scholarly journals Numerical Simulation and Linearized Theory of Vortex Waves in a Viscoelastic, Polymeric Fluid

Fluids ◽  
2021 ◽  
Vol 6 (9) ◽  
pp. 325
Author(s):  
Robert Handler ◽  
Michael Buckingham
Keyword(s):  
Linear Theory ◽  
Elastic Energy ◽  
Time Scale ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Short Wave ◽  
Solution Viscosity ◽  
Wave Speeds ◽  
Polymeric Fluid ◽  
Linearized Theory

In a high viscosity, polymeric fluid initially at rest, the release of elastic energy produces vorticity in the form of coherent motions (vortex rings). Such behavior may enhance mixing in the low Reynolds number flows encountered in microfluidic applications. In this work, we develop a theory for such flows by linearizing the governing equations of motion. The linear theory predicts that when elastic energy is released in a symmetric manner, a wave of vorticity is produced with two distinct periods of wave motion: (1) a period of wave expansion and growth extending over a transition time scale, followed by (2) a period of wave translation and viscous decay. The vortex wave speeds are predicted to be proportional to the square root of the initial fluid tension, and the fluid tension itself scales as the viscosity. Besides verifying the predictions of the linearized theory, numerical solutions of the equations of motion for the velocity field, obtained using a pseudo-spectral method, show that the flow is composed of right- and left-traveling columnar vortex pairs, called vortex waves for short. Wave speeds obtained from the numerical simulations are within 1.5% of those from the linear theory when the assumption of linearity holds. Vortex waves are found to decay on a time scale of the order of the vortex size divided by the solution viscosity, in reasonable agreement with the analytical solution of the linearized model for damped vortex waves. When the viscoelastic fluid is governed by a nonlinear spring model, as represented by the Peterlin function, wave speeds are found to be larger than the predictions of the linear theory for small polymer extension lengths.

Dynamic Characteristics of a Hydrostatic Gas Bearing Driven by Oscillating Exhaust Pressure

1984 ◽  
Vol 106 (4) ◽  
pp. 477-483 ◽  
Author(s):  
C. B. Watkins ◽  
H. D. Branch ◽  
I. E. Eronini
Keyword(s):  
Reynolds Equation ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
Source Model ◽  
Regular Perturbation ◽  
Response Characteristics ◽  
Finite Difference Approximations ◽  
Bode Plots ◽  
Gas Bearing

Vibration of a statically loaded, inherently compensated hydrostatic journal bearing due to oscillating exhaust pressure is investigated. Both angular and radial vibration modes are analyzed. The time-dependent Reynolds equation governing the pressure distribution between the oscillating journal and sleeve is solved together with the journal equation of motion to obtain the response characteristics of the bearing. The Reynolds equation and the equation of motion are simplified by applying regular perturbation theory for small displacements. The numerical solutions of the perturbation equations are obtained by discretizing the pressure field using finite-difference approximations with a discrete, nonuniform line-source model which excludes effects due to feeding hole volume. An iterative scheme is used to simultaneously satisfy the equations of motion for the journal. The results presented include Bode plots of bearing-oscillation gain and phase for a particular bearing configuration for various combinations of parameters over a range of frequencies, including the resonant frequency.

scholarly journals Dynamics of nanodust particles emitted from elongated initial orbits

2018 ◽  
Vol 617 ◽  
pp. A43 ◽  
Author(s):  
A. Czechowski ◽  
I. Mann
Keyword(s):  
Solar Wind ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Dust Cloud ◽  
Theoretical Models ◽  
Ecliptic Plane ◽  
Parent Body ◽  
The Sun ◽  
Trapping Region ◽  
Eccentric Orbits

