Re-entrant corner behavior of the PTT fluid

We integrate the constitutive equation of the Phan-Thien-Tanner (PTT) fluid near a re-entrant 270° corner. The velocity field is assumed given (Newtonian). In contrast to the case of the upper convected Maxwell (UCM) fluid, we find the following features: (1) The elastic stresses near the corner are less singular than Newtonian stresses; (2) Boundary layers near the walls are much less sharp than for the UCM fluid; (3) There are no spurious stresses due to downstream instabilities.     

The stresses of an upper convected Maxwell fluid in a Newtonian velocity field near a re-entrant corner

We investigate the stresses of an upper convected Maxwell fluid in the neighborhood of a re-entrant 270° corner. It is assumed (incorrectly, of course) that the velocity field is Newtonian. Both asymptotic analysis and numerical solutions are presented. It is found that, for a fixed angle, the stresses behave approximately as r^−0.74, which contrasts with a behavior as r^−0.91 at the walls (the latter is simply the square of the Newtonian shear rate at the wall, where the flow is viscometric). The analysis shows that there are boundary layers near the walls, in which there is a transition from the viscometric behavior at the wall to a core region which the behavior is dominated by the convected derivative in the constitutive equation. Moreover, our computations show large spurious stresses downstream resulting from numerical errors.     

Investigation of a re-entrant jet reflection at an inclined cavity closure line

Cavitation on two-dimensional hydrofoils with swept leading edges always displays some 3-dimensional effects. It is well known that the cavity closure on such hydrofoil is not perpendicular to the channel walls, but is curved in a distinctive pattern. The cavitation pocket is longer in the region where the hydrofoil is the shortest. Also the dynamics of cavitation is very distinctive. In the region where the hydrofoil is the longest attached and steady cavitation with no cloud separation exists. On the other side, where the hydrofoil is the shortest, cavitation cloud separations occur.Different explanations for this pattern were proposed in the past but they have not jet been clearly confirmed neither experimentally nor by numerical simulation.In the present paper a clear explanation supported by the numerical simulation and also by experimental measurements, is given.     

Asymptotic theory of laminar flow separation at a corner

The development of the reverse flow structure in the neighborhood of a corner in a viscous incompressible laminar flow at high Reynolds numbers is investigated numerically. It is found that as the angle of inclination increases the internal structure of the reverse flow zone becomes more complex as a result of secondary separation. The effect of the curvature of the surface is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 180–182, January–February, 1991.     

A limit form of the equations for immiscible displacement in a fractured reservoir

We study a model for simulating the flow of an immiscible displacement (waterflooding) of one incompressible fluid by another in a naturally fractured petroleum reservoir when the matrix blocks are quite small. This model is equivalent to a transformed one for immiscible flow in an unfractured reservoir with a reduced saturation and a saturation-dependent porosity. Existence and uniqueness of classical solutions are established. We present some numerical results and a comparison with a single porosity model.     

The formation of a secondary shock wave behind a shock wave diffracting at a convex corner

This paper deals with the formation of a secondary shock wave behind the shock wave diffracting at a two-dimensional convex corner for incident shock Mach numbers ranging from 1.03 to 1.74 in air. Experiments were carried out using a 60 mm 150 mm shock tube equipped with holographic interferometry. The threshold incident shock wave Mach number () at which a secondary shock wave appeared was found to be = 1.32 at an 81° corner and = 1.33 at a 120° corner. These secondary shock waves are formed due to the existence of a locally supersonic flow behind the diffracting shock wave. Behind the diffracting shock wave, the subsonic flow is accelerated and eventually becomes locally supersonic. A simple unsteady flow analysis revealed that for gases with specific heats ratio the threshold shock wave Mach number was = 1.346. When the value of is less than this, the vortex is formed at the corner without any discontinuous waves accompanying above the slip line. The viscosity was found to be less effective on the threshold of the secondary shock wave, although it attenuated the pressure jump at the secondary shock wave. This is well understood by the consideration of the effect of the wall friction in one-dimensional duct flows. In order to interpret the experimental results a numerical simulation using a shock adaptive unstructured grid Eulerian solver was also carried out. Received 1 May 1996 / Accepted 12 September 1996     

Formal solutions of partial differential equations and the projective limit

Based on stratification theory, the existence theorems of formal solutions of partial differential equation (PDE) are given . And the relationship between formal solutions and protective limit of Ehresmann chain is presented .     

