Plane Rigid Wall Research Articles

Superposition of longitudinal and plane flows of a non-Newtonian fluid between eccentric cylinders

Results obtained in previous papers on the flow of an incompressible non-Newtonian fluid, satisfying the second-order Rivlin—Ericksen constitutive equation, in the annular region between infinite circular cylinders with parallel axes, which results from the longitudinal steady motion of the inner cylinder, or the existence of a longitudinal pressure gradient, or the steady rotation of the cylinders about their axes, are used to calculate the effect on the forces exerted by the fluid on the cylinders when certain of these flows are superposed. As an asymptotic case, the forces exerted by the fluid on an infinite cylinder immersed in a half-space of the fluid bounder by a plane rigid wall are obtained when the cylinder moves parallel to the wall with a constant velocity inclined at an arbitrary angle to its length.

Micropolar fluids as models for suspensions of rigid spheres

The question of the approppropriate boundary conditions for a micropolar fluid is considered with respect to the representation of a suspension of solid particles in a Newtonian fluid. The case of a dilute suspension of rigid force-free spheres in slow viscous Couette flow parallel to a rigid plane wall is treated in an exact manner by classical continuum mechanics. The analogous micropolar fluid can be adjusted to reproduce particle rotations with good approximation, but the perturbation of the mean velocity of the micropolar fluid does not coincide with the perturbation of the mean velocity of the suspension.

Memory effect in a turbulent boundary-layer flow due to a relatively strong axial variation of the mean-velocity gradient

Measurements have been made of the distributions of mean velocity, turbulence intensities and turbulence shear-stress in a turbulent boundary-layer downstream of a hemi-spherical cap attached onto the plane rigid wall. The eddy-viscosity, when computed in the classical way according to Boussinesq's concept from the lateral gradient of the mean velocity and the turbulence shear-stress, showed a very strong non-uniform lateral distribution, also across the outer region of the boundary-layer. More over , the non-dimensional values of the eddy viscosity, using the wall-friction velocity and the boundary-layer thickness as the velocity scale and length scale respectively, were higher than those for the boundary-layer when not disturbed by the wake of the spherical cap. However, when account is taken of an axial memory effect of the streamwise variation of the lateral gradient of the mean-velocity, the values of the non-dimensional eddy viscosity are close to those for the undisturbed boundary-layer.

Wall-Pinning Model of Magnetic Hysteresis

A simple model for the motion of a domain wall in most magnetic materials is presented. The real flexible wall is represented by an equivalent rigid plane wall. The motion of the plane is impeded by springs which are attached to the wall and to defects in the material in the wake of the plane. A spring breaks when the force it exerts on the wall reaches a value characteristic of the defect, and the energy stored in it is lost to the motion. The model is similar to one developed earlier but is based on a more plausible view of the interaction of a wall with a defect. In common with the earlier model, it describes both reversible and irreversible flux changes; it predicts a small-signal hysteresis loop which is similar to, but significantly different from, the Rayleigh loop; it explains the frictional nature of magnetic hysteresis without invoking any ad hoc energy-loss mechanisms. It is sufficiently simple to be applied to complex problems. It agrees with the results of exact dynamic calculations to good accuracy and is exact when applied to quasistatic problems. The model is derived by considering the motion of an anisotropic flexible wall pinned by a periodic array of defects. The springs of the model represent the distortions of the wall in the vicinity of the defects. It is shown that the force f exerted by the defect on the wall and the energy W per defect stored in wall distortions are related approximately to the average displacement z̄ of the wall from the defect by the equations f(t)=kz̄(t), W(t)=kz̄2(t)/2,where k is a spring constant associated with wall deformation. These relations justify the spring model. Finally, a comparison is made among f, W, and z̄ calculated as functions of time for a step of applied field from both the exact equations and from the spring model. The agreement is, in all cases, better than 10% of the final values. After the wall breaks free from a defect the energy stored in distortion is dissipated in viscous damping. It does not affect the average wall position and is thus lost to the motion in agreement with the model.