Context. Because of high charge-to-mass ratio, the nanodust dynamics near the Sun is determined by interplay between the gravity and the electromagnetic forces. Depending on the point where it was created, a nanodust particle can either be trapped in a non-Keplerian orbit, or escape away from the Sun, reaching large velocity. The main source of nanodust is collisional fragmentation of larger dust grains, moving in approximately circular orbits inside the circumsolar dust cloud. Nanodust can also be released from cometary bodies, with highly elongated orbits. Aims. We use numerical simulations and theoretical models to study the dynamics of nanodust particles released from the parent bodies moving in elongated orbits around the Sun. We attempt to find out whether these particles can contribute to the trapped nanodust population. Methods. We use two methods: the motion of nanodust is described either by numerical solutions of full equations of motion, or by a two-dimensional (heliocentric distance vs. radial velocity) model based on the guiding-center approximation. Three models of the solar wind are employed, with different velocity profiles. Poynting–Robertson and the ion drag are included. Results. We find that the nanodust emitted from highly eccentric orbits with large aphelium distance, like those of sungrazing comets, is unlikely to be trapped. Some nanodust particles emitted from the inbound branch of such orbits can approach the Sun to within much shorter distances than the perihelium of the parent body. Unless destroyed by sublimation or other processes, these particles ultimately escape away from the Sun. Nanodust from highly eccentric orbits can be trapped if the orbits are contained within the boundary of the trapping region (for orbits close to ecliptic plane, within ~0.16 AU from the Sun). Particles that avoid trapping escape to large distances, gaining velocities comparable to that of the solar wind.

scholarly journals Perturbation Solutions For Magnetohydrodynamics (Mhd) Flow of in a Non-Newtonian Fluid Between Concentric Cylinders

2019 ◽  
Vol 24 (1) ◽  
pp. 199-211
Author(s):  
M. Yürüsoy ◽  
Ö.F. Güler
Keyword(s):  
Analytical Solutions ◽  
Numerical Solutions ◽  
Variable Viscosity ◽  
Perturbation Technique ◽  
Equations Of Motion ◽  
Mhd Flow ◽  
Model Space ◽  
Viscosity Model ◽  
Concentric Cylinders ◽  
Approximate Analytical Solutions

Abstract The steady-state magnetohydrodynamics (MHD) flow of a third-grade fluid with a variable viscosity parameter between concentric cylinders (annular pipe) with heat transfer is examined. The temperature of annular pipes is assumed to be higher than the temperature of the fluid. Three types of viscosity models were used, i.e., the constant viscosity model, space dependent viscosity model and the Reynolds viscosity model which is dependent on temperature in an exponential manner. Approximate analytical solutions are presented by using the perturbation technique. The variation of velocity and temperature profile in the fluid is analytically calculated. In addition, equations of motion are solved numerically. The numerical solutions obtained are compared with analytical solutions. Thus, the validity intervals of the analytical solutions are determined.

Leading-Edge Corrections to the Linear Theory of Partially Cavitating Hydrofoils

1991 ◽  
Vol 35 (01) ◽  
pp. 15-27
Author(s):  
Spyros A. Kinnas
Keyword(s):  
Integral Equations ◽  
Linear Theory ◽  
Leading Edge ◽  
Foil Thickness ◽  
System Of Integral Equations ◽  
Cavitation Inception ◽  
Perturbation Velocity ◽  
Linearized Theory ◽  
Work First ◽  
Cavitating Hydrofoil

In this work, first, the partially cavitating hydrofoil problem is formulated in linear theory in terms of vorticity and source distributions on the projection of the hydrofoil to the free-stream direction. The resulting system of integral equations is inverted and the solution is expressed in terms of integrals of the horizontal perturbation velocity in fully wetted flow, multiplied by weighting functions that are independent of the shape of the hydrofoil. Second, the linearized dynamic boundary condition on the cavity is modified so that the total velocity on the cavity as predicted by applying Lighthill's leading-edge corrections is a constant. This results in a varying horizontal perturbation velocity on the cavity rather than a constant as required by conventional linear theory. The modified system of integral equations is inverted and the solution is expressed in terms of integrals of known quantities. The present linearized theory with the leading-edge corrections included, predicts a finite cavitation inception number as well as the correct effect of foil thickness on cavity size.

