The influence of Prandtl number on free convection in a rectangular cavity

Natural convection in a rectangular cavity is considered for the problem where one vertical wall is heated and the other is cooled. The boundary layer flow is solved using a modified Oseen technique in a manner similar to Gill's solution. Temperature and velocity profiles in the core, and the Nusselt number, are found as functions of the Rayleigh and Prandtl numbers and the length ratio. The solution indicates that for a Prandtl number less than 1/7, a midsection shear layer develops.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/24514/1/0000792.pd

A galerkin method for calculation of laminar thermal boundary layers

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/24419/1/0000690.pd

Elasticity of tissues involved in accommodation

Uniaxial loading of human lens, zonules, ciliary muscle, and choroid shows a nonlinear relationship between stress and deformation of the specimen, and hysteresis on unloading. Spring constants, at 10% elongation, have been calculated for all specimens and for several combinations. Except for zonules they are significantly correlated with age. They will provide a basis for a model of accommodation. The mean force required for 10% elongation of the lens, at mean age 43, is about 22 x greater than for the zonules; and the mean elongation of the zonules is nearly 7 x that of the lens at a load of 0.01 N (~ 1 g). Ranges of forces and pressures in the accommodated and nonaccommodated eye are estimated.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/29667/1/0000756.pd

The Second Stokes Problem with Specular - Diffusive Boundary Conditions in Kinetic Theory

The second Stokes problem with specular - diffusive boundary conditions of the kinetic theory is considered. The new method of the decision of the boundary problems of the kinetic theory is applied. The method allows to receive the decision with any degree of accuracy. At the basis of a method lays the idea of representation of a boundary condition on distribution function in the form of a source in the kinetic equation. By means of integrals Fourier the kinetic equation with a source is reduced to the integral equation of Fredholm type of the second kind. The decision is received in the form of Neumann's series.Comment: 20 pages, 2 figure

Multi-Dimensional, Compressible Viscous Flow on a Moving Voronoi Mesh

Numerous formulations of finite volume schemes for the Euler and Navier-Stokes equations exist, but in the majority of cases they have been developed for structured and stationary meshes. In many applications, more flexible mesh geometries that can dynamically adjust to the problem at hand and move with the flow in a (quasi) Lagrangian fashion would, however, be highly desirable, as this can allow a significant reduction of advection errors and an accurate realization of curved and moving boundary conditions. Here we describe a novel formulation of viscous continuum hydrodynamics that solves the equations of motion on a Voronoi mesh created by a set of mesh-generating points. The points can move in an arbitrary manner, but the most natural motion is that given by the fluid velocity itself, such that the mesh dynamically adjusts to the flow. Owing to the mathematical properties of the Voronoi tessellation, pathological mesh-twisting effects are avoided. Our implementation considers the full Navier-Stokes equations and has been realized in the AREPO code both in 2D and 3D. We propose a new approach to compute accurate viscous fluxes for a dynamic Voronoi mesh, and use this to formulate a finite volume solver of the Navier-Stokes equations. Through a number of test problems, including circular Couette flow and flow past a cylindrical obstacle, we show that our new scheme combines good accuracy with geometric flexibility, and hence promises to be competitive with other highly refined Eulerian methods. This will in particular allow astrophysical applications of the AREPO code where physical viscosity is important, such as in the hot plasma in galaxy clusters, or for viscous accretion disk models.Comment: 26 pages, 21 figures. Submitted to MNRA

Transient Stage Comparison of Couette Flow under Step Shear Stress and Step Velocity Boundary Conditions

Couette flow has been widely used in many industrial and research processes, such as viscosity measurement. For the study on thixotropic viscosity, step-loading, which includes (1) step shear stress and (2) step velocity conditions, is widely used. Transient stages of Couette flow under both step wall shear stress and step wall velocity conditions were investigated. The relative coefficient of viscosity was proposed to reflect the transient process. Relative coefficients of viscosity, dimensionless velocities and dimensionless development times were derived and calculated numerically. This article quantifies the relative coefficients of viscosity as functions of dimensionless time and step ratios when the boundary is subjected to step changes. As expected, in the absence of step changes, the expressions reduce to being functions of dimensionless time. When step wall shear stresses are imposed, the relative coefficients of viscosity changes from the values of the step ratios to their steady-state value of 1. but With step-increasing wall velocities, the relative coefficients of viscosity decrease from positive infinity to 1. The relative coefficients of viscosity increase from negative infinity to 1 under the step-decreasing wall velocity condition. During the very initial stage, the relative coefficients of viscosity under step wall velocity conditions is further from 1 than the one under step wall shear stress conditions but the former reaches 1 faster. Dimensionless development times grow with the step ratio under the step-rising conditions and approaches the constant value of 1.785 under the step wall shear stress condition, and 0.537 under the step wall velocity condition respectively. The development times under the imposed step wall shear stress conditions are always larger than the same under the imposed step wall velocity conditions

Does transient increase in axial length during accommodation attenuate with age?

Background: The aim was to profile transient accommodative axial length (AXL) changes from early adulthood to advanced presbyopia and to determine whether any differences exist between the responses of myopic and emmetropic individuals. Methods: Ocular biometry was measured by the LenStar biometer (Haag-Streit, Switzerland) in response to 0.00, 3.00 and 4.50 D accommodative stimuli in 35 emmetropes and 37 myopes, aged 18 to 60 years. All results were corrected to reduce errors arising from the increase in crystalline lens thickness with accommodation. Accommodative responses were measured sequentially by the WAM 5500 Auto Ref/Keratometer (Grand Seiko, Japan). Results: AXL increased significantly with accommodation (p<0.001), with a mean corrected AXL elongation of 2 ± 18 µm and 8 ± 16 µm observed at 3.00 D and 4.50 D, respectively. The magnitude of accommodative AXL change was not dependent on refractive error classification (p=0.959), however a significant reduction in the magnitude and variance of AXL change was evident after 43-44 years of age (p<0.002). Conclusion: The negative association between transient AXL elongation and age, in combination with reduced variance of data after age 43-44 years, is consistent with a significant increase in posterior ocular rigidity, which may be influential in the development of presbyopia