scholarly journals A note on non-symmetric flow: surface shrinking in mutually orthogonal directions

Meccanica ◽  
2021 ◽  
Author(s):  
J. H. Merkin ◽  
Y. Y. Lok ◽  
I. Pop
Keyword(s):  
Critical Points ◽  
Numerical Solutions ◽  
Permeable Surface ◽  
Asymptotic Solutions ◽  
Symmetric Flow ◽  
Flow Surface ◽  
Fluid Transfer

AbstractIn this note, we extend the problem treated in (Lok, Math Modelling Anal 24:617–634 (2019)) to the case of permeable surface which is shrinking in mutually orthogonal directions. Both numerical and asymptotic solutions are obtained for two important governing parameters, $$\gamma $$ γ the shrinking rate and S characterizing the fluid transfer through the boundary. In this problem, a restriction on S is required for a solution to exist. This contrasts with the problem in (Lok, Math Modelling Anal 24:617–634 (2019)) where no restriction on S is needed. Numerical solutions show that for a fixed value of S, two critical points $$\gamma _c$$ γ c are observed for $$S > 2$$ S > 2 . Conversely, two critical points $$S_c$$ S c are found for a given value of $$\gamma $$ γ when $$S > 2$$ S > 2 . A discussion on the nonexistence of solution for $$S = 2$$ S = 2 is given and asymptotic solutions for S large and $$(S-2)$$ ( S - 2 ) small are also presented.

scholarly journals NON-SYMMETRIC FLOW OVER A STRETCHING/SHRINKING SURFACE WITH MASS TRANSFER

2019 ◽  
Vol 24 (4) ◽  
pp. 617-634 ◽  
Author(s):  
Yian Yian Lok ◽  
John H. Merkin ◽  
Ioan Pop
Keyword(s):  
Mass Transfer ◽  
Numerical Solutions ◽  
Numerical Results ◽  
Critical Values ◽  
Symmetric Flow ◽  
Dimensionless Parameters ◽  
Fluid Transfer ◽  
Shrinking Surface ◽  
Quiescent Fluid ◽  
Asymptotic Forms

The non-symmetric flow over a stretching/shrinking surface in an other-wise quiescent fluid is considered under the assumption that the surface can stretch orshrink in one direction and stretch in a direction perpendicular to this. The problem is reduced to similarity form, being described by two dimensionless parameters, γ the relative stretching/shrinking rate and S characterizing the fluid transfer throughthe boundary. Numerical solutions are obtained for representative values of γ and S, a feature of which are the existence of critical values γc of γ dependent on S, these being determined numerically. Asymptotic forms for large γ and S, for both fluid withdrawal, S > 0 and injection S < 0 are obtained and compared with the corresponding numerical results.

Topological vortex dynamics in axisymmetric viscous flows

1994 ◽  
Vol 260 ◽  
pp. 57-80 ◽  
Author(s):  
Mogens V. Melander ◽  
Fazle Hussain
Keyword(s):  
Phase Portrait ◽  
Critical Points ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Saddle Points ◽  
Time History ◽  
Vorticity Field ◽  
Vortex Lines ◽  
Short Period ◽  
Virtual Velocity

The topology of vortex lines and surfaces is examined in incompressible viscous axisymmetric flows with swirl. We argue that the evolving topology of the vorticity field must be examined in terms of axisymmetric vortex surfaces rather than lines, because only the surfaces enjoy structural stability. The meridional cross-sections of these surfaces are the orbits of a dynamical system with the azimuthal circulation being a Hamiltonian H and with time as a bifurcation parameter μ. The dependence of H on μ is governed by the Navier–Stokes equations; their numerical solutions provide H. The level curves of H establish a time history for the motion of vortex surfaces, so that the circulation they contain remains constant. Equivalently, there exists a virtual velocity field in which the motion of the vortex surfaces is frozen almost everywhere; the exceptions occur at critical points in the phase portrait where the virtual velocity is singular. The separatrices emerging from saddle points partition the phase portrait into islands; each island corresponds to a structurally stable vortex structure. By using the flux of the meridional vorticity field, we obtain a precise definition of reconnection: the transfer of flux between islands. Local analysis near critical points shows that the virtual velocity (because of its singular behaviour) performs ‘cut-and-connect’ of vortex surfaces with the correct rate of circulation transfer - thereby validating the long-standing viscous ‘cut-and-connect’ scenario which implicitly assumes that vortex surfaces (and vortex lines) can be followed over a short period of time in a viscous fluid. Bifurcations in the phase portrait represent (contrary to reconnection) changes in the topology of the vorticity field, where islands spontaneously appear or disappear. Often such topology changes are catastrophic, because islands emerge or perish with finite circulation. These and other phenomena are illustrated by direct numerical simulations of vortex rings at a Reynolds number of 800.

