Plane constantly inclined magnetogasdynamic flows

Compressible MHD constantly inclined steady plane flows are studied by using the formulation in terms of streamfunction and magnetic flux function. This approach is illustrated through solutions of four problems.     

Dense inclined flows of inelastic spheres

We outline an extension of the hydrodynamic equations for dense flows of identical, inelastic spheres that incorporates an additional length scale in the expression for the collisional rate of dissipation. This length scale is identified with the length of a particle chain. In steady, fully developed inclined flows, the resulting theory predicts that at a given angle of inclination a range of flow depths is possible, that such flows possess a region of uniform volume fraction, and that this volume fraction decreases as the angle of inclination increases. The balance of particle fluctuation energy, integrated through the depth of a flow, results in a relation between the mean flow velocity, the depth, and the angle of inclination that collapses experimental data taken over a range of inclination angle.     

Kinetic theory applied to inclined flows

We apply the continuum equations of a kinetic theory to predict the features of uniform, steady, inclined flows of identical, frictional, inelastic spheres over a rigid, bumpy base between vertical, frictional side walls. Numerical solutions of these equations over a range of mass flow rates exhibit features seen in physical experiments and numerical solutions in the absence of side walls. For the densest flows, we employ a phenomenological extension of kinetic theory that involves a length scale associated with particle correlations. When a dense flow is thick enough, an algebraic balance between the production and dissipation of fluctuation energy reproduces the relation between mass flow rate and mass hold-up obtained when solving the boundary-value problem of the extended theory.     

Drag coefficients for a sphere rolling down an inclined channel

Abstract

Forces acting on a sphere rolling down an inclined smooth or rough channel include submerged weight of the sphere, fluid drag, fluid lift, and rolling resistance from the channel bed. When the acting forces are in equilibrium, the sphere rolls down the inclined channel with a terminal velocity. The steady movements of such a sphere rolling down smooth and rough channel beds with the terminal velocity are studied through laboratory experiments and a simplified theoretical analysis. A method is proposed to evaluate the rolling resistance that consists of collision and friction parts of resistance. The coefficients of rolling resistance and the drag coefficients are determined by regression analysis with experimental data. The relationships of the drag coefficients and the Reynolds numbers for a sphere rolling over smooth and rough channel beds are also obtained. It is found that at the same Reynolds number, the drag coefficient for a sphere rolling down a rough bed is larger than that for a sphere down a smooth one, and that both are much larger than that of a sphere in the free fall.     

Similarity flow in interaction of a shock wave with an inclined heated channel

A study is made of gasdynamic flow that initiates when a shock wave propagates along a thin heated channel. Analytical conditions of the onset of an unsteady flow precursor are obtained. The flow similarity is proved experimentally; precursor characteristics vs shock wave and heated channel parameters are analyzed.Institute of Geosphere Dynamics, Russian Academy of Sciences, Moscow. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 54, No. 5, pp. 543–547, May, 1993.     

Stability of a layer of liquid flowing down an inclined plane

Summary The stability of the flow down an inclined plane is studied for small angles of inclination β. The same problem has been studied by S. P. Lin, however using an incorrect boundary condition. The correctly formulated eigenvalueproblem is solved by a numerical integration of the Orr-Sommerfeld equation employing the orthonormalization technique. It is shown that in the range 3′<β<1° a decrease in β means a decrease in the critical Reynolds number for the hard mode, which is a shear wave modified by the presence of the free surface. In that range the stability is still more or less governed by the stability of the soft waves, which are essentially surface waves modified by the presence of shear. For values of β<1′ the stability is governed by the hard mode, contradictory to Lin's statements. In that case instability occurs at high Reynolds numbers.     

