Application of Third‐Order Constitutive Relations to Poiseuille Flow of a Rarefied Gas

In this paper, a finite element model for efficient nonlinear analysis of the mechanical response of viscoelastic beams is presented. The principle of virtual work is utilized in conjunction with the third-order beam theory to develop displacement-based, weak-form Galerkin finite element model for both quasi-static and fully-transient analysis. The displacement field is assumed such that the third-order beam theory admits C0 Lagrange interpolation of all dependent variables and the constitutive equation can be that of an isotropic material. Also, higher-order interpolation functions of spectral/hp type are employed to efficiently eliminate numerical locking. The mechanical properties are considered to be linear viscoelastic while the beam may undergo von Kármán nonlinear geometric deformations. The constitutive equations are modeled using Prony exponential series with general n-parameter Kelvin chain as its mechanical analogy for quasi-static cases and a simple two-element Maxwell model for dynamic cases. The fully discretized finite element equations are obtained by approximating the convolution integrals from the viscous part of the constitutive relations using a trapezoidal rule. A two-point recurrence scheme is developed that uses the approximation of relaxation moduli with Prony series. This necessitates the data storage for only the last time step and not for the entire deformation history.

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Poiseuille flow in a capillary at all regimes of rarefied gas flows

10.1063/1.4769628 ◽

2012 ◽

Cited By ~ 2

Author(s):

V. V. Zhvick ◽

O. G. Friedlander

Keyword(s):

Poiseuille Flow ◽

Rarefied Gas ◽

Gas Flows ◽

Rarefied Gas Flows

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Plane Thermal Transpiration of a Rarefied Gas Between Two Walls of Maxwell-Type Boundaries With Different Accommodation Coefficients

Journal of Fluids Engineering ◽

10.1115/1.4026926 ◽

2014 ◽

Vol 136 (8) ◽

Cited By ~ 7

Author(s):

Toshiyuki Doi

Keyword(s):

Boltzmann Equation ◽

Mass Flow Rate ◽

Knudsen Number ◽

Mass Flow ◽

Flow Rate ◽

Poiseuille Flow ◽

Rarefied Gas ◽

Thermal Transpiration ◽

Wide Range ◽

Accommodation Coefficients

Plane thermal transpiration of a rarefied gas between two walls of Maxwell-type boundaries with different accommodation coefficients is studied based on the linearized Boltzmann equation for a hard-sphere molecular gas. The Boltzmann equation is solved numerically using a finite difference method, in which the collision integral is evaluated by the numerical kernel method. The detailed numerical data, including the mass and heat flow rates of the gas, are provided over a wide range of the Knudsen number and the entire range of the accommodation coefficients. Unlike in the plane Poiseuille flow, the dependence of the mass flow rate on the accommodation coefficients shows different characteristics depending on the Knudsen number. When the Knudsen number is relatively large, the mass flow rate of the gas increases monotonically with the decrease in either of the accommodation coefficients like in Poiseuille flow. When the Knudsen number is small, in contrast, the mass flow rate does not vary monotonically but exhibits a minimum with the decrease in either of the accommodation coefficients. The mechanism of this phenomenon is discussed based on the flow field of the gas.

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Poiseuille Flow and Thermal Transpiration of a Rarefied Gas Between Parallel Plates: Effect of Nonuniform Surface Properties of the Plates in the Transverse Direction

Journal of Fluids Engineering ◽

10.1115/1.4030490 ◽

2015 ◽

Vol 137 (10) ◽

Cited By ~ 6

Author(s):

Toshiyuki Doi

Keyword(s):

Distribution Function ◽

Flow Rate ◽

Poiseuille Flow ◽

Transverse Direction ◽

Rarefied Gas ◽

Mean Free Path ◽

Accommodation Coefficient ◽

Parallel Plates ◽

Nonuniform Surface ◽

The Mean

Poiseuille flow and thermal transpiration of a rarefied gas between parallel plates with nonuniform surface properties in the transverse direction are studied based on kinetic theory. We considered a simplified model in which one wall is a diffuse reflection boundary and the other wall is a Maxwell-type boundary on which the accommodation coefficient varies periodically and smoothly in the transverse direction. The spatially two-dimensional (2D) problem in the cross section is studied numerically based on the linearized Bhatnagar–Gross–Krook–Welander (BGKW) model of the Boltzmann equation. The flow behavior, i.e., the macroscopic flow velocity and the mass flow rate of the gas as well as the velocity distribution function, is studied over a wide range of the mean free path of the gas and the parameters of the distribution of the accommodation coefficient. The mass flow rate of the gas is approximated by a simple formula consisting of the data of the spatially one-dimensional (1D) problems. When the mean free path is large, the distribution function assumes a wavy variation in the molecular velocity space due to the effect of a nonuniform surface property of the plate.

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