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The steady secondary flow around a circular cylinder oscillating rotationally about its axis in a uniform flow is investigated with the successive approximation based on the Oseen approximation. Th...

The hydrodynamic interaction between two cylinders with rotational motion through an inviscid and incompressible fluid is investigated theoretically. The dynamical behaviors of an elliptic cylinder...

The secondary flow around an oscillating circular cylinder in a uniform flow is investigated with the successive approximation of the Navier-Stokes equation for low Reynolds numbers. The approximation is based on the Oseen's one and the equation and the solution are expanded in the power series of the Reynolds number. It is shown that the secondary flow is obtained in the order of the square of the Reynolds number and directs to the upstream for low frequencies of the oscillation. The drag due to the secondary flow is expressed in the velocity and the oscillation amplitudes and takes positive values for low values of the Reynolds numbers and the frequencies.

The flow around a source doublet in a two-dimensional shear flow along a flat plane is investigated analytically with an incompressible inviscid fluid. The velocity field is obtained by the linearized Euler equation. It is shown that the shear suppresses the three-dimensionality of the disturbance and makes the flow be rather two-dimensional one that parallel to the plane. A deformed vortex line consists of the looped line and the straight line of the shear flow.

An exact solution of the Navier-Stokes equations is established for a general initial/boundary-value problem which describes the development of an incompressible circulatory flow outside an infinite circular cylinder with uniform suction or injection. An asymptotic evaluation of the solution, valid after a long time, reveals the effects of suction or injection upon the ultimate flow behavior; for R >2, the circulation at infinity remains unchanged throughout the process if it is finite initially; for R ≤1, all of the vorticity, even if its amount is infinite, is blown away to infinity and the flow becomes irrotational in the final state; intermediate between them is the case of 1< R ≤2. Here R = V a /ν, with V , a and ν being the suction velocity, radius of the cylinder and the kinematic viscosity, respectively.

Polymer melts exhibit flow-history-dependent behavior. To clearly show this behavior, we have investigated the flow behavior of an entangled polymer melt around two cylinders placed in tandem along the flow direction in a two-dimensional periodic system. In this system, polymer states around a cylinder on the downstream side differ from those around a cylinder on the upstream side because the former have a memory of the strain they experienced when passing around the cylinder on the upstream side but the latter have no such memory. Therefore, the shear stress distributions around two cylinders are found to differ. Moreover, we have found that the mean flow velocity decreases accordingly with increasing distance between the two cylinders, whereas the applied external force is constant. Although this behavior is consistent with that of the Newtonian fluid, the flow-history-dependent behavior is found to enhance the reduction in the flow resistance. We have also discussed the microscopic states during the fl...

In this paper steady viscous flow between two infinitely long circular cylinders produced by small-amplitude transverse vibrations of the inner cylinder about the axis of the outer cylinder and torsional oscillation of the outer cylinder about its axis is studied. The Vishik–Lyusternik method is employed to construct an asymptotic expansion of the solution of the Navier–Stokes equations in the limit of high-frequency vibrations for Reynolds numbers of order of unity. The effect of the Stokes drift of fluid particles is also studied. It is shown that it is nonzero not only within the boundary layers but also in higher order terms of the expansion of the averaged outer flow.

This paper deals theoretically with the two-dimensional flow driven by a symmetrical Joukowski profile oscillating harmonically along its axis of symmetry in an unbounded viscous fluid otherwise at rest. Slow steady secondary flow which occurs outside the boundary layer around the profile is studied in detail. It is found that a uniform flow is induced at infinity owing to the asymmetry of the profile in the direction of oscillation.

The paper develops an exact analysis of MHD Falkner-Skan flow of an electrically conducting, incompressible viscous fluid. The existence of similarity solutions is demonstrated when the applied magnetic field is inversely proportional to the boundary layer thickness. Numerical solutions for velocity, temperature, skin-friction and rate of heat transfer are obtained. The numerical values of skin-friction and rate of heat transfer are tabulated and the velocity and temperature are graphically exhibited.

