Asymptotic Methods for an Infinite Slider Bearing With a Discontinuity in Film Slope

1976 ◽  
Vol 98 (3) ◽  
pp. 446-452 ◽  
Author(s):  
J. A. Schmitt ◽  
R. C. DiPrima
Keyword(s):  
Boundary Layer ◽  
Film Thickness ◽  
Asymptotic Expansions ◽  
Asymptotic Expression ◽  
Asymptotic Methods ◽  
Matched Asymptotic Expansions ◽  
Slider Bearing ◽  
Trailing Edge ◽  
Slider Bearings ◽  
Special Cases

The method of matched asymptotic expansions is used to develop an asymptotic expression for the pressure for large bearing numbers for the case of an infinite slider bearing with a general film thickness that has a discontinuous slope at a point. It is shown that, in addition to the boundary layer of the pressure at the trailing edge, there is also a boundary layer in the derivative of the pressure at the point of discontinuity. The corresponding load formula is also derived. The special cases of the taper-flat and taper-taper slider bearings are discussed.

Asymptotic Methods for a General Finite Width Gas Slider Bearing

1978 ◽  
Vol 100 (2) ◽  
pp. 254-260 ◽  
Author(s):  
J. A. Schmitt ◽  
R. C. DiPrima
Keyword(s):  
Film Thickness ◽  
Asymptotic Expression ◽  
Asymptotic Methods ◽  
Finite Width ◽  
Slider Bearing ◽  
Sliding Direction ◽  
Special Cases ◽  
Load Carrying ◽  
The Individual ◽  
Bearing Number

The method of matched asymptotic expansions is used to develop an asymptotic expression for the load-carrying capacity of a finite width gas slider bearing for large bearing numbers and for film thicknesses varying both in the sliding and transverse directions. The individual terms in the formula for the load are independent of the bearing number and are related to the interior portion, the side edge boundary layers, and the trailing edge boundary layer of the bearing. Only the terms associated with the side leakage phenomena must be computed numerically. Two special cases are discussed: (i) the film thickness varying only in the sliding direction, and (ii) the film thickness having linear or parabolic variation in the sliding direction and parabolic variation in the transverse direction.

Trailing edge flows

1969 ◽  
Vol 39 (1) ◽  
pp. 193-207 ◽  
Author(s):  
N. Riley ◽  
K. Stewartson
Keyword(s):  
Boundary Layer ◽  
Asymptotic Expansions ◽  
Pressure Field ◽  
Matched Asymptotic Expansions ◽  
Trailing Edge

The flow is examined in the neighbourhood of the trailing edge of slender aerodynamic shapes which terminate in either a cusp or a wedge. The manner in which the boundary layer reacts to the rapidly varying pressure field in such regions is analyzed using the method of matched asymptotic expansions. The case of a wedge is examined in greater detail and a criterion for separation to occur is established.

Nearly equilibrium dissociating boundary-layer flows by the method of matched asymptotic expansions

1966 ◽  
Vol 26 (4) ◽  
pp. 793-806 ◽  
Author(s):  
George R. Inger
Keyword(s):  
Boundary Layer ◽  
Asymptotic Expansions ◽  
High Speed ◽  
Numerical Solutions ◽  
The Body ◽  
Matched Asymptotic Expansions ◽  
Form Analysis ◽  
Perturbation Problem ◽  
Valid Solution ◽  
Non Equilibrium

The approach to equilibrium in a non-equilibrium-dissociating boundary-layer flow along a catalytic or non-catalytic surface is treated from the standpoint of a singular perturbation problem, using the method of matched asymptotic expansions. Based on a linearized reaction rate model for a diatomic gas which facilitates closed-form analysis, a uniformly valid solution for the near equilibrium behaviour is obtained as the composite of appropriate outer and inner solutions. It is shown that, under near equilibrium conditions, the primary non-equilibrium effects are buried in a thin sublayer near the body surface that is described by the inner solution. Applications of the theory are made to the calculation of heat transfer and atom concentrations for blunt body stagnation point and high-speed flat-plate flows; the results are in qualitative agreement with the near equilibrium behaviour predicted by numerical solutions.

On the shape of small sessile and pendant drops by singular perturbation techniques

1991 ◽  
Vol 233 ◽  
pp. 519-537 ◽  
Author(s):  
S. B. G. O'Brien
Keyword(s):  
Boundary Layer ◽  
Singular Perturbation ◽  
Asymptotic Expansions ◽  
Numerical Results ◽  
Matched Asymptotic Expansions ◽  
Perturbation Techniques ◽  
Pendant Drop ◽  
Asymptotic Expressions ◽  
Good Agreement ◽  
Pendant Drops

The problem of obtaining asymptotic expressions describing the shape of small sessile and pendant drops is revisited. Both cases display boundary-layer behaviour and the method of matched asymptotic expansions is used to obtain solutions. These give good agreement when compared with numerical results. The sessile solutions are relatively straightforward, while the pendant drop displays a behaviour which is both rich and interesting.