Dynamic response of a pre-stressed bi-layered plate-strip subjected to an arbitrary inclined time-harmonic force

2017 ◽  
Vol 26 (3) ◽  
pp. 255-262
Author(s):  
AHMET DASDEMIR ◽  
Keyword(s):  
Dynamic Response ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Layered Plate ◽  
Field Problem ◽  
Time Harmonic ◽  
Governing System ◽  
Plate Strip ◽  
Linearized Theory ◽  
Complete Contact

Within the scope of the piecewise homogeneous body model with utilizing of the three dimensional linearized theory of elastic waves in initially stressed bodies the dynamical stress field problem in a bi-layered plate-strip with initial stress under the action of an arbitrary inclined timeharmonic force resting on a rigid foundation is investigated. The concrete materials such as a pair of Aluminum and Steel are selected. It is assumed that there exists a complete contact interaction between the layers. The mathematical modeling of the problem under consideration is carved out, and the governing system of the partial differential equations of motion is approximately solved by employing Finite Element Method. The numerical results related to the influence of certain parameters on the dynamic response of the plate-strip are presented.

scholarly journals Resonant Stellar Orbits in Spiral Galaxies

1975 ◽  
Vol 69 ◽  
pp. 237-244
Author(s):  
P. O. Vandervoort
Keyword(s):  
Excellent Agreement ◽  
Spiral Galaxy ◽  
Harmonic Balance ◽  
Spiral Galaxies ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Analytic Solutions ◽  
Stellar Orbits ◽  
Method Of Harmonic Balance ◽  
Resonance Phenomena

This paper reviews a series of investigations of the orbits of stars in the regions of the Lindblad resonances of a spiral galaxy. The analysis is formulated in an epicyclic approximation. Analytic solutions of the epicyclic equations of motion are obtained by the method of harmonic balance of Bogoliubov and Mitropolsky. These solutions represent the resonance phenomena exhibited by the orbits in generally excellent agreement with numerical solutions.

scholarly journals The Dynamical Behavior of a Rigid Body Relative Equilibrium Position

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
T. S. Amer
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Relative Equilibrium ◽  
Dynamical Behavior ◽  
Vertical Plane ◽  
The Body ◽  
Physical Parameters ◽  
Pendulum Model

In this paper, we will focus on the dynamical behavior of a rigid body suspended on an elastic spring as a pendulum model with three degrees of freedom. It is assumed that the body moves in a rotating vertical plane uniformly with an arbitrary angular velocity. The relative periodic motions of this model are considered. The governing equations of motion are obtained using Lagrange’s equations and represent a nonlinear system of second-order differential equations that can be solved in terms of generalized coordinates. The numerical solutions are investigated using the fourth-order Runge-Kutta algorithms through Matlab packages. These solutions are represented graphically in order to describe and discuss the behavior of the body at any instant for different values of the physical parameters of the body. The obtained results have been discussed and compared with some previous published works. Some concluding remarks have been presented at the end of this work. The importance of this work is due to its numerous applications in life such as the vibrations that occur in buildings and structures.

Numerical solutions for the spin-up of a stratified fluid

1982 ◽  
Vol 117 ◽  
pp. 71-90 ◽  
Author(s):  
Jae Min Hyun ◽  
William W. Fowlis ◽  
Alex Warn-Varnas
Keyword(s):  
Time Scale ◽  
Numerical Solutions ◽  
Side Wall ◽  
Stratified Fluid ◽  
Meridional Flow ◽  
Laser Doppler Velocimeter ◽  
Numerical Work ◽  
Diffusion Effects ◽  
Rate Of Decay ◽  
Flow Gradients

