scholarly journals Control of the motion of a circular cylinder in an ideal fluid using a source

2020 ◽  
Vol 30 (4) ◽  
pp. 604-617
Author(s):  
E.M. Artemova ◽  
E.V. Vetchanin
Keyword(s):  
Fixed Point ◽  
Circular Cylinder ◽  
Level Set ◽  
Ideal Fluid ◽  
Equations Of Motion ◽  
Energy Integral ◽  
Integral Conditions ◽  
Unstable Fixed Point ◽  
Constant Strength ◽  
Momentum Integral

The motion of a circular cylinder in an ideal fluid in the field of a fixed source is considered. It is shown that, when the source has constant strength, the system possesses a momentum integral and an energy integral. Conditions are found under which the equations of motion reduced to the level set of the momentum integral admit an unstable fixed point. This fixed point corresponds to circular motion of the cylinder about the source. A feedback is constructed which ensures stabilization of the above-mentioned fixed point by changing the strength of the source.

Consistent section-averaged equations of quasi-one-dimensional laminar flow

2010 ◽  
Vol 656 ◽  
pp. 337-341 ◽  
Author(s):  
PAOLO LUCHINI ◽  
FRANÇOIS CHARRU
Keyword(s):  
Equations Of Motion ◽  
Stability Result ◽  
Momentum Equation ◽  
Dissipation Function ◽  
Dimensional Solution ◽  
First Order ◽  
Averaged Equations ◽  
Classical Stability ◽  
Momentum Integral ◽  
Free Falling

Section-averaged equations of motion, widely adopted for slowly varying flows in pipes, channels and thin films, are usually derived from the momentum integral on a heuristic basis, although this formulation is affected by known inconsistencies. We show that starting from the energy rather than the momentum equation makes it become consistent to first order in the slowness parameter, giving the same results that have been provided until today only by a much more laborious two-dimensional solution. The kinetic-energy equation correctly provides the pressure gradient because with a suitable normalization the first-order correction to the dissipation function is identically zero. The momentum equation then correctly provides the wall shear stress. As an example, the classical stability result for a free falling liquid film is recovered straightforwardly.

scholarly journals Chaotic Itinerancy Observed in Mutually Coupled Gaussian Maps

2018 ◽  
Vol 28 (04) ◽  
pp. 1830011
Author(s):  
Mio Kobayashi ◽  
Tetsuya Yoshinaga
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Chaotic Behavior ◽  
Periodic Points ◽  
Two Dimensional ◽  
Stable Fixed Point ◽  
Bifurcation Structure ◽  
Unstable Fixed Point ◽  
Gaussian Map ◽  
Gaussian Maps

A one-dimensional Gaussian map defined by a Gaussian function describes a discrete-time dynamical system. Chaotic behavior can be observed in both Gaussian and logistic maps. This study analyzes the bifurcation structure corresponding to the fixed and periodic points of a coupled system comprising two Gaussian maps. The bifurcation structure of a mutually coupled Gaussian map is more complex than that of a mutually coupled logistic map. In a coupled Gaussian map, it was confirmed that after a stable fixed point or stable periodic points became unstable through the bifurcation, the points were able to recover their stability while the system parameters were changing. Moreover, we investigated a parameter region in which symmetric and asymmetric stable fixed points coexisted. Asymmetric unstable fixed point was generated by the [Formula: see text]-type branching of a symmetric stable fixed point. The stability of the unstable fixed point could be recovered through period-doubling and tangent bifurcations. Furthermore, a homoclinic structure related to the occurrence of chaotic behavior and invariant closed curves caused by two-periodic points was observed. The mutually coupled Gaussian map was merely a two-dimensional dynamical system; however, chaotic itinerancy, known to be a characteristic property associated with high-dimensional dynamical systems, was observed. The bifurcation structure of the mutually coupled Gaussian map clearly elucidates the mechanism of chaotic itinerancy generation in the two-dimensional coupled map. We discussed this mechanism by comparing the bifurcation structures of the Gaussian and logistic maps.

The Isobars in Boundary Layers at Supersonic Speeds

1968 ◽  
Vol 19 (2) ◽  
pp. 105-126 ◽  
Author(s):  
D. F. Myring ◽  
A. D. Young
Keyword(s):  
Boundary Layer ◽  
Boundary Layers ◽  
Equations Of Motion ◽  
Normal Pressure ◽  
Simple Extension ◽  
Pressure Gradients ◽  
Boundary Layer Flows ◽  
Hyperbolic Flow ◽  
The Individual ◽  
Momentum Integral

SummaryFor boundary layer flows over curved surfaces at moderately high supersonic speeds the existence of normal pressure gradients within the boundary layer becomes important even for small curvatures and they cannot be ignored. The describing equations are basically parabolic in form so that the simplifications inherent in hyperbolic flows would not at first sight seem to be relevant. However, the equations of motion for a two-dimensional, supersonic, rotational, viscous flow are analysed along the lines of a hyperbolic flow and the individual effects of viscosity and vorticity are examined with regard to the isobar distributions. It is found that these two properties have compensating effects and the experimental evidence presented confirms the conclusion that inside the boundary layer the isobars follow much the same rules as those which determine the isobars in the external hyperbolic flow. Since for turbulent boundary layers the fullness of the Mach number profile produces almost linear Mach lines in the boundary layer, this provides a simple extension to the methods of analysis, and the momentum integral equation is reformulated using a swept element bounded by linear isobars. The final equation is similar in form to the conventional one except that the momentum and displacement thicknesses are now defined by integrals along the swept isobars, and all normal pressure gradients due to centrifugal effects are accounted for.