Experimental evidence is reported that a two-dimensional vortex is formed in front of a half-submerged circular cylinder, which is supported horizontally near the surface of water in a towing tank and moved in the direction normal to its axis. The experiment was mode at Reynolds numbers (based on the velocity and the radius of the cylinder) less than about 5×10 3 . The upstream vortex seems to be due to the forward movement of the surface layer of water driven by the cylinder.

When a circular cylinder performs the saw-toothed oscillations, the induced steady streaming is the sum of the circulatory streaming due to the sinusoidal oscillations and an additional flow. The additional flow is approximately unidirectional in the direction of oscillation, and its velocity increases with the increase of S , but is independent of D , where S =ω s 2 / v , D =ω d 2 / v , ω is the angular frequency of the cylinder motion, s the oscillation amplitude, v the kinematic viscosity of the working fluid and d the diameter of the cylinder.

This paper deals theoretically with the two-dimensional flow driven by a circular cylinder which performs unharmonically translational oscillation in an unbounded viscous incompressible fluid otherwise at rest. Slow steady secondary flow occurs outside the oscillatory boundary layer around the cylinder, provided that the non-dimensional amplitude of the oscillation e and the boundary layer thickness are both extremely small. It is found that induced steady flow of O(e) is symmetric with respect to the direction of the oscillation and vanishes at infinity; however, in O(e 2 ) asymmetric steady streaming is induced in the main field and remains at infinity as a uniform flow in the direction of oscillation, due to the effect of the asymmetry in the motion of the circular cylinder.

A viscous two-dimensional flow of shear layers superimposed on a stagnation-point flow is investigated. This situation allows an exact solution of unsteady Navier-Stokes equation of an incompressible fluid in free space for a general initial condition. The solution is exemplified for several sorts of initial condition. One of them represents a flow in a balance between viscous diffusion and convective confinement of vorticity in the final asymptotic state. Another shows a flow field of collision of two shear layers of opposite senses, which is forced to come into contact by the imposed flow, and this collision results in `pair annihilation' of the vortical layers. The decay of the vortex strength of the layer shows a similarlity behaviour for different Reynolds numbers. A comment is given about a possible dissipation mechanism in free flows.

The two-dimensional unsteady flow of an incompressible viscous fluid around a circular cylinder, which is started impulsively from rest, is considered. Numerical solutions of unsteady boundary-layer equation are sought by employing the local non-similarity solution method presented by Sparrow, Quack and Boerner (Amer. Inst. Aero. Astronaut. J. 8 (1970) 1936), together with the difference-differential method. Numerical solutions are given for the velocity profile, wall shear and the time when the separation of the boundary layer occurs; the development of the boundary layer and its separation are discussed.

An analytic solution of non-axisymmetric flow in a rotating cylinder with a sloping bottom and a flat top, which is rotating slightly farster than the other walls, is obtained for compressible fluid, and is compared with the imcompressible counterpart considered by Pedlosky and Greenspan. In the incompressible fluid the flow field is z -independent due to Taylor-Proudman theorem, and the phenomenon so called westward intensification is observed. In the compressible case, on the other hand, horizontal flow lines are circular, while non-axisymmetric weak z -motion is induced by the bottom slope.

The axisymmetric steady flow of an infinite or cylindrically bounded, inviscid, compressible fluid is investigated assuming that the magnetic Reynolds number and the interaction parameter are small. For the unbounded flow the radial velocity alone is evaluated with reference to the potential theory. For the channel flow, the solutions are obtained using the finite Hankel transform and they consist of two parts, namely, the convolution integral term and the magnetic field integral term. In the subsonic case, the first term is negligible because of its exponential damping so that the flow field is explicitly determined by the magnetic field applied. In the supersonic case, the first term is rather dominant and periodic so that the flow field remains oscillatory to infinity downstream. This oscillatory behavior results from the existence of discontinuity (Mach cones or lines) continuing by reflection off the wall of the cylinder.

When there is a linear shear flow along a plane wall with a cylindrical depression, flow penetrating into the cavity separates from the wall, if the ratio of the depth of the cavity to the width of its mouth exceeds a critical value which is about 0.32. The pressure becomes infinite at the sharp edges of the cavity, while for the flow over a projection the pressure is a continuous function which on the boundary has two extremum values symmetrically about the cylindrical portion. As another example of separation in Stokes flows, the flow generated by the rotation of a circular cylinder which is eccentrically encircled with a larger cylinder or placed near a plane is considered.

