One-dimensional finite element method in hydrodynamic stability

The stability of plane Poiseuille flow and circular Couette flow are examined with respect to linear azimuthally periodic disturbances by the finite element method. In the case of Couette motion, solutions are obtained for a narrow gap, a wide gap and a dilute polymer solution with an elongational viscosity in the narrow gap limit when both cylinders rotate at almost equal speed in the same direction. Results are in good agreement with previous calculations by other numerical methods.     

BEM approaches and simplified kinetic models for the analysis of damping in deformable MEMS

Microflows occurring in MEMS are addressed by applying a continuum BEM code for quasi-static Stokes flow accounting for slip boundary condition. Working conditions typical of transition and rarefied flows are treated in a simplified manner by employing a viscosity corrected according to semi-analytical solutions for the linearized BGK model of Couette and Poiseuille flows. Numerical results are compared with experiments showing excellent agreement     

Flow of a binary mixture of incompressible Newtonian fluids in a rectangular channel

The problems concerning some simple steady and unsteady flows of a mixture composed of two incompressible Newtonian fluids in an infinitely long channel of rectangular cross-section are examined. By means of finite Fourier sine transforms, the exact solutions of the field equations are obtained for the following four problems: (i) steady Couette flow in a rectangular channel, (ii) unsteady Couette flow in a rectangular channel, (iii) steady Poiseuille flow in a rectangular channel, (iv) unsteady Poiseuille flow in a rectangular channel.     

On the explicit analytic solutions of an Oldroyd 6-constant fluid

This paper deals with some steady unidirectional flows of an Oldroyd 6-constant fluid. The modelled differential equation is non-linear and proposes some new mathematical difficulties. The governing non-linear equation for the steady flow of an Oldroyd 6-constant fluid is different than from the Newtonian fluid and involves the non-Newtonian features. Moreover, the exact analytic solutions are obtained for Couette, Poiseuille and generalized Couette flows for all values of non-Newtonian parameters. The homotopy analysis method (HAM) is used to construct the solutions.     

SPH Simulation of Low Reynolds Number Planar Shear Flow and Heat Convection

The present study reports on a set of computer programmable SPH formulations, which are used to simulate transient planar shear flows, and in particular Poiseuille flow and Couette flow with different types of body forces. The flows examined have Reynolds numbers within the range 0.05∼50. SPH results agree well with analytical solutions for those situations amenable to an analytical treatment, with the largest deviation being less than 2.0 %. The accuracy of a SPH formulation for heat convection, with particular emphasis in the viscous heat dissipation, is also tested via a steady convective heat transfer case for a combined Poiseuille and Couette flow.     

Unsteady flows of a binary mixture of incompressible Newtonian fluids in an annulus

This paper is concerned with applying the mixture theory of two chemically inert incompressible Newtonian fluids to some simple unsteady flows in the annular region between two infinitely long coaxial cylinders. The equations governing the motion of the binary mixture under discussion are reduced to a system of coupled partial differential equations. With the help of finite Hankel transforms, the exact solutions of these equations are obtained in series form for the following three problems: (i) unsteady axial Couette flow in an annulus, (ii) unsteady Poiseuille flow in an annulus, (iii) unsteady circular Couette flow in an annulus.     

Simulation of bead-and-spring chain models for semidilute polymer solutions in shear flow

We report preliminary results of simulations of the steady-state rheological behavior for semidilute polymer solutions of head-and-spring chain models in planar Couette now. The simulations include examination of the effects of excluded volume. hydrodynamic interactions and density. Hydrodynamic interactions are modeled by the Rotne -Prager Yamakawa tensor. The simulations are based on the nonequilibrium Brownian dynamics algorithm of Ermak and McCammon. In addition to the spring potential between neighboring beads in the chain. the interaction between any two beads in the solution is modeled using a shifted, repulsive Leonard-Jones potential. Lees Edward sliding brick boundary conditions are used for consistency with the Couette flow field.Paper presented at the Twelfth Symposium on Thermophysical Properties. June 19–24. 1994, Boulder. Colorado. U.S.A.     

Similarity solutions to rotating Couette flow with power law fluids

Summary The nonlinear rheological effects of a power law fluid in rotating Couette flow are addressed. The governing equations are derived and exact similarity solutions to the Cauchy problem for the angular velocity and shear stress distributions are shown and discussed. The case when the flow is generated by a line impulse of angular momentum is considered. The existence of a travelling wave solution for a shear thickening fluid is shown.     