scholarly journals Multiple Positive Solutions of Third-Order BVP with Advanced Arguments and Stieltjes Integral Conditions

2016 ◽  
Vol 2016 ◽  
pp. 1-12
Author(s):  
Jian Chang ◽  
Jian-Ping Sun ◽  
Ya-Hong Zhao
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Positive Solutions ◽  
Integral Boundary Conditions ◽  
Multiple Positive Solutions ◽  
Stieltjes Integral ◽  
Integral Conditions ◽  
Third Order ◽  
Integral Boundary

We consider the following third-order boundary value problem with advanced arguments and Stieltjes integral boundary conditions:u′′′t+ft,uαt=0,  t∈0,1,  u0=γuη1+λ1uandu′′0=0,  u1=βuη2+λ2u, where0<η1<η2<1,0≤γ,β≤1,α:[0,1]→[0,1]is continuous,α(t)≥tfort∈[0,1], andα(t)≤η2fort∈[η1,η2]. Under some suitable conditions, by applying a fixed point theorem due to Avery and Peterson, we obtain the existence of multiple positive solutions to the above problem. An example is also included to illustrate the main results obtained.

The thermodynamic stability of three near-degenerate phases of platinum dioxide

2004 ◽  
Vol 848 ◽  
Author(s):  
Shuping Zhuo ◽  
Karl Sohlberg
Keyword(s):  
High Pressure ◽  
Fixed Point ◽  
Low Temperature ◽  
Thermodynamic Stability ◽  
First Principles ◽  
Energy Surface ◽  
Stable Phase ◽  
Free Energies ◽  
Rutile Structure ◽  
Unstable Fixed Point

ABSTRACTThe thermodynamic stability of the three nearly energy degenerate crystal structures of PtO2 is studied here with first-principles-based calculations of their free energies. For P = 0 the α-(CdI2) structure is the thermodynamically stable phase at low temperature, while the β-(CaCl2) structure is stable at high pressure. The β'-(rutile) structure represents an unstable fixed point on the potential energy surface, or is possibly just barely bound. These results reconcile seemingly contradictory findings and answer longstanding questions about PtO2.

Motion of a circular cylinder on a smooth surface

2009 ◽  
Vol 87 (6) ◽  
pp. 607-614 ◽  
Author(s):  
Mark R.A. Shegelski ◽  
Ian Kellett ◽  
Hal Friesen ◽  
Crystal Lind
Keyword(s):  
Circular Cylinder ◽  
Undergraduate Students ◽  
Smooth Surface ◽  
Equations Of Motion ◽  
Static Friction ◽  
Circular Motion ◽  
Pedagogical Treatment ◽  
Newton’S Laws ◽  
Made In

We present a pedagogical treatment of a circular cylinder moving over a smooth surface with one point of the cylinder making contact with the surface. We derive the equations of motion using Newton’s laws. The simplicity of this approach makes the results easily within reach of undergraduate students. A careful derivation of the equations is presented first, and then we show how easily one could make an error. We illustrate how instructors could use this calculation to teach students how to detect errors and critically examine assumptions, including those that seem beyond question. Circular motion with no slipping is examined and we demonstrate the extent of possible motions for static friction. Some calculations are listed that instructors can assign to students to teach the points made in this paper.

scholarly journals Existence of Solutions ofα∈(2,3]Order Fractional Three-Point Boundary Value Problems with Integral Conditions

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
N. I. Mahmudov ◽  
S. Unul
Keyword(s):  
Fixed Point ◽  
Existence And Uniqueness ◽  
Fractional Derivatives ◽  
Fixed Point Theorems ◽  
Existence Of Solutions ◽  
Point Boundary ◽  
Integral Conditions ◽  
Uniqueness Of Solutions ◽  
Fractional Differential ◽  
Derivatives Of

Existence and uniqueness of solutions forα∈(2,3]order fractional differential equations with three-point fractional boundary and integral conditions involving the nonlinearity depending on the fractional derivatives of the unknown function are discussed. The results are obtained by using fixed point theorems. Two examples are given to illustrate the results.

Transient motion of a dipole in a rotating flow

1969 ◽  
Vol 39 (3) ◽  
pp. 433-442 ◽  
Author(s):  
John W. Miles
Keyword(s):  
Fixed Point ◽  
Axial Velocity ◽  
Potential Flow ◽  
Equations Of Motion ◽  
Rotating Flow ◽  
Rossby Number ◽  
Laboratory Measurements ◽  
Transient Motion ◽  
Experimental Scatter ◽  
Upstream Waves

The question of whether or not waves exist upstream of an obstacle that moves uniformly through an unbounded, incompressible, inviscid, unseparated, rotating flow is addressed by considering the development of the disturbed flow induced by a weak, moving dipole that is introduced into an axisymmetric, rotating flow that is initially undisturbed. Starting from the linearized equations of motion, it is shown that the flow tends asymptotically to the steady flow determined on the hypothesis of no upstream waves and that the transient at a fixed point is O(1/t). It also is shown that the axial velocity upstream (x < 0) of the dipole as x → − ∞ with t fixed is O(|x|−3), as in potential flow, but is O(|x|−1) as t → ∞ with |x| fixed. The results extend directly to closed obstacles of sufficiently small transverse dimensions and suggest the existence of a finite, parametric domain of no upstream waves for smooth, slender obstacles. The axial velocity in front of a small, moving sphere at a given instant in the transient régime is calculated and compared with Pritchard's laboratory measurements. The agreement is within the experimental scatter for Rossby numbers greater than about 0·3 even though the equivalence between sphere and dipole is exact only for infinite Rossby number.

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