scholarly journals Numerical solutions for unsteady gravity-driven flows in collapsible tubes: evolution and roll-wave instability of a steady state

1999 ◽  
Vol 396 ◽  
pp. 223-256 ◽  
Author(s):  
B. S. BROOK ◽  
S. A. E. G. FALLE ◽  
T. J. PEDLEY
Keyword(s):  
Shallow Water ◽  
Numerical Solutions ◽  
Godunov Method ◽  
Test Case ◽  
Interesting Phenomenon ◽  
One Dimensional ◽  
Collapsible Tubes ◽  
Highly Nonlinear ◽  
Pressure Area ◽  
The One

Unsteady flow in collapsible tubes has been widely studied for a number of different physiological applications; the principal motivation for the work of this paper is the study of blood flow in the jugular vein of an upright, long-necked subject (a giraffe). The one-dimensional equations governing gravity- or pressure-driven flow in collapsible tubes have been solved in the past using finite-difference (MacCormack) methods. Such schemes, however, produce numerical artifacts near discontinuities such as elastic jumps. This paper describes a numerical scheme developed to solve the one-dimensional equations using a more accurate upwind finite volume (Godunov) scheme that has been used successfully in gas dynamics and shallow water wave problems. The adapatation of the Godunov method to the present application is non-trivial due to the highly nonlinear nature of the pressure–area relation for collapsible tubes.The code is tested by comparing both unsteady and converged solutions with analytical solutions where available. Further tests include comparison with solutions obtained from MacCormack methods which illustrate the accuracy of the present method.Finally the possibility of roll waves occurring in collapsible tubes is also considered, both as a test case for the scheme and as an interesting phenomenon in its own right, arising out of the similarity of the collapsible tube equations to those governing shallow water flow.

Diffusion-Limited Cell Dehydration: Analytical and Numerical Solutions for a Planar Model

1999 ◽  
Author(s):  
Alexander V. Kasharin ◽  
Jens O. M. Karlsson
Keyword(s):  
Analytical Solution ◽  
Numerical Solutions ◽  
Planar System ◽  
One Dimensional ◽  
Diffusion Limited ◽  
Dimensional Diffusion ◽  
Constant Diffusivity ◽  
A Cell ◽  
Cell Dehydration ◽  
The One

Abstract The process of diffusion-limited cell dehydration is modeled for a planar system by writing the one-dimensional diffusion-equation for a cell with moving, semipermeable boundaries. For the simplifying case of isothermal dehydration with constant diffusivity, an approximate analytical solution is obtained by linearizing the governing partial differential equations. The general problem must be solved numerically. The Forward Time Center Space (FTCS) and Crank-Nicholson differencing schemes are implemented, and evaluated by comparison with the analytical solution. Putative stability criteria for the two algorithms are proposed based on numerical experiments, and the Crank-Nicholson method is shown to be accurate for a mesh with as few as six nodes.

Vibration of an elastic strip with varying curvature

1992 ◽  
Vol 339 (1655) ◽  
pp. 587-625 ◽  
Keyword(s):  
Total Internal Reflection ◽  
Normal Modes ◽  
Mode Shapes ◽  
Elastic Strip ◽  
Interesting Phenomenon ◽  
One Dimensional ◽  
End Conditions ◽  
The One ◽  
Fixed End ◽  
Good Agreement

The vibrational behaviour of an elastic strip with varying curvature is investigated. The case of vibration which is predominantly transverse is considered, and it is shown that when the strip is S-shaped, certain of the normal modes may be confined to the vicinity of the inflection point of the S by a process of total internal reflection from points where the curvature reaches critical values. This confinement can produce modes with extraordinarily low damping factors. Asymptotic analysis is compared with experimental measurements on a strip in several S-shaped configurations, and very good agreement is demonstrated for modal frequencies and shapes. Mathematically, the lower modes turn out to be analogous to those of the one-dimensional harmonic oscillator in quantum mechanics. This mode confinement behaviour occurs for all waveguide branches except the lowest, ‘bending beam ’, branch. In this particular case, wave propagation is insensitive to curvature. However, an interesting phenomenon associated with curvature is found : the successive mode shapes do not display the normal alternation of symmetry and antisymmetry with respect to the centre of the strip. The effect is shown to result from the constraint on axial movement produced by fixed end conditions. For the geometry of the experiments, this constraint raises the frequencies of antisymmetric modes in a characteristic way while leaving the symmetric modes unaltered, thus changing the mode sequence. Theory is developed which gives reasonable quantitive agreement with the observations.

scholarly journals FULLY DISCRETE SCHEMES FOR THE SCHRÖDINGER EQUATION: DISPERSIVE PROPERTIES

2007 ◽  
Vol 17 (04) ◽  
pp. 567-591 ◽  
Author(s):  
LIVIU I. IGNAT
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Numerical Solutions ◽  
Continuous Model ◽  
Local Smoothing ◽  
Euler Approximation ◽  
One Dimensional ◽  
Fully Discrete ◽  
Backward Euler ◽  
The One

We consider fully discrete schemes for the one-dimensional linear Schrödinger equation and analyze whether the classical dispersive properties of the continuous model are presented in these approximations. In particular, Strichartz estimates and the local smoothing of the numerical solutions are analyzed. Using a backward Euler approximation of the linear semigroup we introduce a convergent scheme for the nonlinear Schrödinger equation with nonlinearities which cannot be treated by energy methods.

