A systematic approach to the search for variational principles for plane steady flows

Using Vainberg's theorem of nonlinear potential operators, alternate potential principles associated with the differential equations governing the gasdynamics of plane steady irrotational diabatic flow and isoenergetic rotational adiabatic flow are formulated and their equivalence with Bateman's principle is established. Further, the advantage usefulness of treating a single nonlinear equation for the existence and hence formulation of a functional over the equivalent system for the same problem is brought into sharper focus.     

Surface curvature of steady granular flows

Laboratory experiments have shown that the steady flow of granular material down a rough inclined plane has a surface that is not parallel to the plane, but has a curvature across the slope with the height increasing toward the middle of the flow. We study this observation by postulating a new granular rheology, similar to that of a second order fluid. This model is applied to the experiments using a shallow water approximation, given that the depth of the flow is much smaller than the width. The model predicts that a second normal stress difference allows cross-slope height variations to develop in regions with considerable cross-slope velocity shear, consistent with the experiments. The model also predicts the development of lateral eddies, which are yet to be observed.     

Fracture prediction in steady ideal plastic flows

The theory of ideal plastic flows is a powerful tool for the preliminary design of metal forming operations. The present paper develops an approach to include a fracture criterion in this design procedure. The fracture criterion is based on the workability diagram and an average value of the triaxiality factor. In the case of three dimensional flows, this fracture criterion is modified which leads to mathematical advantages. It is shown that in many cases the modified criterion coincides with the original criterion. Using the proposed approach fracture initiation in drawing and extrusion processes is predicted. In particular, it is demonstrated that fracture predictions may be carried out without having the solution for stress and strain in the plastic zone (it is only necessary to know that it exists). The maximum possible area reduction with no fracture is given by the solution to a quadratic equation (in general, it might be a transcendental equation).     

The stability chart of double-diffusive processes with parallel flows — construction by the Galerkin and continuation methods

Summary The stability of an infinite fluid layer subject to arbitrary horizontal flow and to arbitrary vertical temperature and salinity distributions is considered. Linear stability analysis is used to investigate the stability under general three-dimensional perturbations.A general discussion of the properties of the stability chart, using only the governing equations, but not their solution for any particular case, is presented in [1]. This paper contributes a numerical scheme, based on a combination of the Galerkin and the continuation methods, to obtain the stability chart from the characteristic equation. The method is applied to an example with a parabolic velocity distribution and linear temperature and salinity fields. The stability chart in the plane of the Rayleigh numbers is obtained for the region corresponding to solar ponds.     

The rheological modelling of carbon nanotube (CNT) suspensions in steady shear flows

This paper is concerned with the rheological modelling of both chemically treated and untreated carbon nanotube (CNT) suspended in a Newtonian epoxy resin. CNT suspensions generally exhibited shear-thinning characteristic—the apparent viscosity decreases as shear rate increases—when subject to steady shear flows. Chemically treated CNT suspensions with little optical microstructure were found to exhibit a less significant shear-thinning effect compared with untreated CNT suspensions where clear optical aggregates were observed. In the case of treated CNT suspensions, the shear-thinning characteristic could be described using a Fokker–Planck based orientation model. The model assumed that the treated CNTs behaved as high aspect ratio rods and that shear flow was able to align the CNTs in the flow direction, thereby resulting in a decrease in the shear viscosity. Despite the success in describing the rheological response of treated CNT in steady shear flows, the orientation model failed to explain the more pronounced shear-thinning effect observed in untreated CNT suspensions having a hierarchy of aggregate structures. A new model called the aggregation/orientation (AO) model was formulated by modifying the Fokker–Planck equation. The AO model considered elements of aggregation as well as CNT orientation and it was capable of capturing the steady shear response of untreated CNT suspensions.     

The stability of swirling flows with a heat source

The absolute instability of a Rankine vortex with an axial flow and paraxial heat source is investigated. The dispersion relation for vortex modes is derived analytically. The dependence of dispersion properties of the media on control parameters such as swirl parameter S, velocity a, and heat source power (density parameter Q) is studied. The frequency of helical waves increases and the increment decreases with increasing heat source power, accompanied by a decrease in the width of the neutral stability region. Numerical analysis also suggests that one of the dispersion curve branches could include an instability region of a parametric nature.     

Inhomogeneous shear flows of linear polymers

Solutions of a system of equations of nonlinear viscoelastic fluid motion describing inhomogeneous shear flows of linear polymers are indicated.Notation _ij stress tensor - p pressure - F_i mass force vector - _ij Kronecker delta - coefficient of shear viscosity - relaxation time - _ij inner parameter - _ij=vi/x_j velocity gradient tensor - ₀ initial value of the shear viscosity coefficient - ₀ initial value of the relaxation time - D dimensionless first invariant of the additional stress tensor - A, B, C constants of integration - f(D) universal function characterizing the material - r, , z cylindrical coordinates - u=v_z axial component of the velocity vector - v=v circumferential component of the velocity vector - ₁, ₂ first and second differences of the normal stress - Q volume mass flow rate - R radius of a circular tube - R₁, R₂ radii of the inner and outer cylinders, respectively - M moment per unit length Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 41, No. 3, pp. 449–456, September, 1981.     