倾斜转子-滑动轴承系统周期解稳定性研究

摘　要：针对倾斜转子-滑动轴承系统的周期解稳定性问题,考虑倾斜状态下滑动轴承径向载荷的变化,基于有限元法建立了系统的动力学方程。使用Newmark方法仿真得到了倾斜转子-轴承系统的动力学响应,并分析了倾斜角度变化对转子周期解失稳阈速和失稳区宽度的影响。仿真结果表明,转子倾斜角度增大会导致转子失稳阈速减小,失稳宽度增大,但对周期解失稳区域上边界影响不大。倾斜转子的升速实验验证了仿真结果。     

Dense inclined flows of inelastic spheres: tests of an extension of kinetic theory

Using the results of recent numerical simulations, we extend an existing kinetic theory for dense flows of identical, nearly elastic, frictionless spheres to identical, very dissipative, frictional spheres. The existing theory incorporates an additional length scale in the expression for the collisional rate of dissipation; this length scale is identified with the size of a cluster of correlated particles. Parameters of the theory for very dissipative, frictional spheres are set using the results of physical experiments on inclined flows of spheres over a rigid, bumpy base in the absence of sidewalls. The resulting theory is then tested against the results of physical experiments on flows of the same material over the surface of an erodible heap when frictional sidewalls are present.     

Turbulent spots in channel flows

A number of investigations into the formation and development of turbulent spots in plane Poiseuille, plane Couette and watertable flows are reviewed. Three main observations are drawn from this work. Firstly, the initial development is associated with the transient growth due to the three-dimensional lift-up effect. Secondly, the spreading and propagation velocities of the different spots are quite similar. Thirdly, the velocity field inside the spots show essentially all the characteristics of fully developed turbulent flow, albeit at low Reynolds numbers. The mechanism behind the rapid spreading of the spots, however, is not yet fully understood, although observations of wave activity, modifications of the surrounding flow field and stability calculations point in the direction of growth by destabilization.     

Turbulent fluid flows in a circular pipe and plane channel and models of mesoscale turbulence

Based on the model of anisotropic wall turbulence in the near-wall layer and the momentum model in the flow core, velocity profiles in the entire region of the turbulent flow of an incompressible fluid in a circular pipe and plane channel have been obtained. The differences in the profiles among the layers are due to the change in the structure of turbulent vortices.     

Stability of a thin viscous fluid film flowing down a rotating non-uniformly heated inclined plane

A thin viscous liquid film flowing down a rotating non-uniformly heated inclined plane is investigated. It is assumed that the rotation is small and the region of investigation is very far from the axis of rotation, so that the centrifugal force has a dominant role in the instability of the investigated region. Therefore, we have derived a Benney-like free surface evolution equation on the basis of a long-wave (small wave number) approximation and not included the Coriolis effects in the expansion of the dependent variables. Further, a linear and a weakly nonlinear study have been carried out. The linear study reveals that the growth of linear perturbation is independent of the Rossby number Ro, that is invariant with the Coriolis effect, while the linear phase speed c _r depends on Ro as well as on the Taylor number Ta. It also reveals that as Ta increases the stable zone decreases, and the influence of Ta is stronger for greater inclination of the rotating inclined plane. Again, it is found that the Marangoni number Mn has similar qualitative (destabilizing) behavior as Ta, but the destabilizing behavior of Ta is more at high Mn than at low Mn. The relation between the parameters Ta and Ro gives a unified parameter Ta_rot, which reflects the effect of rotation, and we found that the linear phase speed c _r first decreases with Ta_rot up to a critical value and then increases with Ta_rot. This is due to the fact that the Coriolis force is dominant at very small rotation, while for relatively large rotation the centrifugal force dominates the flow field. The weakly nonlinear study reveals that the effect of rotation appears in the form of both Coriolis and centrifugal force into the growth of finite amplitude perturbations in contrast to the growth of the infinitesimal perturbations where the rotation arrives in the form of centrifugal force only. Also, it plays significant role in the different stability zones, amplitudes of sub/super critical disturbances and nonlinear phase speed.     