The slow motion of a sphere through a viscous fluid towards a plane surface—II Small gap widths, including inertial effects

Singular perturbation techniques are employed to calculate the hydrodynamic force experienced by a sphere moving, at small Reynolds numbers, perpendicular to a solid plane wall bounding a semi-infinite viscous fluid, for the limiting case where the gap width between the sphere and plane tends to zero. Two distinct, but related, analyses of the problem are presented. In the first analysis, the exact bipolar-coordinate expression for the force given independently by M aude [1] and B renner [2] for the quasistatic Stokes flow case is expanded via a novel asymptotic procedure. The second analysis, which is somewhat more general in scope, provides a perturbation solution of the unsteady Navier—Stokes equations for a more general axisymmetric particle than a sphere, taking accounts of the finiteness of the Reynolds number. When the Reynolds number is taken into account, the force on the particle differs, according as it moves towards or away from the wall. Using the techniques developed in the first portion of this paper, formulas are derived for the Stokes couple required to maintain the symmetrical rotation of a dumbbell in an unbounded fluid, as well as the couples required to maintain the symmetrical rotation of a sphere touching a rigid plane wall and touching a planar free surface.

Particle motions in sheared suspensions XXI: Interactions of rigid spheres (theoretical)

Based on a method developed byFaxen, approximate expressions were developed for the rotational and translational velocities of a single rigid sphere near a plane rigid wall bounding Couette flow, and near another sphere of either different or the same size in Couette flow. The solutions to the latter problem provide equations for the orbits of each sphere about the other, some of which are closed. In many respects the results are similar to the corresponding two-dimensional problems for cylinders.

Measurement of the Transmission Loss of a Finite-Depth Aperture

To provide experimental verification of an approximate solution for the diffraction of sound waves by a circular aperture in a plane rigid wall of finite thickness, a series of experiments was conducted on the transmission of reverberant sound through steel pipes inserted individually into a thick wall between two reverberation chambers. The usually recommended methods of evaluating the transmission loss proved inadequate because long, narrow apertures have sharp resonant peaks. By electronically averaging and equalizing the outputs of four microphones in each chamber and using a slowly sweeping sine wave with a small super-imposed warble, it was possible to record continuous plots of the aperture transmission loss with high resolution of frequency and amplitude. Experimental accuracy of 1 dB was possible over most of the frequency range of 300–12 000 cps, diverging to 2 dB at the frequency extremes, and for apertures with sharp resonances, diverging to about 4 dB at the peaks. The accuracy was limited by reading the “average” of the recorded fluctuating sound level, which was most difficult at resonant peaks. For larger-diameter apertures, which gave data relatively uninfluenced by flanking sound, the agreement with theoretical results was within 1 dB over most of the frequency range.

Approximation to the Diffraction of Sound by a Circular Aperture in a Rigid Wall of Finite Thickness

An approximate solution for the diffraction of a plane sound wave indicent normally on a circular aperture in a plane rigid wall of finite thickness is obtained by postulating rigid, massless, infinitely thin plane pistons in each end of the aperture, whose motions simulate the movement of the air particles at these positions under acoustic excitation. Plane longitudinal waves are assumed inside the aperture. Numerical solutions obtained on an IBM-7090 computer very closely coincide over a wide range of frequencies with those of much more complicated existing exact solutions. Furthermore, extensive experiments were conducted on the transmission of reverberant sound through steel pipes of various lengths and diameters inserted individually into a heavy wall between two concrete chambers of 1750 ft3 each. Measured values of reverberant sound-transmission loss through these apertures were found to agree within experimental error with those calculated from the approximate theory. Experimental results indicated that the assumption of plane-wave propagation in the aperture was valid up to values of d/λ exceeding unity, where d is the pipe diameter and λ is the wavelength of incident sound.