Thermo-Mechanical Stress near Apex of a Bi-Material Wedge by a Novel Finite Element Analysis

2012 ◽  
Vol 525-526 ◽  
pp. 93-96
Author(s):  
Xue Cheng Ping ◽  
Lin Leng ◽  
Si Hai Wu
Keyword(s):  
Finite Element Method ◽  
Finite Element Analysis ◽  
Finite Element ◽  
Finite Elements ◽  
Mechanical Stress ◽  
Displacement Field ◽  
Thermal Loading ◽  
Numerical Solutions ◽  
Asymptotic Solutions ◽  
Element Analysis

A super wedge tip element for application to a bi-material wedge is develop utilizing the thermo-mechanical stress and displacement field solutions in which the singular parts are numerical solutions. Singular stresses near apex of an arbitrary bi-material wedge under mechanical and thermal loading can be obtained from the coupling between the super wedge tip element and conventional finite elements. The validity of this novel finite element method is established through existing asymptotic solutions and conventional detailed finite element analysis.

On the Accuracy of Some Shell Solutions

1959 ◽  
Vol 26 (4) ◽  
pp. 577-583
Author(s):  
G. D. Galletly ◽  
J. R. M. Radok
Keyword(s):  
Negative Curvature ◽  
Numerical Solutions ◽  
Asymptotic Solutions ◽  
Thin Shells ◽  
Shells Of Revolution ◽  
Equilibrium Equations ◽  
Influence Coefficients ◽  
Bending Loads ◽  
Edge Influence

Abstract R. B. Dingle’s method [1] for finding asymptotic solutions of ordinary differential equations of a type such as occur in the bending theory of thin shells of revolution is presented in outline. This method leads to the same results as R. E. Langer’s method [2], recently used for problems of this kind, and permits a simple analytical and less formal interpretation of the asymptotic treatment of such equations. A comparison is given of edge influence coefficients due to bending loads, obtained by use of these asymptotic solutions and numerical integration of the equilibrium equations, respectively. The particular shells investigated are of the open-crown, ellipsoidal, and negative-curvature toroidal types. The results indicate that the agreement between these solutions is satisfactory. In the presence of uniform pressure, the use of the membrane solutions for the determination of the particular integrals appears to lead to acceptable results in the case of ellipsoidal shells. However, in the case of toroidal shells, the membrane and the numerical solutions disagree significantly.

Ein einfaches Inkrementenverfahren zur Bestimmung der Ladungen in nichtaromatischen Molekiilen

1984 ◽  
Vol 39 (11) ◽  
pp. 1082-1088
Author(s):  
Wolfgang Schleker ◽  
Burghard Schmidt ◽  
Wilhelm Mertens ◽  
Jörg Fleischhauer
Keyword(s):  
Small Perturbation ◽  
Perturbation Analysis ◽  
Numerical Solutions ◽  
Asymptotic Solutions ◽  
Polar Coordinates ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Whitham Equations

The Chester-Chisnell-Whitham equations in Cartesian and polar coordinates are written out for the two-dimensional case. The solution expanded near a point is obtained. The asymptotic solutions of order 2 near focus are presented. Linear small perturbation analysis for a converging cylindrical shock is given. To study the development of perturbations numerical solutions are carried out.

scholarly journals On the response of a reservoir sidearm to diurnal heating and cooling

1993 ◽  
Vol 246 ◽  
pp. 143-161 ◽  
Author(s):  
D. E. Farrow ◽  
J. C. Patterson
Keyword(s):  
Unit Volume ◽  
Numerical Solutions ◽  
Asymptotic Solutions ◽  
Internal Heating ◽  
Periodic Forcing ◽  
Horizontal Pressure Gradient ◽  
Heating And Cooling ◽  
Flow Response ◽  
Horizontal Pressure ◽  
Temperature Equation

During the day, the shallower regions of a reservoir sidearm absorb more heat per unit volume than the deeper parts, leading to a horizontal pressure gradient that drives a circulation in the sidearm. At night, the shallow regions cool more rapidly, leading to a circulation in the opposite direction. Since the spin-up time of a typical sidearm is at least of the same order as a day, the flow within a diurnally forced sidearm is principally an inertia–buoyancy balance. In this paper, a diurnally forced sidearm is modelled by periodically forced natural convection in a triangular cavity. The periodic forcing enters the model via an internal heating/cooling term in the temperature equation. Reservoir sidearms typically have small bottom slopes and this fact can be exploited to obtain asymptotic solutions of the resulting equations. These solutions clearly demonstrate the transition from the viscous-dominated flow in the shallows to the inertia-dominated flow in the deeper parts. In the inertia-dominated region, the flow response significantly lags the forcing. Numerical solutions of the full nonlinear problem are consistent with the asymptotic solutions.