BEM solution of thin film flows on an inclined plane with a bottom outlet

This article considers the numerical analysis of thin films flowing on an inclined plane with an opening (a gap) at the bottom of the plane, representing an outlet. The film is subject to gravitational force and a constant surface shear stress at the film surface. The present problem can be regarded as a special draining-type problem, in which there is a flow bifurcation from one side of the domain (inlet flux) to the outlet and the other side of the domain. The thin film flow is modelled as governed by the two-dimensional Stokes system of equations and an investigation into the effects of surface tension, surface shear stress and geometry variation on the flow of the film into an outlet is attempted. Numerical simulations are obtained in terms of a direct boundary element method (BEM) with primitive variables of velocity and surface traction.     

Exact solutions for some simple flows of an Oldroyd-B fluid

Summary We present two simple but elegant solutions for the flow of an Oldroyd-B fluid. First, we consider the flow past an infinite porous plate and find that the problem admits an asymptotically decaying solution in the case of suction at the plate, and that in the case of blowing it admits no such solution. Second, we study the longitudinal and torsional oscillations of an infinitely long rod of finite radius. The solutions are found in terms of Bessel functions.     

Mass transfer from a wall to a gas-liquid flow in an inclined flat channel

Experimental study is made of mass transfer from a wall to a gas-liquid flow in an inclined flat channel. Measurements are made by the electrodiffusional method. It is shown that maximum values of mass transfer correspond to intermediate ones of inclination angles of the channel. The effect of the gas phase structure on mass transfer is examined.Thermophysics Institute, Novosibirsk. Uniprommed Institute, Ekaterinburg. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 64, No. 5, pp. 523–528, May, 1993.     

Long gravity waves on a viscous fluid flowing down an inclined plane

Summary The properties of permanent waves on a running stream of viscous fluid flowing down an inclined plane are investigated for small Reynolds number. The waves considered have a long wavelength compared with the depth of liquid. The approximation for the free surface in this Stokes flow problem is taken to the third order and extends work initiated by Mei [1]. This extension introduces additional permanent waves including cnoidal and solitary waves and also transverse strips of liquid running down the plane which appear as liquid drops in section. A qualitative analysis of the differential equation is obtained using phase-plane techniques.     

Hodograph transformations and solutions in variably inclined MHD plane flows

Summary By a variably inclined MHD plane flow we mean a flow in which the magnetic and velocity fields are coplanar, the angle between these vector fields is variable and all the flow variables are functions of two coordinates and time. No work seems to have been done for these general plane MHD flows, even in the steady case.In the present paper the work in steady, viscous, incompressible flows is extended to general variably inclined, but nowhere aligned, flows with the objective of obtaining some exact solutions. We employ the hodograph transformation, one of the strong analytical methods, to find these solutions.     

Hodograph transformations and solutions in constantly inclined MHD plane flows

Summary In this work we develop a technique, based on the hodograph method, for the study of steady plane, viscous, incompressible constantly inclined MHD flows.The equation describing the diffusion of magnetic field is used to write the magnetic field vector in terms of the velocity vector field and the angle between the two vector fields. The hodograph method (and its modifications) is applied to reduce the problem to that of determining the Legendre transform of the stream function. The resulting partial differential equation is studied for several flow problems to illustrate the advantages of the theory.This paper also employs a similar approach as the above to study flows in the magnetograph plane.     

Homotopy analysis of MHD flows of an Oldroyd 8-constant fluid

Summary. This paper deals with some steady unidirectional flows of an Oldroyd 8-constant magnetohydrodynamic (MHD) fluid in bounded domains. The fluid is electrically conducting in the presence of a uniform magnetic field. Three nonlinear flows are produced by the motion of a boundary or by sudden application of a constant pressure gradient or by the motion of a boundary and pressure gradient. The governing nonlinear differential equations are solved analytically using homotopy analysis method (HAM). Expressions for the velocity distribution are given. It is noted that for steady flow the solutions are strongly dependent on the non–Newtonian and magnetic parameters. The MHD solutions for a Newtonian fluid, as well as those corresponding to the Oldroyd 3 and 6-constant fluids, a Maxwell fluid and a second grade one, appear as limiting cases of our solutions. Finally, a physical interpretation of the results is given with the help of several graphs.