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  • 1
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    Double-diffusive convection with large variable gradients (1982)
    Cambridge University Press
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    Publication Date: 1982-12-01
    Description: The onset of double-diffusive convection is discussed for a layer of fluid in which the vertical salinity gradient varies with depth and for which the thermal and saline Rayleigh numbers R and Rs are large. These conditions are similar to those that exist in a solar pond prior to the onset of any instability. It is shown that when convection occurs it takes the form of an overstable mode and is essentially confined to a narrow region of vertical extent Rs- 1/4 x depth of the fluid layer, centred at the critical depth where the salt gradient is smallest. The leading terms in asymptotic expansions of the ratio R/Rs, the frequency of oscillation p and the horizontal wavenumber a are determined for Rs 1. The results predicted by the theory are shown to be in good agreement with numerical results and with observations of solar ponds. © 1982, Cambridge University Press. All rights reserved.
    Print ISSN: 0022-1120
    Electronic ISSN: 1469-7645
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
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  • 2
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    The axial flow of a Bingham plastic in a narrow eccentric annulus (1991)
    Cambridge University Press
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    Publication Date: 1991-01-01
    Description: This paper describes analytical and numerical solutions for the flow of a Bingham plastic in an eccentric annulus. Analytical solutions are obtained by expanding in powers of δ, the ratio of the difference in radii of the bounding cylinders to their mean. The solution over most of the annulus is similar to that in a slot of uniform width, containing a central plug-like region over which the velocity is independent of the radial variable. However, unlike the uniform-slot solution, the velocity in the plug varies around the annulus and the stress exceeds the yield stress. This simple structure is supplemented by true plugs (over which the velocity is constant and the stress is below the yield stress) at the widest and, in some cases, the narrowest parts of the annulus. A simple criterion is given for conditions under which the fluid ceases to flow on the narrow side and bounds are obtained for the extent of the motionless region and for the true plugs. The predictions of the theory have been compared to numerical results over a wide range of eccentricities, radius ratios, fluid properties and flow parameters. Good quantitative agreement has been reached for radius ratios in excess of about 0.7. In particular the extent and location of pseudo-plugs and true plugs are confirmed. © 1991, Cambridge University Press. All rights reserved.
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  • 3
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    Nonlinear processes in rotating fluids: a report on Euromech 56 (1975)
    Cambridge University Press
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    Publication Date: 1975-09-23
    Description: A European Mechanics Colloquium (Euromech 56) on ‘Nonlinear processes in rotating fluids’ was held at University College, London from 14–18 April 1975. It was attended by 77 mathematicians, engineers and physicists from 14 countries, and 33 papers were presented and discussed. The topics included motions in homogeneous and stratified rotating fluids, with applications to the atmosphere, the oceans, the earth's core and centrifuges. Brief summaries of the papers presented are contained in this report. © 1975, Cambridge University Press. All rights reserved.
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  • 4
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    The effect of a shear flow on convection in a layer heated non-uniformly from below (1985)
    Cambridge University Press
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    Publication Date: 1985-05-01
    Description: In an earlier paper (Walton 1982) we showed that, when a fluid layer is heated non-uniformly from below in such a way that the vertical temperature difference maintained across the layer is a slowly varying monotonic function of a horizontal coordinate x, then convection occurs for x 〉 xc where xcis the point where the local Rayleigh number is equal to the critical value for a uniformly heated layer. Furthermore, the amplitude of the convection increases smoothly from exponentially small values for x ≪ xcand asymptotes to a value given by Stuart—Watson theory for x ≫ xc. At the present time no solutions of this kind are available for a class of problems in which the onset of instability is heavily influenced by a shear flow (e.g. Gortler vortices in a boundary layer on a curved wall, convection in a heated Blasius boundary layer). In a first step to bridge the gap between these problems and in order to elucidate the difficulties associated with the presence of a shear flow, we investigate the effect of a (weak) shear flow on our earlier convection problem. We show that the onset of convection is delayed and that it appears more suddenly, but still smoothly. The role of horizontal diffusion is shown to be of paramount importance in enabling a solution of this kind to be found, and the implications of these results for instabilities in higher-speed shear flows are discussed. © 1985, Cambridge University Press. All rights reserved.
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  • 5
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    The viscous nonlinear symmetric baroclinic instability of a zonal shear flow (1975)
    Cambridge University Press
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    Publication Date: 1975-04-29
    Description: The stability of a baroclinic zonal current to symmetric perturbations on a meridionally unbounded f-plane is considered. The lower boundary is at rest but the upper one moves with a constant velocity in keeping with the velocity of the zonal current. Following Stone (1966) a horizontal length scale O(Ro) is taken, where Ro is the Rossby number, with the Richardson number Ri = O(1). Instability sets in when the wavelength is O(E1/3), where E is the Ekman number based on the distance between the rigid horizontal boundaries, which corresponds to Stone's inviscid value zero, and to McIntyre's (1970) value infinity on a length scale O(E½). A nonlinear analysis about the point of onset of instability yields the result that for the monotonic mode zonal momentum is convected polewards. The possible implications of this result for the dynamics of Jupiter's atmosphere are discussed. © 1975, Cambridge University Press. All rights reserved.