This paper deals theoretically with the two-dimensional flow induced by a circular cylinder translationally oscillating in an unbounded visous fluid otherwise at rest. The oscillation is composed of a harmonic oscillation with a frequency normal to the axis of the cylinder and another harmonic one with double frequency perpendicular to the former and the axis. Steady secondary flow is studied in detail by the method of successive approximations. It is shown that a uniform flow is induced at infinity. The direction of the uniform flow can change according as a Reynolds number. This is qualitatively similar to the result of the available experimental result on the flow around a circular cylinder which performs an eccentric relational oscillation. The direction depends ends also on the type of the motion of the cylinder, which traces, in general, a Lissajous's figure.

Numerical soilutons for the steady flow of a viscous fluid driven by a slowly rotating disk in a cylinder of finite aspect ratio are presented. Based on the comprehensive flow data acquired, the torque exerted on the disk is computed. The radial and axial structures of both the azimuthal and meridional flow details are exhibited. The effect of small, but non-zero, Reynolds number is discussed. The torque increases rapidly as the disk-cylinder radius ratio approaches unity because of the concentrated flow gradients near the tip of the disk. As the cylinder aspect ratio increases, the torque decreases. The flow can be approximated by the infinite cylinder model if the aspect ratio is greater than about 2.0. The theoretical torque estimate by using the infinite cylinder configuration is shown to be in reasonable agreement with the numerical results for large aspect ratios.

Finite-difference solutions of the Navier-Stokes equations are given in stream function-vorticity formulation for a series of Reynolds numbers and for varied positions of the cylinder in the channel. For flows in a doubly connected region, the value of the stream function on the body cannot always be given in advance. A new iteration scheme to solve such a problem is proposed for getting the proper solution by imposing a condition that the pressure is single-valued. The force and the moment of force acting on the cylinder are computed and their dependence on the Reynolds number as well as on the position of the cylinder is studied. The numerical errors caused by the existence of corners on the boundary are checked for Stokes flow by comparison with exact solutions, and their dependence on the mesh length is shown numerically.

Gaseous flow in a rapidly rotating short cylinder is studied within the framework of a linear treatment. Since the aspect ratio of the cylinder is assumed to be of the order of E 1/2 , where E is the Ekman number, a viscous force is dominant throughout the flow region. Meridional flow is induced either by a temperature difference of the top and bottom end plates or by a differential rotation of the cylinder walls.The flow region is divided into an Ekman-layer like region and a corner region with the dimension of E 1/2 × E 1/2 . The equation governing the latter corner region is ( Δ 3 + a 4 ∂ 2 /∂ 2 )ψ=0, which is solved by an expansion technique.

Possibility of occurrence of secondary flow in boundary layer around a circular cylinder in rotatory oscillation is studied by means of linear theory. The secondary flow can occur above a certain critical Reynolds number, and the condition of occurrence of the secondary flow is shown by curves in the Reynolds and wave number diagrams with a nondimensional thickness of the boundary layer as a parameter. Satisfactory agreement of the results with experimental ones by Taneda et al. is obtained.

Theoretical test for the accuracy of the M 2 -expansion solutions to subsonic flow problems has been made by taking the simplest and basic case of the flow past a circular cylinder without circulation. Imai's equation of two-dimensional compressible flow is used for this purpose and the velocity q is assumed to bear the characteristics of the M 2 -expansion solutions of this particular flow problem. It is shown then that, under a certain condition, all q 's satisfying the above equation approximately do not differ much as to their magnitudes. This property which is called the `appropriateness' of the equation is utilized to check the accuracy of the M 2 -expansion solutions, leading to an error estimate in terms of the free stream Mach number M and the adiabatic index γ.

Separated flow past a two-dimensional smooth curved bluff body can be solved by the use of a free-streamline theory so long as the base pressure and the separation position are prescribed as empirical parameters. The purpose of the present paper is to eliminate one of these parameters by determining the separation position as a function of the base pressure with the help of the boundary layer theory. Calculations were performed for cases of a subcritical and a transcritical flow past a circular cylinder. Although some discrepancy is seen to exist between the theory and the experimental data, it is shown that solutions of Kawaguti, and of Bluston and Paulson can be included in the present theory as special examples. Furthermore, a conclusion of the latter paper is criticised from the viewpoint of the present theory.

