Free surface on a simple fluid between rotating eccentric cylinders. Part II: Experiments

This paper summarizes the results from an experimental investigation of the effects of eccentricity and rotational speed on the free surface shape on a viscoelastic liquid between eccentric cylinders. In the experimental geometry, the inner cylinder rotates and the outer cylinder is stationary. The experiments show that there is a circumferential pressure gradient (the lubrication effect) which has a dominant influence on the free surface shape at all eccentricities and rotational speeds. For a liquid with small normal stress effects, the normal-stress induced component of the deformation tends to be overwhelmed by the lubrication effect, whereas a liguid with large normal stress effects exhibits characteristics normal-stress induced deformations at small eccentricities and rotational speeds. There is good agreement between experiment and second order predictions for the large normal stress liquid under these conditions. The ranges of eccentricities and rotational speeds for which second order theory describes the low normal stress liquid appear to be much more limited and are difficult to reproduce experimentally.     

URANS simulations for a high-speed transom stern ship with breaking waves

An exploratory study of high-speed surface ship flows is performed to identify modelling and numerical issues, to test the predictive capability of an unsteady RANS method for such flows, to explain flow features observed experimentally, and to document results obtained in conjunction with the 2005 ONR Wave Breaking Workshop. Simulations are performed for a high-speed transom stern ship (R/V Athena I) at three speeds Froude number (Fr) = 0.25, 0.43 and 0.62 with the URANS code CFDSHIP-IOWA, which utilizes a single-phase level set method for free surface modelling. The two largest Fr are considered to be high-speed cases and exhibit strong breaking plunging bow waves. Structured overset grids are used for local refinement of the unsteady transom flow at medium speed and for small scale breaking bow and transom waves at high-speeds. All simulations are performed in a time accurate manner and an examination of time histories of resistance and free surface contours is used to assess the degree to which the solutions reach a steady state. The medium speed simulation shows a classical steady Kelvin wave pattern without breaking and a wetted naturally unsteady transom flow with shedding of vortices from the transom corner. At higher speeds, the solutions reach an essentially steady state and display intense bow wave breaking with repeated reconnection of the plunging breaker with the free surface, resulting in multiple free surface scars. The high-speed simulations also show a dry transom and an inboard breaking wave, followed by outboard breaking waves downstream. In comparison to an earlier dataset, resistance is well predicted over the three speeds. The free surface predictions are compared with recent measurements at the two lowest speeds and show good agreement for both non-breaking and breaking waves.     

Primary and Secondary Infiltration of Wetting Liquid Sessile Droplet into Porous Medium

The infiltration of a wetting droplet into the porous medium is a two-step process referred to as primary and secondary infiltration. In the primary infiltration there is a free liquid present at the porous medium surface, and when no fluid is left on the surface, the secondary infiltration is initiated. In both situations the driving force is the capillary pressure that is influenced by the local medium heterogeneities. A capillary network model based on the micro-force balance is developed with the same formulation applied to both infiltrations. The only difference between the two is that the net liquid flow into the porous medium in the secondary infiltration is equal to zero. The primary infiltration starts as a single-phase (fully saturated) flow and may proceed as a multiphase flow. The multiphase flow emerges as the interface (flow front) becomes irregular in shape. The immobile clusters of the originally present phase can be locally formed due to entrapment. Throughout the infiltration, it was found that portions of the liquid phase can be detached from the main body of the liquid phase forming some isolated liquid ganglia that increase in number and decrease in size. The termination of the secondary infiltration occurs once the ganglia become immobile due to their reduction in size. From the transient solution, the changes in the liquid saturation and capillary pressure during the droplet infiltration are determined. The solution developed in this study is used to investigate the droplet infiltration dynamics. However, the solution can be used to study the flow in fuel cell, nano-arrays, composites, and printing.     

On Temporal Stability Analysis for Hydromagnetic Flow in a Channel Filled with a Saturated Porous Medium

In this paper, a linear stability analysis is presented to trace the time evolution of an infinitesimal, two-dimensional disturbance imposed on the base flow of an electrically conducting fluid in a channel filled with a saturated porous medium under the influence of a transversely imposed magnetic field. An eigenvalue problem is obtained and solved numerically using the Chebyshev collocation spectral method. The critical Reynolds number Re _c, the critical wave number α _c and the critical wave speed c _c are obtained for a wide range of the porous medium shape factor parameter S and Hartmann number H. It is found that an increase in the magnetic field intensity and a decrease in porous medium permeability have a stabilizing effect on the fluid flow.     

