On intrinsic properties of steady gas flows

Summary Considering the geometric theory of triply orthogonal spatial curves, the basic equations governing a steady gas flow are transformed into the intrinsic form and the results obtained are:(1) The pressure is uniform along the binormal to the stream line and the radius of curvature varies as the square of the velocity along it, for the baratropic fluids.(2) Acceleration is irrotational field when the fluid is compressible but baratropic or incompressible, in which case the relations existing between the flow quantities, curvature and torsions of the curves under consideration are obtained.(3) Considering incompressible flows, it is observed that either velocity in magnitude is uniform or the vorticity lies in the normal plane, in which case the stream lines are orthogonal to the vortex lines.Stream lines are observed to be either right circular helices or circles or straight lines.If the stream lines are not straight then the torsions of the binormal congruences and stream lines are equal.(4) The compatibility conditions of Berker¹) are transformed into intrinsic form, involving the curvatures and torsions of the above curves.     

Calculation of aerodynamic characteristics of a wing profile with discharge of a controllable jet into the external flow

A wing profile of infinite span, whose lower surface is replaced by a system of guide vanes, is placed in a flow of an ideal incompressible fluid. Fluid flows out through the system of guide vanes from the internal cavity of the wing into the external stream, forming a jet in the wake (Fig. 1). The total pressure in the wing cavity and in the jet differs from the total pressure in the outer free stream. The jet boundaries are streamlines extending to infinity, along which there is a discontinuity of the velocity value. The flow of fluid in the internal wing cavity is simulated by a flow caused by a system of suitably located sources, and the system of guide vanes is replaced by discrete vortices.The form of the profile arc is selected so that the fluid flow from the sources in the direction which is nearly opposite the direction of the freestream velocity is restrained by the segment of the contour with high curvature in the vicinity of the leading edge. We consider the flow regime about the profile with an exhausting jet for which the two ends of the arc the points of detachment of the stream and the velocity discontinuity line (profile arc, jet boundary) is a smooth curve, which imposes an additional condition on the magnitude of the circulation. As the model for the study of the flow about a profile with jet blowing we take the arc of a logarithmic spiral.Formulas are obtained for determining the over-all characteristics of the stream forces acting on the profile in the presence of the jet and the total pressure discontinuity. On the basis of the calculations made for a thin wing a qualitative analysis is made for the stream force acting on the profile.The authors wish to thank S. A. Khristianovich for formulating the problem and for his advice.     

Simulation of peristaltic flow of chyme in small intestine for couple stress fluid

In this article couple stress fluid have been considered for the peristaltic flow of chyme in intestine. Problem under consideration have been formulated assuming that two non-periodic sinusoidal wave of different wavelength propagate with same speed c along the outer wall of the tube. Governing equations have been simplified under the assumptions of long wavelength and low Reynolds number approximation (such assumption are consistent that Re (Reynold number) is very small and long wavelength approximation also exists in the small intestine). Exact solutions have been evaluated for velocity and pressure rise. Physical behaviour of different parameter of couple stress fluid have been presented graphically for velocity, pressure rise, pressure gradient and frictional forces. The stream lines are also made against different parameters.     

Simultaneous heat and mass transfer under unsteady mixed convection along a vertical slender cylinder embedded in a porous medium

Unsteady laminar mixed convection flow (combined free and forced convection flow) along a vertical slender cylinder embedded in a porous medium under the combined buoyancy effect of thermal and species diffusion has been studied. The effect of the permeability of the medium as well as the magnetic field has been included in the analysis. The partial differential equations with three independent variables governing the flow have been solved numerically using a implicit finite difference scheme in combination with the quasilinearization technique. Computations have been carried out for accelerating, decelerating and oscillatory free stream velocity distributions. The effects of the permeability of the medium, buoyancy forces, transverse curvature and magnetic field on skin friction, heat transfer and mass transfer have been studied. It is found that the effect of free stream velocity distribution is more pronounced on the skin friction than on the heat and mass transfer. The permeability and magnetic parameters increase the skin friction, but reduce the heat and mass transfer. The skin friction, heat transfer and mass transfer are enhanced due to the buoyancy forces and curvature parameter. The heat transfer is strongly dependent on the viscous dissipation parameter and the Prandtl number, and the mass transfer on the Schmidt number.     