A steady axisymmetric source-sink flow of an incompressible viscous fluid in a rotating circular cylinder is studied. The source-sink distribution is idealized as the line distribution along the concentric circles on the top and the bottom. The effect of the discontinuity of the angular velocity across the same circles is also examined. The method of analysis is the boundary layer analysis of the linearized basic equations. The attention is concentrated on the details of the flow structure in the shear layers, and several numberical results are shown.

The flow around a circular cylinder performing a rotatory oscillation about its axis in a uniform flow has been investigated experimentally by means of flow visualization methods at Reynolds numbers between 30 and 300 and Strouhal numbers between 0 and 55. It has been found that the dead water region is reduced as the frequency of the oscillation is increased, and that the critical Strouhal number at which the dead water region vanishes is approximately proportional to the reciprocal of the angular amplitude of oscillation, but is independent of the Reynolds number. It has also been found that when the Strouhal number is much higher than the critical Strouhal number the flow past the cylinder is strongly accelerated.

The velocity field around a solid sphere, performing harmonic oscillations in an unbounded viscous fluid otherwise at rest, is exactly determined to the second order for small values of the Reynolds number. The streamline pattern of the steady streaming and the resultant instantaneous streamlines of the complete flow are shown. The drag exerted on the surface of the sphere is also evaluated.

An analytical investigation of the second order thermal boundary layer in an incompressible plane flow past a curved surface with uniform heat flux is carried out to study the effects of body curvature and free-stream vorticity on heat transfer. The application of the present theory to heat transfer problems is shown by considering the symmetric uniform shear flow past a flat plate and the potential flow past a circular cylinder.

The steady flow around a circular cylinder is numerically investigated. The solutions are obtained at Reynolds numbers of 40, 100, 200, 400, 500, 700 and 1000 by an implicit finite-difference approximation. The similarity of the shape of standing vortices is confirmed at Reynolds numbers from 20 to 100, and the similarity of the velocity distribution along the flow axis in the standing vortices is confirmed at Reynolds numbers from 15 to 100.

The steady flow generated by an oscillating triangular cylinder immersed in a viscous, incompressible fluid is investigated experimentally. When an equilateral triangular cylinder is oscillated perpendicularly to one of its sides, the flow pattern in the vicinity of the cylinder is symmetric with respect to the axis of oscillation but asymmetric with respect to the axis perpendicular to it. The flow patterns are classified into two types on the basis of the values of two non-dimensional parameters, \(\textbf{\itshape r}_{0}\sqrt{\omega/\nu}\) and s / r 0 , where r 0 is half the side length of the cylinder section, ω the angular frequency, ν the kinematic viscosity and s the oscillation amplitude. On the other hand, when the triangular cylinder is oscillated parallel to one of its sides, the flow pattern is asymmetric with respect to the axis of oscillation but symmetric with respect to the axis perpendicular to it. The flow patterns are classified into three types.

The steady streaming induced by a circular cylinder performing a rotatory oscillation about an eccentric axis in a fluid which is otherwise at rest is investigated by means of a flow visualization method. The solution of glycerine in water is used as the working fluid. The aluminum dust method is used for flow visualization. It is found that when δ 0.2 the inner layer spreads over the whole flow field, where δ is the non-dimensional thickness of the Stokes layer. It is also found that when δ

A steady two-dimensional MHD flow of an incompressible, inviscid, electrically conducting fluid past an elliptic cylinder (of small eccentricity) is considered for the aligned field case when the magnetic diffusivity (η m ) is large. Attempt is made to find the interaction of the field and the motion, and also the drag force experienced by the cylinder by retaining the non-linear terms. It is found, to a first approximation, that the drag force is independent of the fluid density; the drag force for a circular cylinder is also deduced.