MODEL EXPERIMENT ABOUT RESPONSE OF FLOATING ICE SHEET SUBJECTED TO MOVING AIR CUSHION VEHICLE

基于弹性薄板振动微分方程和相似理论,建立了气垫船在浮冰上运动的原型系统与模型系统参数之间的相似对应关系. 导出了浮冰层自由振动波形传播的相速度和群速度计算公式,明确了相速度的极小值和浅水波传播速度即为气垫船的第一、第二临界速度. 根据研制的高精度非接触式激光位移测量系统,在变水深拖曳水槽中, 开展了不同速度移动气垫载荷激励薄膜变形响应的系列实验,证实了存在使薄膜变形达到最大的移动气垫载荷临界速度. 第一临界速度使气垫载荷之后的薄膜产生最大的下陷变形,第二临界速度使气垫载荷之前的薄膜产生最大的上凸变形. 通过实验结果进一步分析了气垫速度、高度、压力及水深等参数对薄膜变形和临界速度的影响,揭示了移动气垫载荷激励薄膜变形响应的聚能共振增幅机理,为利用气垫船实施有效破冰提供了依据.     

航行气垫船激励浮冰响应的模型实验研究

基于弹性薄板振动微分方程和相似理论,建立了气垫船在浮冰上运动的原型系统与模型系统参数之间的相似对应关系. 导出了浮冰层自由振动波形传播的相速度和群速度计算公式,明确了相速度的极小值和浅水波传播速度即为气垫船的第一、第二临界速度. 根据研制的高精度非接触式激光位移测量系统,在变水深拖曳水槽中, 开展了不同速度移动气垫载荷激励薄膜变形响应的系列实验,证实了存在使薄膜变形达到最大的移动气垫载荷临界速度. 第一临界速度使气垫载荷之后的薄膜产生最大的下陷变形,第二临界速度使气垫载荷之前的薄膜产生最大的上凸变形. 通过实验结果进一步分析了气垫速度、高度、压力及水深等参数对薄膜变形和临界速度的影响,揭示了移动气垫载荷激励薄膜变形响应的聚能共振增幅机理,为利用气垫船实施有效破冰提供了依据.      

On the Chebyshev collocation spectral approach to stability of fluid flow in a porous medium

In this paper, the temporal development of small disturbances in a pressure‐driven fluid flow through a channel filled with a saturated porous medium is investigated. The Brinkman flow model is employed in order to obtain the basic flow velocity distribution. Under normal mode assumption, the linearized governing equations for disturbances yield a fourth‐order eigenvalue problem, which reduces to the well‐known Orr–Sommerfeld equation in some limiting cases solved numerically by a spectral collocation technique with expansions in Chebyshev polynomials. The critical Reynolds number Re_c, the critical wave number α_c, and the critical wave speed c_c are obtained for a wide range of the porous medium shape factor parameter S. It is found that a decrease in porous medium permeability has a stabilizing effect on the fluid flow. Copyright © 2008 John Wiley & Sons, Ltd.     

Free surface deformation due to a source or a sink in Stokes flow

Free surface shape and cusp formation are analyzed by considering two-dimensional viscous flow due to a line source or a line sink below the free surface where the strength of source/sink is given arbitrarily. In the analysis, the Stokes' approximation is used and surface tension effects are included, but gravity is neglected. The solution is obtained analytically by using conformal mapping and complex function theory. From the solution, shapes of the free surface are shown and the formation of a cusp on the free surface is discussed. As the capillary number decreases in negative, the free surface shape becomes singular and in a real fluid a cusp should form on the free surface below some negative critical capillary number. Typically, streamline patterns for some capillary numbers are also shown. As the small capillary number vanishes, the solution is reduced to a linearized potential flow solution.     

