Non-Newtonian magnetohydrodynamic flow over a stretching surface with heat and mass transfer

An analysis is carried out to study the momentum, mass and heat transfer characteristics on the flow of visco-elastic fluid (Walter's liquid-B model) past a stretching sheet in the presence of a transverse magnetic field.In heat transfer, two cases are considered:

1.

The sheet with prescribed surface temperature (PST case); and

2.

The sheet with prescribed wall heat flux (PHF case).

The solution of equations of momentum, mass and heat transfer are obtained analytically. Emphasis has been laid to study the effects of various parameters like magnetic parameter Mn, visco-elastic parameter k₁, Schmidt number Sc, and Prandtl number Pr on flow, heat and mass transfer characteristics.     

A one-dimensional theory of strain-gradient plasticity: Formulation, analysis, numerical results

This study develops a one-dimensional theory of strain-gradient plasticity based on: (i) a system of microstresses consistent with a microforce balance; (ii) a mechanical version of the second law that includes, via microstresses, work performed during viscoplastic flow; (iii) a constitutive theory that allows

•

the free-energy to depend on the gradient of the plastic strain, and

•

the microstresses to depend on the gradient of the plastic strain-rate.

The constitutive equations, whose rate-dependence is of power-law form, are endowed with energetic and dissipative gradient length-scales L and l, respectively, and allow for a gradient-dependent generalization of standard internal-variable hardening. The microforce balance when augmented by the constitutive relations for the microstresses results in a nonlocal flow rule in the form of a partial differential equation for the plastic strain. Typical macroscopic boundary conditions are supplemented by nonstandard microscopic boundary conditions associated with flow, and properties of the resulting boundary-value problem are studied both analytically and numerically. The resulting solutions are shown to exhibit three distinct physical phenomena:

(i)

standard (isotropic) internal-variable hardening;

(ii)

energetic hardening, with concomitant back stress, associated with plastic-strain gradients and resulting in boundary layer effects;

(iii)

dissipative strengthening associated with plastic strain-rate gradients and resulting in a size-dependent increase in yield strength.

     

Simulation of polycrystal deformation with grain and grain boundary effects

Modeling the strengthening effect of grain boundaries (Hall-Petch effect) in metallic polycrystals in a physically consistent way, and without invoking arbitrary length scales, is a long-standing, unsolved problem. A two-scale method to treat predictively the interactions of large numbers of dislocations with grain boundaries has been developed, implemented, and tested. At the first scale, a standard grain-scale simulation (GSS) based on a finite element (FE) formulation makes use of recently proposed dislocation-density-based single-crystal constitutive equations (“SCCE-D”) to determine local stresses, strains, and slip magnitudes. At the second scale, a novel meso-scale simulation (MSS) redistributes the mobile part of the dislocation density within grains consistent with the plastic strain, computes the associated inter-dislocation back stress, and enforces local slip transmission criteria at grain boundaries.Compared with a standard crystal plasticity finite element (FE) model (CP-FEM), the two-scale model required only 5% more CPU time, making it suitable for practical material design. The model confers new capabilities as follows:

(1)

The two-scale method reproduced the dislocation densities predicted by analytical solutions of single pile-ups.

(2)

Two-scale simulations of 2D and 3D arrays of regular grains predicted Hall-Petch slopes for iron of 1.2 ± 0.3 MN/m^3/2 and 1.5 ± 0.3 MN/m^3/2, in agreement with a measured slope of 0.9 ± 0.1 MN/m^3/2.

(3)

The tensile stress-strain response of coarse-grained Fe multi-crystals (9-39 grains) was predicted 2-4 times more accurately by the two-scale model as compared with CP-FEM or Taylor-type texture models.

(4)

The lattice curvature of a deformed Fe-3% Si columnar multi-crystal was predicted and measured. The measured maximum lattice curvature near grain boundaries agreed with model predictions within the experimental scatter.

     

Modeling plastic shocks in periodic laminates with gradient plasticity theories

Steady plastic shocks generated by planar impact on metal-polymer laminate composites are analyzed in the framework of gradient plasticity theories. The laminate material has a periodic structure with a unit cell composed of two layers of different materials. First- and second-order gradient plasticity theories are used to model the structure of steady plastic shocks. In both theories, the effect of the internal structure is accounted for at the macroscopic level by two material parameters that are dependent upon the layer's thickness and the properties of constituents. These two structure parameters are shown to be uniquely determined from experimental data. Theoretical predictions are compared with experiments for different cell sizes and for various shock intensities. In particular, the following experimental features are well-reproduced by the modeling:

•

the shock width is proportional to the cell size;

•

the magnitude of strain rate is inversely proportional to cell size and increases with the amplitude of applied stress following a power law.