Evaporative instability at the superheat limit

The explosive vaporization of a single bubble inside a droplet of butane heated to the limit of superheat has been investigated experimentally using short-exposure photographs and fast-response pressure measurements. An interfacial instability driven by rapid evaporation has been observed on the surface of the bubbles. It is proposed that the Landau mechanism of instability, originally described in connection with the instability of laminar flames, also applies to rapid evaporation at the superheat limit. Calculations suggest that other technically important fluids may be even more unstable when boiling at the superheat limit. The rate of evaporation after the onset of instability is estimated from the experimental measurements to be two orders of magnitude greater than would be predicted by conventional bubble-growth theories that do not account for the effects of instability. An estimate of the mean density within the bubbles during the evaporative stage indicates that it is nearly equal to the critical density of butane.     

Closed system of limit equilibrium equations for axisymmetric shells

We consider rigid-plastic axisymmetric shells and use methods of control theory to construct the carrying capacity loss condition for such shells in formalized form. We show that solving limit equilibrium problems for such structures can be reduced to solving a multipoint boundary value problem for a system of nonlinear differential-algebraic equations with unknown matching boundaries between different plastic modes as well as between rigid and plastic domains. We present a complete system of resolving equations for the problem on the carrying capacity of axisymmetric shells, including the matching conditions for domains in different states.     

Initiation of fracture at the interface corner of bi-material joints

Within the context of linear elasticity, a stress singularity of the form Hr^λ−1 may exist at the interface corner of a bi-material joint, where r is the radial distance from the corner, H is the stress intensity factor and λ−1 is the order of the singularity. Recent experimental results in the literature support the use of a critical value of the intensity factor H=H_c as a fracture initiation criterion at the interface corner. In this paper, we examine the validity and limitations of this criterion for predicting the onset of fracture in a butt joint consisting of a thin layer of an elastic-plastic adhesive layer sandwiched between two elastic adherends. The evolution of plastic deformation at the corner is determined theoretically and by the finite element method, and the solution is compared with the extent of the elastic singular field. It is shown that H_c is a valid fracture parameter if h>B(H_c/σ_Y)^1/(1−λ) where the non-dimensional constant B=100 for β=0 and B=13 for β=α/4. Here, h is the thickness of the adhesive layer, σ_Y is the uniaxial yield stress of the bulk adhesive and (α,β) are Dundurs’ parameters (Dundurs, J., J. Appl. Mech. 36 (1969) 650). Experimental results for aluminium/epoxy/aluminium and brass/solder/brass sandwiched joints are used to assess the role of plastic deformation on the validity of the failure criterion.     

Analysis of the plastic zone at the corner point of interface

A symmetric problem of elasticity is formulated to analyze the plastic zone at the corner point of the interface between two isotropic media. The piecewise-homogeneous isotropic body with an interface in the form of angle sides consists of different elastic parts joined by a thin elastoplastic layer. The plastic zone is modeled by discontinuity lines of tangential displacement, which are located at the interface. The exact solution of the problem is found using the Wiener–Hopf method and is then used to determine the length of the plastic zone. The stress at the corner point is analyzed Translated from Prikladnaya Mekhanika, Vol. 45, No. 2, pp. 59–69, February 2009.     

An experimental study of stress singularities at a sharp corner in a contact problem

The photoelastic method was used to investigate the nature of the local stress field at a sharp corner of a wedge that was compressed against a larger body. Planar wedge specimens made of photoelastic material were compressed against a half plane (larger body) of identical material at various load levels. Several wedge angles were studied. The nature of the singular stress field postulated by linear elastic analysis was verified and the strength of the singularity was obtained by plotting the variation of fringe order as a function of radial distance from the sharp corner on a logarithmic scale. The experimental results were found to be in good agreement with the theoretical predictions. The effect of interface friction and the effect of rounding off the sharp edge are brietly discussed.