Flow along a horizontal plate near a free surface

1993 ◽  
Vol 252 ◽  
pp. 399-418
Author(s):  
Milan Hofman
Keyword(s):  
Boundary Layer ◽  
Free Surface ◽  
Flat Plate ◽  
Viscous Liquid ◽  
Gravity Waves ◽  
Asymptotic Expansions ◽  
Small Amplitude ◽  
Surface Effect ◽  
Matched Asymptotic Expansions ◽  
Horizontal Plate

The problem of flow along a horizontal semi-infinite flat plate moving in its own plane through a viscous liquid just below the free surface is considered. The method of matched asymptotic expansions is used to analyse the interaction between the free surface and the boundary layer formed on the plate. It is found that, due to viscosity, small-amplitude gravity waves on the free surface can be formed. The formulae for the resistance of the plate containing the free-surface effect and for the lift, appearing as a new phenomenon, are derived.

Theory for Finite-Width High-Speed Self-Acting Gas-Lubricated Slider (and Partial-Arc) Bearings

1969 ◽  
Vol 91 (1) ◽  
pp. 17-24 ◽  
Author(s):  
H. G. Elrod ◽  
J. T. McCabe
Keyword(s):  
Pressure Distribution ◽  
Edge Effects ◽  
High Speed ◽  
Load Capacity ◽  
Asymptotic Methods ◽  
Finite Width ◽  
Trailing Edge ◽  
Slider Bearings ◽  
Edge Conditions

The behavior of the pressure distribution within partial-arc and slider bearings under conditions of high-speed operation is investigated to provide an understanding of the basic phenomenon. Both the trailing-edge conditions and side-leakage effects are treated by asymptotic methods. A design example is given in which the edge effects on load capacity are computed as “corrections” to the infinite speed (Λ → ∞) solution.

Free Fall of a Sphere in a Partially Lubricated Cylinder

1991 ◽  
Vol 58 (3) ◽  
pp. 812-819
Author(s):  
Roger F. Gans
Keyword(s):  
Contact Force ◽  
Closed System ◽  
Experimental Verification ◽  
Asymptotic Expansions ◽  
Total Mass ◽  
Free Fall ◽  
Matched Asymptotic Expansions ◽  
Biological Applications ◽  
Slider Bearing ◽  
Acceleration Of Gravity

The entrainment of lubricant at the entrance of a lubrication zone, such as that of a partially starved slider bearing, is analyzed in a closed system using the method of matched asymptotic expansions. A sphere falling together with a small lens of lubricant in a closely fitting tube is shown to fall under gravity at a speed V=(Mg−Fc)√[(RC−RS)/RC]/(16π2μRC), where M denotes the total mass of the system, sphere plus lubricant, g the acceleration of gravity, Fc the differential contact force, μ the viscosity of the lubricant, and RC and RS the radii of the tube and the sphere, respectively. Potential biological applications and experimental verification are discussed.

scholarly journals Padé Approximation to Solve the Problems of Aerodynamics and Heat Transfer in the Boundary Layer

2020 ◽  
Author(s):  
Igor Andrianov ◽  
Anatoly Shatrov
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Asymptotic Expansions ◽  
Asymptotic Methods ◽  
Padé Approximation ◽  
Mathematical Physics ◽  
Pade Approximation ◽  
Padé Approximations ◽  
Viscous Gas ◽  
Pade Approximations

In this chapter, we describe the applications of asymptotic methods to the problems of mathematical physics and mechanics, primarily, to the solution of nonlinear singular perturbed problems. We also discuss the applications of Padé approximations for the transformation of asymptotic expansions to rational or quasi-fractional functions. The applications of the method of matching of internal and external asymptotics in the problem of boundary layer of viscous gas by means of Padé approximation are considered.

Free convection at low Prandtl numbers

1969 ◽  
Vol 37 (4) ◽  
pp. 785-798 ◽  
Author(s):  
H. K. Kuiken
Keyword(s):  
Boundary Layer ◽  
Free Convection ◽  
Prandtl Number ◽  
Asymptotic Expansions ◽  
Extreme Case ◽  
Matched Asymptotic Expansions ◽  
Convection Boundary Layer ◽  
Convection Boundary ◽  
Prandtl Numbers ◽  
Free Convection Boundary Layer

In this paper it is shown that the free convection boundary layer approaches a singular character if the Prandtl number tends to zero. The method of matched asymptotic expansions is used to integrate the equations for this extreme case. An expression is derived for the Nusselt—Grashof relation and the results are compared with those of previous investigations which attack the problem in a different way.

Matched asymptotics for two-dimensional planing hydrofoils with spoilers

1998 ◽  
Vol 358 ◽  
pp. 259-281 ◽  
Author(s):  
G. M. FRIDMAN
Keyword(s):  
Asymptotic Solution ◽  
Flat Plate ◽  
Linear Theory ◽  
Asymptotic Expansions ◽  
Flow Problem ◽  
Matched Asymptotic Expansions ◽  
Asymptotic Solutions ◽  
Two Dimensional ◽  
Trailing Edge ◽  
Matched Asymptotics

The purpose of the paper is to demonstrate the effectiveness of the matched asymptotic expansions (MAE) method for the planing flow problem. The matched asymptotics, taking into account the flow nonlinearities in those regions where they are most pronounced (i.e. in the vicinity of the edges), are shown to significantly extend the range where the linear theory gives good results. Two model problems are used: the planing flat plate with a spoiler on the trailing edge and the curved planing foil. Asymptotic solutions obtained by the MAE method are compared with those obtained using linear and exact nonlinear theories. Based on the results, the asymptotic solution to the planing problem under the gravity is proposed.

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