Plane slow motion of an incompressible, electrically conducting and viscous fluid past a circular cylinder placed midway between the two parallel walls in a channel in the presence of a uniform magnetic field parallel to the walls is considered; the consideration is restricted to the case when the magnetic diffusivity is large (i.e., when electrical conductivity is small) and investigation is carried out by considering outer and inner solutions on the basis of Stokes approximation. The drag force on the cylinder is also derived. It is found that the walls have significant effect on the drag force.

An experimental study on the flow of an eccentric Couette-Taylor system is presented. In the system, the position of a fixed outer cylinder can be adjusted in relation to a rotating inner cylinder. The following results were obtained: when the displacement parameter was 0 0.3 the flow pattern of a no-axis-symmetric Taylor vortex flow was different from that of a concentric system. For e >0.6, there was no steady vortex flow. A pre-vortex flow changed drastically at about e =0.3, and a confined eddy occured for e >0.3.

Oseen flow past a circular cylinder is studied for the whole range of the Reynolds number R on the basis of an integral equation. Approximate analytical solutions are obtained for both small and large R . A non-trivial solution of the associated homogeneous equation is noticed and discussed. The integral equation is also solved numerically for R =0.2, 2, 20, 200, 2000 and ∞. The value of the drag coefficient is calculated for each case.

This paper treats thermophoresis of a solid circular cylinder in a rarefied gas. The analysis is carried out on the basis of the linearized B-G-K equation. The simultaneous integral equations for the density, flow velocity and temperature are derived from the equation. These are solved numerically over a wide range of the Knudsen number. It is shown that the induced flow around the cylinder does not vanish at infinity but has a constant non-zero value. The cylinder experiences no force in this uniform flow. The variation of the thermophoretic velocity versus the Knudsen number is calculated for any value of the thermal conductivity of the cylinder. The distributions of the flow velocity as well as the temperature are also shown.

An exact solution of the Navier-Stokes equations is presented for an incompressible circulatory flow about an infinite circular cylinder with uniform suction or injection. An asymptotic evaluation of the solution for large time shows that, starting from a typical initial state, the flow ultimately becomes irrotational for R ≦1; for 1 2, the initial vorticity distribution changes with the circulation at infinity remaining constant: Here R = V a /ν, with V , a and ν being the suction velocity, radius of the cylinder and the kinematic viscosity, respectively. The torque on the cylinder has been evaluated for both small and large times.

The flow around a flexible plate performing a swimming motion was investigated using hot-wire anemometers, pressure transistors and flow visualization techniques. It was found that the swimming motion accelerates the flow near the surface in the wave direction, reduces the boundary layer thickness, and suppresses the turbulent fluctuation.

Two-dimensional slow viscous flow due to an arbitrary translation of a cylinder on the center-line of a wedge is studied on the basis of the Stokes approximation. Drag on the cylinder and the velocity field are calculated up to the first-order effect of the presence of boundaries by the use of a two-dimensional method of reflections.

Stratford's theory for the prediction of separation of the turbulent boundary layer is modified so that it may be applicable to the critical flow, in which occurs a separation bubble followed by a turbulent reattachment. Calculations are based on the assumptions that (i) transition occurs at the laminar separation position which is predicted by the Stratford-Curle formula, and (ii) the velocity profile at the transition point is that of a turbulent zero-pressure-gradient flow which is taken as the “comparison” profile. The accuracy of the method is tested for experimental pressure distributions around a circular and an elliptical cylinder. It is seen that the theoretical positions of transition are in remarkably good agreement with the experiments, whereas those of turbulent separation (for critical flows) will tend somewhat to precede the experimental ones. An application to the free-streamline flow past a circular cylinder is also presented.

Steady, two-dimensional stagnation point flow of a highly conducting, inviscid, incompressible fluid with an aligned magnetic field is considered. An approximate solution is obtained by the use of the Karman-Pohlhausen integral method. The solution is compared with the exact ones.

The shear flow of a viscous fluid around a circular cylinder has been studied to Oseen's approximation and has been compared with the author's previous results of Stokes' approximation. The formulae for the drag and the lift have been obtained neglecting the terms of orders O ( R e ), O (α R e -1 ), O (α 2 ), and it has been shown that the drag is not affected by the presence of the shear and the direction of the lift is the same as that for the inviscid shear now within the order of our approximation.