Approximation to the Diffraction of Sound by a Circular Aperture in a Rigid Wall of Finite Thickness

An approximate solution for the diffraction of a plane sound wave incident normally on a circular aperture in a plane rigid wall of finite thickness is obtained by postulating rigid, massless, infinitely thin plane pistons in each end of the aperture, whose motions simulate the movement of the air particles at these positions under acoustic excitation. Plane longitudinal waves are assumed inside the aperture. Numerical solutions obtained on an IBM-7090 computer very closely coincide over a wide range of frequencies with those of much more-complicated existing exact solutions. Furthermore, extensive experiments were conducted on the transmission of reverberant sound through steel pipes of various lengths and diameters inserted individually into a heavy wall between two concrete chambers of 1750 cu ft each. Measured values of reverberant sound-transmission loss through these apertures were found to agree within experimental error with those calculated from the approximate theory. The assumption of plane-wave propagation in the aperture was found to give valid experimental results up to values of d/λ exceeding unity, where d is the pipe diameter and λ is the wave length of incident sound.

Measurement of the Transmission Loss of Finite Depth of Circular Apertures

To provide experimental verification of an approximate solution for the diffraction of sound waves by a circular aperture in a plane rigid wall of finite thickness, a series of experiments was conducted on the transmission of reverberant sound through steel pipes inserted individually into a thick wall between two reverberation chambers. Usually, recommended methods of evaluating the transmission loss proved inadequate because long, narrow apertures have sharp, resonant peaks. By electronically averaging and equalizing the outputs of four microphones in each chamber and using a slowly sweeping sine wave with a small superimposed warble, it was possible to record continuous plots of the aperature transmission loss with high resolution of frequency and amplitude. Experimental acuracy of 1 dB was possible over most of the frequency range of 300–12 000 cps, diverging to 2 dB at the frequency extremes and for apertures with sharp resonances diverging to about 4 dB at the peaks. The accuracy was limited by reading the “average” of the recorded fluctuating sound level, which was most difficult at resonant peaks. For larger-diameter apertures, which gave data relatively uninfluenced by flanking sound, the agreement with theoretical results was within 1 dB over most of the frequency range.

Effects of a flexible boundary on hydrodynamic stability

The theoretical study presented in this paper was inspired by the recent report (Krämer 1960) of experiments showing that considerable reductions in the drag of an underwater solid body were achieved by covering it with a skin of flexible material; apparently this effect was due to the boundary layer being stabilized in the presence of the skin, so that transition to a turbulent condition of flow was prevented or at least delayed. The stability problem for flow past a flexible boundary is here formulated in a general way which allows a full exploration of the possibility of a stabilizing effect without the need to assign specific properties to the flexible medium; the collective properties of possible boundaries are represented by a ‘response coefficient’ α (a sort of ‘effective compliance’) measuring the deflexion of the surface under a travelling sinusoidal distribution of pressure.A remarkably simple analytical connexion is established between the present general problem and the corresponding stability problem for the boundary layer on a rigid plane wall, and hence many details of the existing theory of hydrodynamic stability are immediately useful. However, the presence of the flexible boundary admits possible modes of instability additional to those which already exist when the boundary is rigid, and clearly every mode must be considered with regard to practical measures for stabilization—that is to say, it might be useless to inhibit one mode by a device which lets in another. What is believed to be an essentially complete interpretation of the over-all possibilities is deduced on recognizing three more or less distinct forms of instability. The first comprises waves resembling the unstable waves which can arise in the presence of a rigid boundary, but now being modified by the effects of flexibility. These waves tend to be stabilized when the boundary has a compliant response to them, which means the respective wave velocity has to be less than the velocity of free surface waves on the boundary; but it is found that the effect of internal friction in the flexible medium is actually destabilizing. The second form of instability is essentially a resonance effect and comprises waves travelling at very nearly the velocity of free surface waves. These waves can only be excited when the latter velocity falls below the free-stream velocity; they are scarcely affected by the viscosity of the fluid since the ‘wall friction layer’ is largely cancelled, so that damping due to the medium itself becomes the only stabilizing factor. The third form is akin to Kelvin–Helmholtz instability.This interpretation of the theoretical results seems to point to the essential factors in the operation of a flexible skin as a stabilizing device, and accordingly in the concluding secttion of the paper two alternative sets of criteria are proposed each of which would provide a logical basis for designing such a device. The principle of the first alternative explains the success of Gamer's invention, but the second appears equally promising and the relative advantages of the two can really be proved only by further experiment.