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  • 6
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    Bénard convection in a finite box: secondary and imperfect bifurcations (1979)
    Cambridge University Press
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    Publication Date: 1979-01-29
    Description: In the solution given by Hall & Walton (1977) for Bénard convection in a two-dimensional box with slightly imperfectly insulated side walls it was shown that there were certain critical values 2Lc of the length 2L of the box at which two modes became unstable simultaneously. In this paper we show that in the neighbourhood of Lc a secondary transition takes place which may be a bifurcation or a smooth transition depending on the boundary conditions. © 1979, Cambridge University Press. All rights reserved.
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  • 7
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    Second-order effects in free convection (1974)
    Cambridge University Press
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    Publication Date: 1974-02-27
    Description: The equations of conservation of momentum, energy and mass together with the equations of state are examined for free convection from a vertical paraboloid. A transformation due to Saville & Churchill is applied to the first- and second-order boundary-layer equations, which are then solved using series about the stagnation point, using asymptotic series far up the body and in between by a method due to Merk. The second-order outer inviscid flow is given in terms of infinite integrals as a solution of Laplace's equation in paraboloidal co-ordinates. Eight second-order effects are distinguished, depending on longitudinal and transverse curvatures, the displacement flow, heat flux into the boundary layer and the variation of density, viscosity, thermometric conductivity and the coefficient of expansion with temperature. Expressions for the skin friction, heat-transfer coefficient and various flux thicknesses are obtained and a comparison of the second-order effects is made. © 1974, Cambridge University Press. All rights reserved.
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  • 8
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    The drag on a sphere moving axially in a long rotating container (1979)
    Cambridge University Press
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    Publication Date: 1979-02-01
    Description: A container of viscous incompressible liquid is bounded by rigid parallel planes and rotates steadily about an axis normal to these planes. A rigid sphere moves steadily parallel to the rotation axis and the Rossby and Ekman numbers characterizing the motion are both small. The drag on the sphere is calculated in the case when the length of the Taylor column is comparable to the axial dimension of the container. Viscous effects are allowed for in the boundary of the Taylor column, but the Ekman layers on the sphere and on the bounding planes are shown not to affect the drag to leading order. The determination of the drag involves solving dual integral equations. This is done numerically and, for the limiting cases of long and short containers, analytically. The interaction of the Taylor column and the ends of the container leads to an increase in the drag over its value in an unbounded fluid, but the increase is smaller than that measured by Maxworthy (1970). © 1979, Cambridge University Press. All rights reserved.
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  • 9
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    The linear stability of the flow in a narrow spherical annulus (1978)
    Cambridge University Press
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    Publication Date: 1978-06-28
    Description: The flow of a fluid in a narrow spherical annulus is considered. When the outer sphere remains fixed and the angular velocity of the inner one is increased beyond a critical value an instability resembling Taylor vortices appears. This instability is investigated by expanding the solution in powers of the small parameter ∊, the ratio of the gap thickness to the radius, and assuming that two length scales, O(1) and O(∊), are important in the latitudinal direction. The perturbation then takes the form of cells which are of roughly square cross-section, at least near the equator, but whose amplitude decays rapidly with latitude; it is also subject to a slow spatial modulation. The critical value of the Taylor number at which the instability first appears is shown to be that for infinite concentric cylinders plus a correction O(∊) due to secondary motions and a correction not greater than O(∊) due to the domain being bounded. © 1978, Cambridge University Press. All rights reserved.
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  • 10
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    The onset of cellular convection in a shallow two-dimensional container of fluid heated non-uniformly from below (1983)
    Cambridge University Press
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    Publication Date: 1983-06-01
    Description: A theoretical study is made of the onset of cellular convection in a shallow two-dimensional container of fluid when the temperature difference between the horizontal boundaries is a monotonic function of horizontal distance. The typical lengthscale of the horizontal variation in the temperature difference is taken to be of the same order of magnitude as the length of the container, and both are much longer than the depth of the container. The resulting flow may be regarded as consisting of two parts, a steady base flow and a disturbed flow. It is found that a weak disturbance taking the form of transverse rolls is first set up near the endwalls, but as the temperature difference between the horizontal boundaries is uniformly increased, an instability in the form of longitudinal rolls takes place near the hotter end of the container. This description is in good qualitative agreement with experiment. © 1983, Cambridge University Press. All rights reserved.
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