While these results are equally described by both the plasticity theories, the first gradient plasticity approach seems to be favored when comparing the structure of the shock front to the experimental data.     

Gradient single-crystal plasticity with free energy dependent on dislocation densities

This study develops a small-deformation theory of strain-gradient plasticity for single crystals. The theory is based on: (i) a kinematical notion of a continuous distribution of edge and screw dislocations; (ii) a system of microscopic stresses consistent with a system of microscopic force balances, one balance for each slip system; (iii) a mechanical version of the second law that includes, via the microscopic stresses, work performed during viscoplastic flow; and (iv) a constitutive theory that allows:

•

the free energy to depend on densities of edge and screw dislocations and hence on gradients of (plastic) slip;

•

the microscopic stresses to depend on slip-rate gradients.

The microscopic force balances when augmented by constitutive relations for the microscopic stresses results in a system of nonlocal flow rules in the form of second-order partial differential equations for the slips. When the free energy depends on the dislocation densities the microscopic stresses are partially energetic, and this, in turn, leads to backstresses in the flow rules; on the other hand, a dependence of these stresses on slip-rate gradients leads to a strengthening. The flow rules, being nonlocal, require microscopic boundary conditions; as an aid to numerical solutions a weak (virtual power) formulation of the flow rule is derived.     

A review of some plasticity and viscoplasticity constitutive theories

The purpose of the present review article is twofold:

•

recall elementary notions as well as the main ingredients and assumptions of developing macroscopic inelastic constitutive equations, mainly for metals and low strain cyclic conditions. The explicit models considered have been essentially developed by the author and co-workers, along the past 30 years;     

Some issues related to the modeling of interfacial areas in gas–liquid flows I. The conceptual issues

Two-phase flow modeling has been under constant development for the past forty years. Actually there exists a hierarchy of models which extends from the homogeneous model valid for two-phase flows where the phases are strongly coupled to the two-fluid model valid for two-phase flows where the phases are a priori weakly coupled. However the latter model has been used extensively in computer codes because of its potential in handling many different physical situations.The two-fluid model is based on the balance equations for mass, momentum and energy, averaged in a certain sense and expressed for each phase and for the interface between the phases. The difficulty in using the two-fluid model stems from the closure relations needed to arrive at a complete set of partial differential equations describing the flow. These closure relations should supply the information lost during the averaging of the balance equations and should specify in particular the interactions of mass, momentum and energy between the phases. Another requirement for the interaction terms is that they should satisfy the interfacial balance equations. Some of these terms such as the added mass term or the lift force term do not depend on the interfacial area but some others do, such as the mass transfer term, the drag term or the heat flux term. It is then necessary to model the interfacial area in order to evaluate the corresponding fluxes. Another benefit resulting from the modeling of the interfacial area would be to replace the usual static flow pattern maps which specify the flow configuration by a dynamic follow-up of the flow pattern. All these reasons explain why so much effort has been put during the past twenty years on the modeling and measurement of the interfacial area in two-phase flows.This article contains two parts. The first one deals with the conceptual issues and has the following objectives:

1.
to give precise definitions of the interfacial area concentrations;
2.
to explain the origin of the interfacial area concentration transport equation suggested by M. Ishii in 1975;
3.
to explain some paradoxical behaviors encountered when calculating the interfacial area concentration transport velocity.

     

A theory of strain-gradient plasticity for isotropic, plastically irrotational materials. Part I: Small deformations

This study develops a small-deformation theory of strain-gradient plasticity for isotropic materials in the absence of plastic rotation. The theory is based on a system of microstresses consistent with a microforce balance; a mechanical version of the second law that includes, via microstresses, work performed during viscoplastic flow; a constitutive theory that allows:

•

the microstresses to depend on , the gradient of the plastic strain-rate, and

•

the free energy ψ to depend on the Burgers tensor .

The microforce balance when augmented by constitutive relations for the microstresses results in a nonlocal flow rule in the form of a tensorial second-order partial differential equation for the plastic strain. The microstresses are strictly dissipative when ψ is independent of the Burgers tensor, but when ψ depends on G the gradient microstress is partially energetic, and this, in turn, leads to a back stress and (hence) to Bauschinger-effects in the flow rule. It is further shown that dependencies of the microstresses on lead to strengthening and weakening effects in the flow rule.Typical macroscopic boundary conditions are supplemented by nonstandard microscopic boundary conditions associated with flow, and, as an aid to numerical solutions, a weak (virtual power) formulation of the nonlocal flow rule is derived.     