Thermomechanical coupling in frictionally heated circular Couette flow

The thermomechanical coupling in circular Couette rheometers is investigated Asymptotic solutions of second order in the Brinkman number, Br, are developed for Newtonian fluids whose viscosity and thermal conductivity can be expressed as quadratic functions of temperature. The derived solutions are validated by comparison to previously published series solutions as the limit of planar flow is approached as well as to numerical solutions and are found to be reliable for a practical range of the Nahme number. These solutions are explicit in Br and in the properties of the fluid and thus provide valuable insight into the functionality of the relevant dependences, something that is lost in purely numerical solutions.     

Finite element analysis of the linearized BGK Boltzmann equation

The weighted residual and finite element methods are combined to yield a numerical solution scheme for the linerized ^BGK Boltzmann equation which describes the dynamic behaviour of gases in all flow régimes ranging from the free molecular to the continuum régime. The potential of the proposed method is illustrated by application to the problem of cylindrical Couette flow. Numerical results are compared with alternate numerical solutions and experimental data.     

Numerical Solutions for Unsteady Flows of a Magnetohydrodynamic Jeffrey Fluid Between Parallel Plates Through a Porous Medium

In the present article, the numerical solutions for three fundamental unsteady flows (namely Couette, Poiseuille, and generalized Couette flows) of an incompressible magnetohydrodynamic Jeffrey fluid between two parallel plates through a porous medium are presented using differential quadrature method. The equations governing the flow of Jeffrey fluid are modeled in Cartesian coordinate system. The resulting non-dimensional differential equations are approximated by using a new scheme that is trigonometric B-spline differential quadrature method. The scheme is based on the differential quadrature method in which the weighting coefficients are obtained by using trigonometric B-splines as a set of basis functions. This scheme reduces the equation into the system of first-order ordinary differential equation which is solved by adopting strong stability-preserving time-stepping Runge–Kutta scheme. The effects of the sundry parameters of interest on the velocity profiles are studied and the results are presented through graphs. It is observed that, the velocity increases from the horizontal channel to vertical channel. The velocity is a decreasing function of magnetic parameter. With an increase in time, the velocity increases.     

流体力学问题基于核重构思想的最小二乘配点法

将基于核重构思想的最小二乘配点法应用于流体力学问题,给出了离散二维不可压缩粘性流体非线性偏微分方程的最小二乘配点格式。为了检验该方法的有效性,以二维Stokes问题——Couette流动为典型算例,分别研究了正压与负压两种工况作用下Couette流动的速度分布。数值模拟结果表明,无论离散点是均匀分布还是随机分布,均给出了较准确的数值结果。     

Some simple unsteady unidirectional flows of a binary mixture of incompressible Newtonian fluids

The problems dealing with some simple unsteady unidirectional flows of a mixture of two incompressible Newtonian fluids are investigated. By using the constitutive equations appeared in the literature for binary mixtures of chemically inert incompressible Newtonian fluids, the equations governing the motion of the binary mixture are reduced to a system of coupled partial differential equations. By means of integral transforms, the exact solutions of these equations are obtained for the following three problems: (i) unsteady Couette flow, (ii) unsteady plane Poiseuille flow, (iii) unsteady axisymmetric Poiseuille flow.     

Explicit Corrections for the Effect of Viscous Heating in Circular Couette Viscometers

Based on asymptotic solutions to the problem of coupled flow and heat transfer in circular Couette flow of materials whose viscosity and thermal conductivity are polynomial functions of temperature, we obtain expressions for the effect of viscous heating on the gapwise distribution of shear rate under isothermal and adiabatic wall conditions. These expressions are shown to exhibit the anticipated asymptotic behavior as the gap-to-diameter ratio approaches unity and are in agreement with numerical results for a reasonable range of the Nahme number. Following that, we derive explicit rheological corrections for circular Couette–Hatschek viscometers; these account for the effect of viscous heating in the presence of temperature-dependent fluid properties and are reliable for values of the correction factor down to around 0.8.     

Electrohydrodynamic flow between parallel dielectric channels

Steady, accelerated, and pulsating electro-dynamic flows in a plane dielectric channel are considered, along with Couette flow. It is shown that for these types of electrohydrodynamic flows the effect is concentrated in a thin layer near the walls, which can considerably change the friction stress on the walls. Some exact solutions of the energy equation are obtained.     