Continuity violation trajectories in perfectly plastic bodies

The consistency equations for the increments of small strains in a triorthogonal isostatic coordinate system supplemented with additional relations between the physical components of the inconsistency tensor are considered. There are six significant consistency equations. It is proved that, for the stress states corresponding to the edge of the Coulomb-Tresca prism, there are only three independent consistency equations. Systems of independent consistency equations written in the isostatic coordinate mesh are explicitly indicated and studied. Sufficient conditions for the remaining three consistency equations to be satisfied if the three independent consistency equations are satisfied are obtained. It is shown that the continuity violations on the surface of a perfectly plastic body propagate into the depth of the body along asymptotic lines on the layers of a vector field indicating the directions of the maximum principal normal stress. Since the asymptotic lines are less curved than any other lines on the surface (in the sense that the normal curvature of the asymptotic lines is zero), the continuity violations propagate into a perfectly plastic body along the least curved trajectories, which permits one to speak of the minimum curving of crack propagation trajectories in perfectly plastic rigid bodies.     

Peristaltic motion of a non-Newtonian fluid

Summary To understand theoretically the flow properties of physiological fluids, we have considered as a model the peristaltic motion of a power law fluid in a tube, with a sinusoidal wave of small amplitude travelling down its wall. The solution for the stream function is obtained as a power series in terms of the amplitude of the wave. The stream function and the velocity components are evaluated by solving numerically two point boundary value problems with a singular point at the origin. The influence of the applied pressure gradient along with non-Newtonian parameters on the streamlines and velocity profiles are discussed in detail.     

Peristaltic transport of a Jeffrey fluid under the effect of magnetic field in an asymmetric channel

The peristaltic flow of a Jeffrey fluid in an asymmetric channel is studied under long wavelength and low Reynolds number assumptions. The fluid is electrically conducting by a transverse magnetic field. The channel asymmetry is produced by choosing the peristaltic wave train on the walls to have different amplitudes and phase. The flow is investigated in a wave frame of reference moving with the velocity of the wave. The expressions for stream function, axial velocity and axial pressure gradient have been obtained. The effects of various emerging parameters on the flow characteristics are shown and discussed with the help of graphs. The pumping characteristics, axial pressure gradient and trapping phenomenon have been studied. Comparison of various wave forms (namely sinusoidal, triangular, square and trapezoidal) on the flow is discussed.     

Hypersonic flow of nonequilibrium air over blunt bodies

A numerical method is described for computing nonequilibrium three-dimensional supersonic flow of a gas in the shock layer over the forward surface of blunt bodies with discontinuities of shape. The basic idea is to divide the original system of differential equations into two subsystems, which are solved in succession: first for the gasdynamic variables, the velocity components and the pressure, and then for the relaxation parameters and the enthalpy. To calculate the velocity components and the pressure we use the iterative marching method [1, 2] in the form given in [3]. The relaxation equations and the enthalpy equation are integrated numerically along the stream lines. A discussion is given of the effect of nonequilibrium of physical and chemical reactions on the distribution of parameters in the inviscid shock layer and on the aerodynamic coefficients of blunt bodies in hypersonic air flow. The unsteady aerodynamic coefficients are calculated by the curved body method [4]. The computational algorithm takes the form of a program in “ALGOL-60” for the BéSM-6 computer.     

局部狭窄血管中血液振荡流的速度和切应力

本文建立一种分析局部缓慢狭窄血管中血液振荡流的数学模型，给出了血液的轴向流速，径向流速和切应力的包含压力梯度项的解析表达式，并讨论了血管内由局部狭窄引起的压力梯度沿轴向变化的规律。文章以局部余弦狭窄为例进行数值计算，详细讨论上游均匀管段压力梯度的定常部分和不同次谐波对狭窄管段内流速和切应力的影响。数值结果表明，与均匀管情况相比，在狭窄段内，血液振荡流轴向流速无论平均值还是脉动幅值均明显增大，且径向流速不再为零。但径向流速仍远小于轴向流速。同时，切应力也不再仅由轴向流速梯度提供，径向流速梯度也将产生切应力，但是在计算管壁切向上的切应力时，径向流速梯度的贡献仍相当大。与均匀管管壁切应力沿流运方向保持恒定不同。狭窄管管壁切应力（平均值和脉动值）将随着狭窄高度的增大而增大，在狭窄最大高度处达到最大，因而沿流动方向产生了较大的切应力梯度。     

An experimental study of the effect of the blow-in rate on the size of the flow separation region

Experimental results are presented concerning the effect of the air blow-in rate on the size of the flow separation region downstream of a rearward-facing step at different step heights. The stream function is found from the experimental velocity profiles, the streamline = 0 being taken as the boundary line. It is shown that the separation region increases as the blow-in rate is increased. Generalization of the experimental results for different blow-in rates and step heights has made it possible to obtain an analytical expression describing the location of some characteristic lines in the separation region (boundary streamline, reverse-flow boundary, line of reverse-flow maximum velocity, line of variable-velocity layer thickness, and the displacement thickness line). Velocity profiles are obtained by means of a hot-wire anemometer. Analytical expressions are written as polynomials with unknown coefficients. The results obtained may be helpful in developing techniques for the treatment of heat transfer under flow separation conditions.     