Thermodynamics applied to gradient theories involving the accumulated plastic strain: The theories of Aifantis and Fleck and Hutchinson and their generalization

We discuss the physical nature of flow rules for rate-independent (gradient) plasticity laid down by Aifantis and by Fleck and Hutchinson. As a central result we show that:

•

the flow rule of Fleck and Hutchinson is incompatible with thermodynamics unless its nonlocal term is dropped. If the underlying theory is augmented by a general defect energy dependent on γ^p and ∇γ^p, then compatibility with thermodynamics requires that its flow rule reduce to that of Aifantis.

We establish this result (and others) within a general framework obtained by combining a virtual-power principle of Fleck and Hutchinson with the first two laws of thermodynamics—balance of energy and the Clausius-Duhem inequality—under isothermal conditions.     

Vortical structures behind a transverse jet in a supersonic flow at high jet to crossflow pressure ratios

A three-dimensional supersonic turbulent flow with symmetric normal injection of circular jets from the channel walls is numerically simulated. The initial Favre-averaged Navier–Stokes equations closed by the k–ω turbulence model are solved by an algorithm based on an ENO scheme. The mechanism of the formation of vortical structures due to the interaction of the jet with the free stream is studied for jet to crossflow total pressure ratios ranging from 3 to 50. It is known from experiments reported in the literature that, for n ? 10, mixing of the jet with the high-velocity flow leads to the formation of a pair of vortices and of an additional separation zone near the wall behind the jet. It is demonstrated that the present numerical results are consistent with such findings and that the pressure distribution on the wall ahead of the jet in the plane of symmetry is also in reasonable agreement with available experimental data.     

Direct Numerical Simulation and Experimental Investigation on Suppression of Vortex Induced Vibrations of Circular Cylinders by Radial Water Jets

The present paper deals with suppression of vortex induced vibrations (VIV) by introducing radial water jets from circular openings in the wall of the oscillating cylinder. Overpressure within the water-filled cylinder propels water jets blowing out into the ambient flow as a means to alter the vortex shedding process. This flow will introduce a disturbance that is expected to yield reduced VIV amplitudes. Results are presented from experiments in a towing tank testing a spring-supported cylinder with two straight rows of radial water jets along the the cylinder, located at positions +120° and?120° on the cylinder circumference. A smooth cylinder with no openings is tested for comparison. Direct Numerical Simulations (DNS) have been performed using the Spectral/hp element code Nεκταr. Outflow through openings in the cylinder wall is modeled, and a parameter study is performed where number of jets as well as jet location on the cylinder circumference and jet flow rate are varied.     

Boundary conditions in small-deformation, single-crystal plasticity that account for the Burgers vector

This paper discusses boundary conditions appropriate to a theory of single-crystal plasticity (Gurtin, J. Mech. Phys. Solids 50 (2002) 5) that includes an accounting for the Burgers vector through energetic and dissipative dependences on the tensor G=curlH^p, with H^p the plastic part in the additive decomposition of the displacement gradient into elastic and plastic parts. This theory results in a flow rule in the form of N coupled second-order partial differential equations for the slip-rates , and, consequently, requires higher-order boundary conditions. Motivated by the virtual-power principle in which the external power contains a boundary-integral linear in the slip-rates, hard-slip conditions in which

(A)

on a subsurface S_hard of the boundary

for all slip systems α are proposed. In this paper we develop a theory that is consistent with that of (Gurtin, 2002), but that leads to an external power containing a boundary-integral linear in the tensor , a result that motivates replacing (A) with the microhard condition

(B)

on the subsurface S_hard.

We show that, interestingly, (B) may be interpreted as the requirement that there be no flow of the Burgers vector across S_hard.What is most important, we establish uniqueness for the underlying initial/boundary-value problem associated with (B); since the conditions (A) are generally stronger than the conditions (B), this result indicates lack of existence for problems based on (A). For that reason, the hard-slip conditions (A) would seem inappropriate as boundary conditions.Finally, we discuss conditions at a grain boundary based on the flow of the Burgers vector at and across the boundary surface.     

Experimental exploration of underexpanded supersonic jets

Two underexpanded free jets at fully expanded Mach numbers $M_\mathrm{j}$ = 1.15 and 1.50 are studied. Schlieren visualizations as well as measurements of static pressure, Pitot pressure and velocity are performed. All these experimental techniques are associated to obtain an accurate picture of the jet flow development. In particular, expansion, compression and neutral zones have been identified in each shock cell. Particle lag is considered by integrating the equation of motion for particles in a fluid flow and it is found that the laser Doppler velocimetry is suitable for investigating shock-containing jets. Even downstream of the normal shock arising in the $M_\mathrm{j}$ = 1.50 jet, the measured gradual velocity decrease is shown to be relevant.