Subcritical Flutter in Collapsible Tube Flow: A Model of Expiratory Flow in the Trachea

1991 ◽  
Vol 113 (1) ◽  
pp. 21-26 ◽  
Author(s):  
C. Walsh ◽  
P. A. Sullivan ◽  
J. S. Hansen
Keyword(s):  
Fluid Model ◽  
One Dimensional ◽  
Shell Effects ◽  
Collapsible Tubes ◽  
Dimensional Modeling ◽  
Axial Tension ◽  
Normal Wall ◽  
Axial Curvature ◽  
The One ◽  
Axial Stretching

Using an axisymmetric geometry that retains certain qualitative features of the trachea, we extend one-dimensional modeling of flow in collapsible tubes to include both curved shell effects and, for untethered tubes, wall inertia. A systematic scaling of the finite deformation membrane equations leads to an approximate set which is consistent with the one-dimensional fluid model; axial and normal wall variables are coupled elastically, but only axial inertia is retained. Transverse curvature causes elastic coupling that can give rise to axial wall motion and a flutter instability. The source of instability is the product of a nonzero reference axial curvature with axial tension variation due to axial stretching. The numerical results suggest that this mechanism may be significant even in processes which cannot be assumed one-dimensional.

scholarly journals 1D and 2D Numerical Modeling for Solving Dam-Break Flow Problems Using Finite Volume Method

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Szu-Hsien Peng
Keyword(s):  
Shallow Water ◽  
Finite Volume Method ◽  
Finite Volume ◽  
Shallow Water Equation ◽  
Dam Break ◽  
Flume Experiments ◽  
One Dimensional ◽  
Dam Break Flow ◽  
The One ◽  
Volume Method

The purpose of this study is to model the flow movement in an idealized dam-break configuration. One-dimensional and two-dimensional motion of a shallow flow over a rigid inclined bed is considered. The resulting shallow water equations are solved by finite volumes using the Roe and HLL schemes. At first, the one-dimensional model is considered in the development process. With conservative finite volume method, splitting is applied to manage the combination of hyperbolic term and source term of the shallow water equation and then to promote 1D to 2D. The simulations are validated by the comparison with flume experiments. Unsteady dam-break flow movement is found to be reasonably well captured by the model. The proposed concept could be further developed to the numerical calculation of non-Newtonian fluid or multilayers fluid flow.

Cure Monitoring of Adhesive for Composite/Metal Bonded Structure Based on Highly Nonlinear Solitary Waves

2019 ◽  
Author(s):  
Wu Bin ◽  
Li Mingzhi ◽  
Liu Xiucheng ◽  
Wang Heying ◽  
He Cunfu ◽  
...  
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Curing Process ◽  
One Dimensional ◽  
Granular Chain ◽  
Highly Nonlinear ◽  
Highly Nonlinear Solitary Waves ◽  
Simulation Results ◽  
The One ◽  
Nonlinear Solitary Waves

Abstract In this paper, a nondestructive evaluation technique based on highly nonlinear solitary waves (HNSWs) is proposed to monitor the curing process of adhesive for composite/metal bonded structure. HNSWs are mechanical waves with high energy intensity and non-distortive nature which can form and propagate in a nonlinear system, such as a one-dimensional granular chain. In the present study, a finite element model of the one-dimensional granular chain is established with the commercial software Abaqus, to study the reflection behavior of HNSWs at the interface between the particle at the end of chain and the sample. The simulation results show that the time of flight (TOF) of the primary reflected solitary wave decreases with the stiffness of the sample increases, and the amplitude ratio (AR) between the primary reflected solitary wave and the incident solitary wave increases. An HNSWs transducer based on the one-dimensional granular chain is designed and fabricated. The relationship between the characteristic parameters of the primary reflected solitary wave (TOF and AR) and the curing time of adhesive for a composite/metal bonded structure is experimentally investigated. The experiment results suggest that the TOF decreases and the AR increases as the epoxy cures. The experimental results are in good agreement with the simulation results. This study provides a new characterization method for monitoring the curing process of adhesive for composite/metal bonded structure.

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