The reflexion of sound waves of finite amplitude by a rigid wall

A polytropic gas of adiabatic exponent 5/3 fills the half-space on one side of a rigid plane wall of infinite extent. Initially the gas is at rest and its density is proportional to x 3/2 , where x is the distance from the wall. The gas starts moving towards the wall. It is shown that, although the data are continuous, the problem has no continuous solution, that reflexion at the wall generates a shock wave. The problem is solved completely without recourse to numerical integration.

The Lift On the Symmetrical Joukowski Aerofoil in a Stream bounded by a Plane Wall

In this paper the lift on the symmetrical Joukowski aerofoil of small thickness in the presence of a plane rigid wall is obtained in the form of a power-series, with the intention of discussing the manner in which the ground-effect on the lift of an aerofoil is modified by its thickness. The calculation of the lift is carried out by extending Green's analysis. It is found that in case when the symmetrical Joukowski aerofoil of small thickness is placed in the vicinity of the wall at small angles of attack as used in practice, the wall-effect on the lift is quite similar to that in the case of a plane aerofoil. Namely, as the symmetrical Joukowski aerofoil approaches the wall, the lift first decreases and then tends to increase to values which are greater than the corresponding values for the symmetrical Joukowski aerofoil in an infinite stream, especially when the angle of attack is sufficiently small. Further it is found that for any value of the distance of the aerofoil from the wall, the lift ratio Y / Y 0 decreases as the thickness increases even for small values of the angle of attack.

Note on the Lift acting On a Circular-arc Aerofoil in a Stream bounded by a Plane Wall

Being a continuation of the preceding paper, Green's new method, somewhat extended by the present writer, is applied to the evaluation of the lift acting on a circular-arc aerofoil in the presence of an infinite plane rigid wall. The expression for the lift is obtained in the form of a power series. By performing detailed numerical calculations, a discussion is made on the manner in which the ground-effect on the lift of an aerofoil is modified by its camber. It is thus found that in accordance with the prediction made by Tomotika et al. , as the arc-aerofoil of sufficiently small camber approaches the wall, the lift first decreases and then tends to increase to values which are greater than the corresponding values for the same arc-aerofoil in an unlimited stream, especially when the angle of attack takes small values as used in practice. It is also found that the rate of increase in the lift becomes smaller as the camber of the aerofoil increases.

On the Shock Wave in the Elastic Medium

As the result of the four conservation laws i. e. those of the conservation of mass momentum, energy and the area of the section of the tube of flow, we have obtained following conclusions for the shock wave induced in the elastic medium. 1 . When the wedge moved with the supersonic speed z , the angle between the shock wave and the direction of undisturbed velocity denoted by α is \begin{aligned} \alpha=\text{Sin}^{-1}(\text{M}^{-1})+\theta \end{aligned} \begin{aligned} \text{where} \theta=\frac{3\lambda+2\mu}{\lambda+2\mu} \frac{\text{M}^{4}\alpha^{2}\beta\delta^{2}}{4C_{0}(\text{M}^{2}-1)^{2/3}} \text{and} \text{M}=z/a. \end{aligned} α, β, c 0 , λ and µ, and δ denote the speed of sound, the coefficient of the linear expansion, the specific heat, Lames constants of the elastic medium, and the half vertex angle of the wedge resp.. 2. The plane shock wave is reflected by the plane rigid wall. α and α ′ denoting the angles included between the incident or reflected waves and the wall, we have \begin{ali...

Some Observations on the Acoustic Impedance of Small Orifices

The acoustic impedance of single and multiple small circular orifices is being studied as a function of (a) particle velocity in the orifice, (b) spacing between orifices, (c) orifice dimensions, and (d) eccentricity of orifice position in an impedance tube. The observations and analyses reported here are restricted to wave lengths much greater than the hole diameter. Sivian reported an increase in orifice resistance with particle velocity, but he found no variation of reactance. Using a precision impedance tube, we confirm the increase in resistance, but we find that reactance decreases significantly with increasing particle velocity. The impedance of an orifice is influenced by nearby orifices or bounding walls. Defining interaction impedance as the increase in impedance due to the presence of another orifice or a wall, we have measured and obtained approximate formulas for interaction impedance in certain simple cases. As two orifices in an infinite plate approach each other, the reactance of each becomes more mass-like. An orifice near a rigid plane wall interacts similarly with its image in that wall. For a single orifice centered in a circular (impedance) tube there is a correction for the interaction with the tube walls. As the orifice is moved away from the center of the tube, both the resistance and the reactance vary.