The Entrance Effect on the Electric Potential and Field in the Microchannel With Two Reservoirs

2006 ◽  
Author(s):  
Yong Kweon Suh ◽  
Hyeong Seok Heo ◽  
Jae Hyun Park
Keyword(s):  
Electric Field ◽  
Electric Potential ◽  
Numerical Solutions ◽  
Channel Width ◽  
Asymptotic Solutions ◽  
Channel Size ◽  
Entrance Effect ◽  
Small Channel ◽  
Wide Range ◽  
Channel Region

In this paper we consider the effect of the entrance into the microchannel having two reservoirs on the electric potential and the electric field. For the case when the channel width is much smaller than the reservoir size, as usual, we designed a method to patch the numerical solutions obtained for the reservoir domain and the asymptotic solutions valid for the region near the entrance to the channel which is attached to the reservoir. Based on the assumption of small channel size, we also derived two asymptotic solutions for the potential and the electric field applicable to the reservoir region and to the channel region. Then the working formula has been established which can predict the effect of the electrode and channel size relative to the reservoir on the electric field built inside the channel. It was shown that the working formula is robust and applicable to the wide range of the parameters.

Casson Fluid Flow and Heat Transfer at an Exponentially Stretching Permeable Surface

2013 ◽  
Vol 80 (5) ◽  
Author(s):  
Swati Mukhopadhyay ◽  
Kuppalapalle Vajravelu ◽  
Robert A. Van Gorder
Keyword(s):  
Heat Transfer ◽  
Differential Equations ◽  
Newtonian Fluid ◽  
Numerical Solutions ◽  
Casson Fluid ◽  
Permeable Surface ◽  
Suction Parameter ◽  
Flow And Heat Transfer ◽  
Casson Fluid Model ◽  
Exponentially Stretching

The present paper deals with the boundary layer flow and heat transfer of a non-Newtonian fluid at an exponentially stretching permeable surface. The Casson fluid model is used to characterize the non-Newtonian fluid behavior, due to its various practical applications. With the help of similarity transformations the governing partial differential equations corresponding to the continuity, momentum, and energy equations are converted into nonlinear ordinary differential equations, and numerical solutions to these equations are obtained. Furthermore, in some specific parameter regimes, analytical solutions are found. It is observed that the effect of increasing values of the Casson parameter is to decrease the velocity field while enhancing the temperature field. Furthermore, it is observed that the effect of the increasing values of the suction parameter is to increase the skin-friction coefficient.

Elastic Contact Between a Geometrically Anisotropic Bisinusoidal Surface and a Rigid Base

2015 ◽  
Vol 137 (2) ◽  
Author(s):  
Yang Xu ◽  
Amir Rostami ◽  
Robert L. Jackson
Keyword(s):  
Numerical Solutions ◽  
Numerical Models ◽  
Elastic Half Space ◽  
Asymptotic Solutions ◽  
Rigid Base ◽  
Fe Model ◽  
Linear Elastic ◽  
Principal Directions ◽  
Complete Contact ◽  
Elliptic Contact

In the current study, a semi-analytical model for contact between a homogeneous, isotropic, linear elastic half-space with a geometrically anisotropic (wavelengths are different in the two principal directions) bisinusoidal surface on the boundary and a rigid base is developed. Two asymptotic loads to area relations for early and almost complete contact are derived. The Hertz elliptic contact theory is applied to approximate the load to area relation in the early contact. The noncontact regions occur in the almost complete contact are treated as mode-I cracks. Since those cracks are in compression, an approximate relation between the load and noncontact area can be obtained by setting the corresponding stress intensity factor (SIF) to zero. These two asymptotic solutions are validated by two different numerical models, namely, the fast Fourier transform (FFT) model and the finite element (FE) model. A piecewise equation is fit to the numerical solutions to bridge these two asymptotic solutions.

Asymptotic Solution Near Focus of a Two-dimensional Shock and Instability Near Axis of a Cylindrical Shock with CCW Approach

1984 ◽  
Vol 39 (11) ◽  
pp. 1011-1022
Author(s):  
Shanbing Yu
Keyword(s):  
Asymptotic Solution ◽  
Small Perturbation ◽  
Perturbation Analysis ◽  
Numerical Solutions ◽  
Asymptotic Solutions ◽  
Polar Coordinates ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Whitham Equations

The Chester-Chisnell-Whitham equations in Cartesian and polar coordinates are written out for the two-dimensional case. The solution expanded near a point is obtained. The asymptotic solutions of order 2 near focus are presented. Linear small perturbation analysis for a converging cylindrical shock is given. To study the development of perturbations numerical solutions are carried out.

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