挤出胀大流动的有限元方法研究

本文对Luo-Tanner提出的流线有限元作了重要的改进,提出了沿通过单元高斯点的流线积分本构方程的方法,迴避了速度梯度在单元边界上间断和出口处应力奇点的困难,同时减少了计算量,对比计算表明,采用压力不连续单元来加强不可压缩性限制能使计算质量和收敛性都得到显著的提高,对Maxwell流体的轴对称挤出胀大流动在Weissenberg数1.2下获得了合理的收敛解。     

Determination of gradients of flow variables behind three-dimensional stationary and pseudo stationary curved shock waves in conducting gases

Summary In this paper we have obtained the gradients of magnetic field, velocity, pressure and density behind a shock wave in three dimensional steady motion of a conducting gas. For the shock configuration, we take a continuous differentiable function of coordinates and it is assumed that the components of the magnetic field H ⁱ , velocity components u ⁱ , pressure p and density behind the shock-surface are differentiable functions. Moreover we take H ⁱ , u ⁱ , p and in front of the shock-wave as constant quantities. In § 4 we have obtained the gradients of flow and field quantities behind the pseudostationary shockwave. § 5 is devoted to the calculation of gradients of flow and field quantities in cases where the normal component of the magnetic field is zero on both sides of the shock wave. In § 6 the relation between the curvature k of the shock-surface and the curvature K of the stream line just behind the shock surface in two dimensional steady motion has been derived. § 7 deals with the determination of the ratio K/k for an attached shock in the case of a wedge.     

Flow of Particulate-Fluid Suspension in a Channel with Porous Walls

The problem of the dispersed particulate-fluid two-phase flow in a channel with permeable walls under the effect of the Beavers and Joseph slip boundary condition is concerned in this paper. The analytical solution has been derived for the longitude pressure difference, stream functions, and the velocity distribution with the perturbation method based on a small width to length ratio of the channel. The graphical results for pressure, velocity, and stream function are presented and the effects of geometrical coefficients, the slip parameter and the volume fraction density on the pressure variation, the streamline structure and the velocity distribution are evaluated numerically and discussed. It is shown that the sinusoidal channel, accompanied by a higher friction factor, has higher pressure drop than that of the parallel-plate channel under fully developed flow conditions due to the wall-induced curvature effect. The increment of the channel’s width to the length ratio will remarkably increase the flow rate because of the enlargement of the flow area in the channel. At low Reynolds number ranging from 0 to 65, the fluids move forward smoothly following the shape of the channel. Moreover, the slip boundary condition will notably increase the fluid velocity and the decrease of the slip parameter leads to the increment of the velocity magnitude across the channel. The fluid-phase axial velocity decreases with the increment of the volume fraction density.     

弯曲狭窄管内血液流的分析

与血管狭窄有关的异常血液动力学特征在血管疾病的发生和发展过程中起着重要的作用,由于血管狭窄和弯曲的综合影响,将会出现一系列有趣的流体力学现象,本文研究具有对称狭窄的弯曲小动脉内定常血液流动,在一定的假设条件下,直接从支配血液流动的Navier-Stokes方程求出问题的摄动解,由此求得弯曲狭窄管內血液流动的轴向速度、二次流速度及压力梯度等分析表达式,并进一步求得轴向和周向血管壁切应力。本文的结果是先前有关狭窄直管和弯曲均匀管流动研究的拓广。     

Slip effects on the peristaltic flow of a Jeffrey fluid in an asymmetric channel under the effect of induced magnetic field

In the present study, we investigated the effects of slip and induced magnetic field on the peristaltic flow of a Jeffrey fluid in an asymmetric channel. The governing two‐dimensional equations for momentum, magnetic force function and energy are simplified by using the assumptions of long wavelength and low but finite Reynolds number. The reduced problem has been solved by Adomian decomposition method (ADM) and closed form solutions have been presented. Further, the exact solution of the proposed problem has also been computed and the mathematical comparison shows that both solutions are almost similar. The effects of pertinent parameters on the pressure rise per wavelength are investigated using numerical integration. The expressions for pressure rise, friction force, velocity, temperature, magnetic force function and the stream lines against various physical parameters of interest are shown graphically. Moreover, the behavior of different kinds of wave shape are also discussed. Copyright © 2009 John Wiley & Sons, Ltd.     