Effects of Mineral Dissolution Ratios on Chemical-Dissolution Front Instability in Fluid-Saturated Porous Media

The main purpose of this article is to investigate, both theoretically and computationally, the effects of mineral dissolution ratios on the different respects of chemical-dissolution front instability problems in fluid-saturated porous media. In order to get a better understanding of how the mineral dissolution ratio affects the propagation and evolution of a planar chemical-dissolution front in an infinite space consisting of a fluid-saturated porous medium, the theoretical analysis method is used to derive the generous solution to the propagation speed of the planar chemical-dissolution front, while the computational simulation method is employed to simulate the detailed evolution process when the planar chemical-dissolution front is evolved into complicated morphologies at the supercritical Zhao numbers. The related theoretical results reveal that the mineral dissolution ratio plays an important role in controlling the propagation speed of a planar chemical-dissolution front in the fluid-saturated porous medium. An increase in the value of the mineral dissolution ratio can result in a remarkable decrease in the value of the propagation speed of a planar chemical-dissolution front. On the other hand, the related computational simulation results demonstrate that the mineral dissolution ratio has a considerable influence on the evolution pattern of a planar chemical-dissolution front during its propagation in the fluid-saturated porous medium. An increase in the mineral dissolution ratio can reduce the likelihood for a planar chemical-dissolution front to evolve from the initial planar shape into different morphologies within the fluid-saturated porous medium of finite size.     

Free-surface deformation due to spiral flow owing to a source/sink and a vortex in Stokes flow

The free-surface shape and cusp formation are analyzed by considering a viscous flow arising from the superposition of a source/sink and vortex below the free surface where the strength of the source and vortex are arbitrary. In the analysis, Stokes’ approximation is used and surface tension effects are included, but gravity is neglected. The solution is obtained analytically by using conformal mapping and complex function theory. From the solution, shapes of the free surface are obtained, and the formation of a cusp on the free surface is discussed. Above some critical capillary number with a sink, the free-surface shape becomes singular and an apparent cusp should form on the free surface below a real fluid. On the other hand, no cusp would occur for sources of zero or positive strength. Typical streamline patterns are also shown for some capillary numbers. As the capillary number vanishes, the solution is reduced to a linearized potential flow solution.     

Stability Analysis of Bioconvection of Gyrotactic Motile Microorganisms in a Fluid Saturated Porous Medium

Despite a large number of publications on bioconvection in suspensions of motile microorganisms, bioconvection in a fluid saturated porous medium is a relatively new area of research. This paper is motivated by experimental research by Kessler (1986) who established that a porous medium prevents the development of convection instability in algal suspensions. This suggests that there may exist a critical value of the permeability of a porous medium. If the permeability is smaller than critical, the system is stable and bioconvection does not develop. If the permeability is larger than critical, bioconvection may develop. This paper presents a model of bioconvection of gyrotactic motile microorganisms in a fluid saturated porous medium. The focus of this research is the determination of the critical value of permeability of a porous medium by a linear stability analysis. A simple but elegant analytical solution for the critical Darcy number is obtained.     

Effect of Reactive Surface Areas Associated with Different Particle Shapes on Chemical-Dissolution Front Instability in Fluid-Saturated Porous Rocks

In this article, the effect of reactive surface areas associated with different particle shapes on the reactive infiltration instability in a fluid-saturated porous medium is investigated through analytically deriving the dimensionless pore-fluid pressure-gradient of a coupled system between porosity, pore-fluid flow and reactive chemical-species transport within two idealized porous media consisting of spherical and cubic grains respectively. Compared with the critical dimensionless pore-fluid pressure-gradient of the coupled system, the derived dimensionless pore-fluid pressure-gradient can be used to assess the instability of a chemical dissolution front within the fluid-saturated porous medium. The related theoretical analysis has demonstrated that (1) since the shape coefficient of spherical grains is greater than that of cubic grains, the chemical system consisting of spherical grains is more unstable than that consisting of cubic grains, and (2) the instability likelihood of a natural porous medium, which is comprised of irregular grains, is smaller than that of an idealized porous medium, which is comprised of regular spherical grains. To simulate the complicated morphological evolution of a chemical dissolution front in the case of the chemical dissolution system becoming supercritical, a numerical procedure is proposed for solving this kind of problem. The related numerical results have demonstrated that the reactive surface area associated with different particle shapes can have a significant influence on the morphological evolution of an unstable chemical-dissolution front within fluid-saturated porous rocks.     

The Weissenberg effect in molten polymers

The climb of viscoelastic liquids up a rotating rod, often called the Weissenberg effect, has often been observed in polymer solutions, and several theoretical analyses of this phenomenon have been published. However, no observations of rod-climbing in polymer melts have been reported. In the present work, three commercial polyethylenes were melted in a special chamber under a controlled atmosphere, and a rod was rotated in the pool of molten polymer. The behavior of the free surface was noted as a function of time and rotational speed.In the case of the resin with the lowest molecular weight and the highest density, the free surface reached a steady-state shape within seconds of changing the rotational speed. For the other two resins, the development of the shape of the free surface took place over periods of the order of one hour or more. In the cases where the starting transient was very long-lasting, the melt first climbed the rod and accumulated in a large bulge. Then, the fluid in the bulge flowed down to the bottom of the raised column and a pear-shaped body of liquid was formed whose shape continued to undergo gradual change for periods up to several hours. At high speeds, inertial effects destabilized the flow and led to asymmetry and time unsteadiness in the shape of the raised body of liquid.     