A boundary-element analysis for transient viscoelastic blade coating flow

The one-dimensional plane Couette flow is first determined for a large class of Oldroyd fluids with added viscosity, which typically represent polymer solutions composed of a Newtonian solvent and a polymeric solute. Next, the determined channel velocity profile is used as the boundary condition at the channel exit for the blade coating flow. The free-surface evolution of the flow at the channel exit is simulated using the boundary element method. It is argued that the free surface flow can be assumed to be Newtonian. For the channel flow, the problem is reduced to a non-linear dynamical system using the Galerkin projection method. Stability analysis indicates that the velocity profile at the inlet may be linear or non-linear depending on the range of the Weissenberg number.     

The effect of an inhomogeneous distribution of surface temperature on Couette flow

The Couette flow for surfaces with an inhomogeneous distribution of temperature is considered. It is shown that heat fluxes and friction stresses for Knudsen numbers (Kn) larger than or on the order of unity can be significantly optimized by varying surface temperature distribution at a fixed mean temperature. For Kn ? 1, flows with an inhomogeneously distributed temperature are close to the Couette flow for a surface with corresponding mean temperature.     

Application of the Lambert W function to steady shearing flows of the Papanastasiou model

Using the Lambert W function, the constitutive relation of the Papanastasiou model is inverted so that the second invariant of the first Rivlin-Ericksen tensor can be expressed as a function of the second invariant of the extra stress tensor. In steady shearing flows, this results in the magnitude of the shear rate becoming a function of the magnitude of the shear stress. Since the distribution of the latter is known explicitly in channel, Poiseuille and Couette flows, one can investigate the nature of analytical solutions in these flows. It is shown that explicit answers are found for channel and Poiseuille flows only, with the Couette flow requiring a numerical solution in general. From the channel flow results, it is obvious that there is a great amount of congruence between the predictions of the Papanastasiou model and the Bingham fluid. In turn, this lends further confidence to the application of the Papanastasiou model to study the flows of Bingham fluids.     

Rapid injection linear dichroism for studying the kinetics of biological processes

Linear dichroism is defined as the differential absorbance of linearly polarized light oriented in two orthogonal directions by an aligned sample. The measurement of a linear dichroism (LD) spectrum of a sample provides two key pieces of structural information. First, that the sample and the chromophores within the sample are able to align. Second, given knowledge of the transition polarization directions of the chromophores, the orientation of the chromophores within the aligned sample can be resolved. It has been shown that LD can provide unique information on the structure of some of the more challenging biomolecular complexes. This has included macromolecular protein and peptide fibers such as actin, tubulin, and amyloids as well as protein-membrane complexes and DNA-protein complexes. Much of this work has been enabled by the development of a low volume Couette flow cell that efficiently aligns long molecules in solution. However, the current Couette system is inherently complex to assemble for each experiment and hence not suited to measurement of rapid reactions. In this paper we detail the development of the first rapid injection LD cell. The system utilizes a conventional stopped-flow injection system coupled to a modified low volume Couette cell, where a narrow bore capillary replaces the normal solid central rod. The system is shown to have similar optical characteristics to the conventional LD Couette flow cell but with the added benefit of a much shorter dead time (0.60 s compared to ～60). The rapid injection Couette cell has been used to measure the degradation of DNA by DNA exonuclease I, providing data that would not be available using a conventional system.     

Numerical and experimental study on the flow history effects of axial flow on the Couette–Taylor flow

In this research, experimental and numerical techniques are used to study the flow history effects of axial flow on the Couette–Taylor flow. For the experimental investigation, the flow is visualized using the PIV technique with reflective particles with a density of 1.62 g/cm³. Dispersed in a solution, the particles have a strong refraction index equal to 1.85. In this study, two protocols are adopted to study the effect of an axial flow superimposed on a Couette–Taylor flow, and of the history of the flow. The first one, the direct protocol, consists of imposing an azimuthal flow to the inner cylinder. In this case, when the regime is established, the axial flow is superimposed. The second protocol, the inverse protocol, consists of imposing first the axial flow in the gap of the system, after which an azimuthal flow is conveyed. The Couette–Taylor flow with axial flow is strongly dependent on the flow history (the protocol). Thus, the flow structures and development for different protocols are studied and analyzed here experimentally and numerically. In addition, from the numerical results, mathematical models for the two protocols are presented. For the direct protocol, a new relation between the axial Reynolds number, which stabilizes the Couette–Taylor flow, and the Taylor number is presented; for the inverse protocol, a new mathematical model for the critical Taylor number is developed as a function of the axial Reynolds number and also the first critical Taylor number without axial flow.