Formation of triple shock configurations with negative reflection angle in steady flows

Triple configurations of shock waves with negative reflection angles are considered. These configurations have been observed in quasi-steady cases of shock wave reflection from a planar wedge in real gases, while in steady cases three-shock configurations are only known to occur with positive reflection angles. Boundaries for the appearance of a three-shock configuration with a negative reflection angle in steady cases are analytically determined as dependent on the initial Mach number of the flow, angle of incidence, and adiabatic index. The formation of a three-shock configuration with a negative reflection angle in a steady flow must lead to a change in the character of the wave pattern, and under certain conditions it can lead to instability.     

The linear and nonlinear stability of thread-annular flow

The surgical technique of thread injection of medical implants is modelled by the axial pressure-gradient-driven flow between concentric cylinders with a moving core. The linear stability of the flow to both axisymmetric and asymmetric perturbations is analysed asymptotically at large Reynolds number, and computationally at finite Reynolds number. The existence of multiple regions of instability is predicted and their dependence upon radius ratio and thread velocity is determined. A discrepancy in critical Reynolds numbers and cut-off velocity is found to exist between experimental results and the predictions of the linear theory. In order to account for this discrepancy, the high Reynolds number, nonlinear stability properties of the flow are analysed and a nonlinear, equilibrium critical layer structure is found, which leads to an enhanced correction to the basic flow. The predictions of the nonlinear theory are found to be in good agreement with the experimental data.     

Simple method for checking the steady state of fluctuations in linear electronic systems

A simple criterion is considered for checking the steady state of inherent fluctuations in a linear electronic measurement system. __________ Translated from Izmeritel’naya Tekhnika, No. 6. pp. 13–15, June, 2007.     

On the linear stability of Stokes layers

Oscillatory flows occur naturally, with applications ranging across many disciplines from engineering to physiology. Transition to turbulence in such flows is a topic of practical interest and this article discusses some recent work that has furthered our understanding of the stability of a class of time-periodic fluid motions. Our study starts with an examination of the linear stability of a classical flat Stokes layer. Although experiments conducted over many years have demonstrated conclusively that this layer is unstable at a sufficiently large Reynolds number, it has only been relatively recently that rigorous theoretical confirmation of this behaviour has been obtained. The analysis and numerical calculations for the planar Stokes layer were subsequently extended to flows in channels and pipes and for the flow within a torsionally oscillating circular cylinder. We discuss why our predictions for the onset of instability in these geometries are in disappointingly poor agreement with experimental results. Finally, some suggestions for future experimental work are given and some areas for future theoretical analysis outlined.     

Numerical investigation of the onset of instability of triple shock configurations in steady supersonic gas flows

Triple configurations of shock waves that arise in a steady supersonic flow of a real gas have been numerically simulated. Previously, we have theoretically predicted that a new triple shock configuration with a negative reflection angle can appear in a gas flow at large Mach numbers and small adiabatic indices. This configuration is now obtained for the first time in numerical experiments. It is shown that the formation of this triple shock configuration leads to instability of the entire flow pattern.     

Application of the Lambert W function to steady shearing flows of the Papanastasiou model

Using the Lambert W function, the constitutive relation of the Papanastasiou model is inverted so that the second invariant of the first Rivlin-Ericksen tensor can be expressed as a function of the second invariant of the extra stress tensor. In steady shearing flows, this results in the magnitude of the shear rate becoming a function of the magnitude of the shear stress. Since the distribution of the latter is known explicitly in channel, Poiseuille and Couette flows, one can investigate the nature of analytical solutions in these flows. It is shown that explicit answers are found for channel and Poiseuille flows only, with the Couette flow requiring a numerical solution in general. From the channel flow results, it is obvious that there is a great amount of congruence between the predictions of the Papanastasiou model and the Bingham fluid. In turn, this lends further confidence to the application of the Papanastasiou model to study the flows of Bingham fluids.     

Hydromagnetic stability of spiral flows with density stratification

The present paper deals with the effect of gravity on the hydromagnetic stability of spiral flows. By discussing the various models, it has been established that the density decreasing in the radial direction has a stabilizing effect in the present situation.     

On the spatial stability of free-convection flows in a saturated porous medium

We consider the free-convection boundary-layer flow in a saturated porous medium adjacent to an impermeable vertical surface. It is assumed that the surface is supplying heat to the porous medium in a prescribed way, which varies along the surface. The problem, which relates to the spatial stability of the known similarity solutions of the boundary-layer equations, is formulated and certain analytical results presented for special cases. For this special class of flows we are able to determine analytically the first eigenvalue for all relevant parameter values and thereby show that such flows are spatially stable.