Numerical solutions for the impulsively started spin-up of a thermally stratified fluid in a cylinder with an insulating side wall are presented. Previous experimental and numerical work on stratified spin-up had not provided a comprehensive and accurate set of flow-field data. Further, comparisons of this work with theory showed, in general, a substantial discrepancy. The theory was scaled using the homogeneous meridional-flow spin-up time scale and thus viscous-diffusion effects were excluded from the interior. It was anticipated that these effects could only be significant on the larger viscous-diffusion time scale. However, the comparisons with theory showed a faster rate of decay for the measurements even over the shorter meridional-flow spin-up time scale. Previous workers had suggested a number of explanations but the cause of the discrepancy was still unresolved. To provide data to extend the previous work, a numerical model was used. The model was first checked against accurate experimental measurements of stratified spin-up made using a laser-Doppler velocimeter. New accurate results which cover ranges of Ekman number (5·92 × 10−4 ≤ E ≤ 7·24 × 10−4), Rossby number (0·019 ≤ ε ≤ 0·220), stratification parameter (0·0 ≤ Sa−1 ≤ 1·03), and Prandtl number (5·68 ≤ σ ≤ 7·10) are presented. These results show the radial and vertical structure of the decaying azimuthal and meridional flows. The inertial–internal gravity oscillations excited by the impulsive spin-up are clearly seen. By making use of conclusions from the previous work and the results presented in this paper, it is established that viscous diffusion in the interior is the cause of the discrepancy with theory. Stratification causes the meridional spin-up flow to be confined closer to the boundary disks. This results in non-uniform spin-up of the interior and hence flow gradients in the interior. These gradients introduce viscous diffusion into the interior sooner than anticipated by the theory. A previous suggestion that the faster decay rate is due to angular momentum being injected into the interior from an oscillation of the meridional corner-jet flow is shown to be untenable.

Concurrent Multiple-Time-Scale Simulations With Improved Numerical Dissipation for Structural Dynamics

2013 ◽  
Author(s):  
Tejas Ruparel ◽  
Azim Eskandarian ◽  
James Lee
Keyword(s):  
Time Scale ◽  
Equations Of Motion ◽  
Time Integration ◽  
Nonholonomic Constraints ◽  
Numerical Dissipation ◽  
Multiple Time ◽  
Multiple Time Scale ◽  
Constraint Forces ◽  
Conservation Of Energy ◽  
Coupled Equations

Work presented in this paper describes the formulation for implementation of a concurrent multiple-time-scale integration method with improved numerical dissipation capabilities. This approach generalizes the previous Multiple Grid and Multiple Time-Scale (MGMT) Method [1] implemented for the Newmark family of algorithms. The framework is largely based upon the fundamental principles of Lagrange multipliers used to enforce workless nonholonomic constraints and Domain Decomposition Methods (DDM) to obtain coupled equations of motion for distinct regions of a continuous domain. These methods when combined together systematically yield constraint forces that not only ensure conservation of energy but also enforce continuity of velocities across the interfaces. Multiple grid connections between (non-conforming) sub-domains are handled using Mortar elements whereas coupled multiple-time-scale equations are derived for the Generalized-α Method [2]. We show that MGMT Method can be easily extended to incorporate the Generalized-α family of time integration algorithms, hence allowing selective discretization in space and time along with controlled numerical dissipation for distinct grids. We also show that interface energy across connecting sub-domains is identically zero, further assuring global energy balance and continuity of velocities across connecting sub-domains.

Nonlinear Dynamics of Rotating Structures and Helicopter Blades

2018 ◽  
Author(s):  
Matteo Filippi ◽  
Alfonso Pagani ◽  
Erasmo Carrera
Keyword(s):  
Numerical Solutions ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Constitutive Law ◽  
Rotating Blades ◽  
Helicopter Blades ◽  
Strain Components ◽  
Theory Approximation ◽  
Nonlinear Equations Of Motion ◽  
Lagrange Strain

This work explores the effects of geometrical nonlinearities in the vibration analysis of rotating structures and helicopter blades. Structures are modelled via higher-order beam theories with variable kinematics. These theories fall in the domain of the Carrera Unified Formulation (CUF), according to which the nonlinear equations of motion of rotating blades can be written in terms of fundamental nuclei, whose formalism is an invariant of the theory approximation. The inherent three-dimensional nature of CUF enables one to include all Green-Lagrange strain components as well as all coupling effects due to the geometrical features and the three-dimensional constitutive law. Numerical solutions are considered and opportunely discussed. Also, linearized and full nonlinear solutions for vibrating rotating blades are compared both in case of small amplitudes and in the large deflections/rotations regime.

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