Surface phenomena in elasticity

The characteristic properties of free surfaces of elastic bodies, such as propagation speeds of characteristic waves and conditions of instability of the surface shape (wrinkling), are considered. The results are generalized whenever a free surface is streamlined by an inviscid flow. The theory is illustrated by example.     

Mixed Convection on a Horizontal Surface Embedded in a Porous Medium: the Structure of a Singularity

The mixed convection boundary-layer flow on a horizontal impermeable surface embedded in a saturated porous medium and driven by a local heat source is considered. Similarity solutions are obtained for specific outer flow variations and these are shown to have a solution only for parameter values greater than some critical value. When this is not the case the solution develops a singularity at a finite distance from the leading edge. The nature of this singularity is also discussed.     

Forced oscillations of a vertical continuous rotor with geometric nonlinearity

Nonlinear forced oscillations of a vertical continuous rotor with distributed mass are discussed. The restoring force of the rotor has geometric stiffening nonlinearity due to the extension of the rotor center line. The possibility of the occurrence of nonlinear forced oscillations at various subcritical speeds and the shapes of resonance curves at the major critical speeds and at some subcritical speeds are investigated theoretically. Consequently, the following is clarified: (a) the shape of resonance curves at the major critical speed becomes a hard spring type, and (b) among various kinds of nonlinear forced oscillations, only some special kinds of combination resonances have possibility of occurrence.     

Finite element analysis of convective heat transfer in porous media

The finite element method is used to analyse convective heat transfer in a porous medium. Convection past a vertical surface embedded in the medium and convection in a confined porous medium enclosure are analysed using the above method. The results are compared with those available in the literature and the agreement is found to be good. The method is applicable for two-dimensional analysis in a porous body of any arbitrary shape. The restriction of the boundary layer assumption is relaxed.     

Combined free and forced convection in a porous medium between two vertical walls with viscous dissipation

Viscous dissipation effects in the problem of a fully-developed combined free and forced convection flow between two symmetrically and asymmetrically heated vertical parallel walls filled with a porous medium is analyzed. The equation of motion contains the modified Rayleigh number for a porous medium and the small-order viscous dissipation parameter. Particular attention is given to the solutions near the critical Rayleigh numbers at which infinite flow rates are predicted. Information concerning the multiplicity of solutions at critical Rayleigh numbers is also deduced from perturbation solutions of the governing equation.     

质量任意分布下的柔性转子过临界点时的瞬态响应

本文首先将Riccati传递矩阵法和正、逆回旋运动分解理论应用于有阻尼的分布质量转子的复特征问题计算，求出系统各阶正、逆回旋临界速度（也称临界点）及相应振型然后作者对广义阻尼模态理论作了引伸和发展，结合Bogoliubov-Mitropolskii渐近法，建立起一阶微分方程组，计算不平衡柔性转子分别在正、逆回旋下通过各阶临界点的非线性、非定常瞬态响应，还深入分析了转子相继通过两十分邻近的临界点时发生的耦合现象。     

Lie-group method of solution for steady two-dimensional boundary-layer stagnation-point flow towards a heated stretching sheet placed in a porous medium

The boundary-layer equations for two-dimensional steady flow of an incompressible, viscous fluid near a stagnation point at a heated stretching sheet placed in a porous medium are considered. We apply Lie-group method for determining symmetry reductions of partial differential equations. Lie-group method starts out with a general infinitesimal group of transformations under which the given partial differential equations are invariant. The determining equations are a set of linear differential equations, the solution of which gives the transformation function or the infinitesimals of the dependent and independent variables. After the group has been determined, a solution to the given partial differential equations may be found from the invariant surface condition such that its solution leads to similarity variables that reduce the number of independent variables of the system. The effect of the velocity parameter λ, which is the ratio of the external free stream velocity to the stretching surface velocity, permeability parameter of the porous medium k ₁, and Prandtl number Pr on the horizontal and transverse velocities, temperature profiles, surface heat flux and the wall shear stress, has been studied.