in terms of Strouhal number, constriction height and length on the flow behaviour have been studied. It is found that the peak value of the wall shear stress has significantly changed with the variation of Reynolds numbers and constriction heights. It is also noted that the Strouhal number and constriction length have little effect on the peak value of the wall shear stress. The flow computation reveals that the peak value of the wall shear stress at maximum flow rate time in pulsatile flow situation is much larger than that due to steady flow. The constriction and the flow pulsation produce flow disturbances at the vicinity of the constriction of the channel in the downstream direction.