Societies and Academies

LONDON. Royal Society, November 4.—Sir J. J. Thomson, president, in the chair.—Prof. H. Lamb: The vibrations of an elastic plate in contact with water. The chief problem considered is that of determining the gravest frequency of a thin elastic diaphragm filling an aperture in a plane rigid wall which is in contact on one side with an unlimited mass of water. This had an interest in connection with submarine signalling. An exact solution is not attempted, but a sufficient approximation for practical purposes is obtained by Rayleigh's method of an assumed type, which gives good results if the type be suitably chosen.—Prof. H. M. Macdonald: The transmission of electric waves around the earth's surface.—Lord Rayleigh: A reexamination of the light scattered by gases in respect of polarisation. II.: Experiments on helium and argon. The light scattered by helium and by argon is investigated. It is found in the case of helium that the total light scattered is in accordance with what would be expected from its refractivity. The polarisation in helium, contrary to what was found in 1918, is approximately complete. No intensity was detected in twenty-four hours of exposure in the component vibrating parallel to the exciting beam, and certainly this component was less than 6.5 per cent. of the other. Argon polarises much more completely than any other gas examined (with the possible exception of helium), the weak component being only 0.4 per cent, of the other.—Prof. C. F. Jenkin: Dilatation and compressibility of liquid carbonic acid. The paper describes the measurement of the dilatation and compressibility of carbonic acid between temperatures of —37° C. and + 30° C. and up to pressures of 1400 lb. per square inch. The measurements were made to supply accurate data for determining the starting point for drawing the θø and Iø diagrams and to replace the approximate results (known to be inaccurate) given in a former paper (Phil. Trans., A, vol. ccxiii., p. 76).—W. T. David: Radiation in explosions of hydrogen and air. This paper contains a record of the results of experiments on the emission of radiation during the explosion and later cooling of mixtures of hydrogen and air contained in a closed vessel. The results of experiments on the transparencv of the exploded mixtures are also recorded. Some of the main conclusions arrived at are as follows: (1) The rate of emission is approximately proportional to the fourth power of the absolute mean gas temperature. (2) The maximum rate of emission occurs at the point of maximum temperature. (3) The exploded mixtures are very transparent throughout cooling to radiation of the same kind as they emit. (4) The intrinsic radiance increases both with the lateral dimensions and with the thickness of the radiating layer of gas. (5) The 2.8 μ band of steam ceases to be emitted when the gas temperature has fallen to about 700° C.—Dr. R. E. Slade and G. I. Higson: Photochemical investigations of the photographic plate. (1) It has been shown that the silver halide grain is the photochemical unit in the photographic plate., (2) A method has been devised whereby the law of photochemical behaviour of these grains can be investigated free from the disturbing effects of development, etc., which occur in the photographic plate itself. (3) From experimental results obtained a formula has been deduced which shows the relation between the behaviour of the silver halide grains, the light intensity to which they have been exposed, and the time of exposure. (4) The results show that it is impossible for the mechanism of the process to be the absorption of light in discrete quanta, and that a given amount of light energy has a greater effect photographically when concentrated into a short range of wave-lengths than when it is distributed over a large range.—Dr. E. E. Chapman: The relationship between pressure, and temperature at the same level in the free atmosphere. The paper deals with the exceptionally high values contained in the table of coefficients of correlation between changes of pressure and changes of temperature at different levels in the atmosphere included in Geophysical Memoir 13 of the Meteorological Office, by W. H. Dines. The coefficients are computed for observations taken at random, and arranged in four groups for the year of three months each. For the layers between 4 km. and 8 km. these coefficients range from 0.75 to 0.92. It is assumed that if the observations were freed entirely from errors of measurement the coefficients would be still higher. A method is therefore worked out for correction of coefficients of correlation for probable errors of observation in measurement.—Prof. J. C. McLennan: Note on vacuum grating spectroscopy.