Prediction of soil reaction forces on a moldboard plow surface

Using mathematical models, approximate trajectories of soil furrow subslices were determined and the forces occurring on the moldboard surface were calculated. The various components of the normal force along the trajectory line of the middle section of the furrow slice were analysed in relation to the velocity of plowing operation and the curvature of the path. The soil reaction forces encountered by the share portion of the plow were also determined by a model which used the method of trial wedges to determine the angle of the failure surface with the horizontal. The sum of the calculated forces was compared with that obtained by actual plowing experiments performed in the laboratory.     

Flow patterns and interfacial velocities near a moving contact line

Experimental data on velocity fields and flow patterns near a moving contact line is shown to be at variance with existing hydrodynamic theories. The discrepancy points to a new hydrodynamic paradox and suggests that the hydrodynamic approach may be incomplete and further parameters or forces affecting the surfaces may have to be included. A contact line is the line of intersection of three phases: (1) a solid, (2) a liquid, and (3) a fluid (liquid or gas) phase. A moving contact line develops when the contact line moves along the solid surface. A flat plate moved up and down, inside and out of a liquid pool defines a simple, reliable experimental model to characterize dynamic contact lines. Highlighted are three important conclusions from the experimental results that should be prominent in the development of new theoretical models for this flow. First, the velocity along the streamline configuring the liquid–fluid interface is remarkably constant within a distance of a couple of millimeters from the contact line. Second, the relative velocity of the liquid–fluid interface, defined as the ratio of the velocity along the interface to the velocity of the solid surface, is independent of the solid surface velocity. Third, the relative interface velocity is a function of the dynamic contact angle.     

A robust algorithm for computing fluid flows on highly non‐smooth staggered grids

A new method for computing the fluid flow in complex geometries using highly non‐smooth and non‐orthogonal staggered grid is presented. In a context of the SIMPLE algorithm, pressure and physical tangential velocity components are used as dependent variables in momentum equations. To reduce the sensitivity of the curvature terms in response to coordinate line orientation change, these terms are exclusively computed using Cartesian velocity components in momentum equations. The method is then used to solve some fairly complicated 2‐D and 3‐D flow field using highly non‐smooth grids. The accuracy of results on rough grids (with sharp grid line orientation change and non‐uniformity) was found to be high and the agreement with previous experimental and numerical results was quite good. Copyright © 2008 John Wiley & Sons, Ltd.     

Flow past a plate lattice with developed cavitation

Problems of streamline cavitation flow past a lattice were studied in [1–8] using the Kirchhoff scheme. In this scheme the magnitude of the velocity at the free surface is equal to the stream velocity behind the lattice, and the cavitation number is zero (for a lattice the relative velocity and the cavitation number are defined from the stream velocity behind the lattice). In [4, 7] a solution is given of the problem of flow past a lattice using a scheme with an Efros-Gilbargreturn streamline, which permits considering arbitrary cavitation numbers; however, a unique solution is not given. Some other streamline schemes are mentioned in [8].In the following we consider the cavitational flow of an ideal incompressible inviscid and weightless fluid past an infinite lattice of flat plates, using the streamline wake model previously utilized by Wu [9] in studying cavitational flow past an isolated obstacle. In accordance with this model, the streamlines which separate from the body and bound the cavity behind it pass into two curvilinear infinitely long walls, along which the pressure increases and approaches the pressure in the undisturbed stream.It is further assumed that in the hodograph plane there corresponds to the curvilinear walls a cut along some line and that the complex potential takes the same values at points lying on opposite sides of the cut. In particular, at the points of contact of the streamlines with the curvilinear walls the complex potential is the same. In the Wu scheme the latter condition leads to vanishing of the velocity circulation along the contour CABC₁ (Fig. 1).In conclusion the author wishes to thank N. V. Yurtaeva for the accurately performed numerical work.     

MHD peristaltic flow of a third order fluid in an asymmetric channel

This study is concerned with peristaltic flow of a magnetohydrodynamic (MHD) fluid in an asymmetric channel. Asymmetry in the flow is induced by waves on the channel walls having different amplitudes and phase. A systematic approach based on an expansion of Deborah number is used for the solution series. Analytic expressions have been developed for the stream function, axial velocity and axial pressure gradient. The pressure rise over a wavelength has been addressed through numerical integration. Particular attention has been given to the effects of Hartman number and Deborah number on the pressure rise over a wavelength and the trapping phenomenon. Several limiting solutions of interest are obtained as the special cases of the presented analysis by taking the appropriate parameter(s) to be zero. Copyright © 2009 John Wiley & Sons, Ltd.