Two-dimensional flow of a viscous incompressible fluid past a rotationally-oscillating circular cylinder is investigated on the basis of Oseen's equations of motion. The cylinder is assumed to oscillate about its axis periodically. It is found that the solution is composed of a steady part and a periodic part, the latter of which is due to the periodic motion of the cylinder. The forces acting on the cylinder are calculated by two different methods; in one method the law of momentum conservation is applied to the flow region surrounded by a large control surface, while in the other the viscous stresses on the surface of the cylinder are integrated. The results by these two methods do not agree with each other, however, and there appears to exist something like Garstang's paradox. The flow field is calculated numerically at several instants of time in one period, and the flow patterns are shown graphically.

For a single spherical drop suspended in an unbounded viscous fluid streaming with arbitrary velocity distribution, a general method is presented to calculate the external and internal flow fields of the drop and to evaluate deviation of the drop interface from sphericity. Based upon Lamb's solution in terms of solid spherical harmonics for Stokes flow, boundary conditions to be satisfied at the interface can be replaced by some simple relations between the solid harmonics, which, in conjunction with Laplace's condition for surface tension, give an expression for the drop deformation without detailed knowledge of the flow fields. As examples, both two-dimensional and axially symmetric Poiseuille flows are considered.

This paper deals with the steady slow motion of a circular cylinder in a semi-infinite viscous and electrically conducting liquid parallel to its bounding plane wall in the presence of a uniform magnetic field, using Stokes approximation. Detailed calculation is carried out for the flow produced by the cylinder in two cases: one in a parallel magnetic field and the other in a transverse magnetic field. The expansion formulae for the drag per unit span of the cylinder are obtained in terms of the Hartmann number in each case.

The steady slow motion of a viscous fluid past two parallel circular cylinders C 1 , C 2 of arbitrary radii is discussed on the basis of Oseen's linearized equations of motion, confining ourselves to the case when their axes are perpendicular to a uniform flow. The forces acting on the cylinder C 1 are calculated to the first approximation. By performing detailed numerical calculations, a discussion is made on the manner in which the forces acting on the cylinder C 1 is modified by the presence of the other cylinder C 2 . It is thus found that, as the radius of the cylinder C 2 increases, the drag on the cylinder C 1 decreases, while the lift increases. It is also found that the cylinder C 1 experiences smaller drag when the two cylinders are in tandem orientation than when they are placed side by side in a uniform stream. Especially, when the cylinder C 1 is situated behind the cylinder C 2 , it experiences smaller drag than when it is situated in front of C 2 .

This paper deals with the two-dimensional flow of a viscous, incompressible and electrically conducting fluid past a cylinder in a uniform magnetic field, using Stokes approximation. Detailed calculation is carried out for the flow past a circular cylinder in two cases: one in a parallel magnetic field and the other in a transverse magnetic field. The expansion formulae for the drag per unit span of the cylinder are obtained in terms of the Hartmann number in each case.

The steady slow motion of a circular cylinder in a viscous liquid bounded by two parallel plane walls is discussed on the basis of Oseen's linearized equations of motion, confining ourselves to the case when the cylinder is moving midway between the bounding walls. The drag experienced by the cylinder is then calculated to Lamb's approximation. It is shown that at sufficiently small Reynolds numbers, the drag on the cylinder is independent of the Reynolds number, namely, it is of the so-called Stokes type, as was shown by White's experiments on wires falling in viscous liquids between two vertical plates. When the ratio of the distance between the walls to the diameter of the cylinder is greater than 20, our theoretical results coincide fairly well with White's experiments, allowing for some experimental errors.

The asymptotic behaviour of the viscous flow past an obstacle for vanishingly small viscosity is discussed on the basis of Prandtl's boundary layer theory together with Kirchhoff's dead water theory. If the pressure distribution around the obstacle is such that the separation of the boundary layer on the obstacle takes place, then the flow pattern should involve free stream-lines extending from the separation points. From such considerations it is concluded that the asymptotic flow pattern around a smooth obstacle is a discontinuous flow with dead water region of the Kirchhoff type such that the free stream-lines leave the obstacle with a finite curvature equal to that of the obstacle at the separation point.