Boussinesq solution

An active abrasive grit is subject not only to a normal load as in static indentation, but also to a tangential load in the direction of motion [29-32]. Analogous to the Boussinesq solution, the elastic stress field due to both a normal force component P and tangential component P acting at a point on the surface has been modeled using the Michell solution [29] ... 

Boussinesq and Cerruti made use of potential theory for the solution of contact problems at the surface of an elastic half space. One of the most important results is the solution to the displacement associated with a concentrated normal point load P applied to the surface of an elastic half space. As presented in Johnson [49]... 

Implicit in all these solutions is the fact that, when two spherical indentors are made to approach one another, the resulting deformed surface is also spherical and is intermediate in curvature between the shape of the two surfaces. Hertz [27] recognized this concept and used it in the development of his theory, yet the concept is a natural consequence of the superposition method based on Boussinesq and Cerutti s formalisms for integration of points loads. A corollary to this concept is that the displacements are additive so that the compliances can be added for materials of differing elastic properties producing the following expressions common to many solutions... 

The velocities and other solution variables are now represented by Reynolds-averaged values, and the effects of turbulence are represented by the Reynolds stresses, (—pu pTl) that are modeled by the Boussinesq hypothesis ... 

Boussinesq (B4) proposed that the lack of internal circulation in bubbles and drops is due to an interfacial monolayer which acts as a viscous membrane. A constitutive equation involving two parameters, surface shear viscosity and surface dilational viscosity, in addition to surface tension, was proposed for the interface. This model, commonly called the Newtonian surface fluid model (W2), has been extended by Scriven (S3). Boussinesq obtained an exact solution to the creeping flow equations, analogous to the Hadamard-Rybczinski result but with surface viscosity included. The resulting terminal velocity is... 

In the first case (Figure 8a), the side walls are adiabatic, and the reactor height (2 cm) is low enough to make natural convection unimportant. The fluid-particle trajectories are not perturbed, except for the gas expansion at the beginning of the reactor that is caused by the thermal expansion of the cold gas upon approaching the hot susceptor. On the basis of the mean temperature, the effective Rayleigh number, Rat, is 596, which is less than the Rayleigh number of 1844 necessary for the existence of a two-dimensional, stable, steady-state solution with flow in the transverse direction that was computed for equivalent Boussinesq conditions (188). 

Pure pressure flow was first formulated and solved by Joseph Boussinesq in 1868, and combined pressure and drag flow in 1922 by Rowell and Finlayson (19) in the first mathematical model of screw-type viscous pumps. The detailed solution by the method of separation of variables is given elsewhere (17c), and the resulting velocity profile is given by... 

Point Load—The general solution of the pressure-distribution in a medium, generated by a point load applied to a packing of unlimited. depth and extent, is due to Boussinesq (1876, 1885)... 

The boundary layer equations can, as previously discussed, only be applied to flows in which the Reynolds number is relatively large and in which there is no significant areas of reversed flow. This, in particular, severely limits the applicability of these equations in situations involving opposing flow. When these conditions are not satisfied, the solution must be obtained using the full governing equations. For example, if the flow can be assumed to be two-dimensional and if the Boussinesq approximations are applicable, the equations governing the flow are Eqs. (9.5) to (9.7). If the x-axis is vertical, these equations become ... 

The numerical convection model that is used to illustrate the visualization and quantification of mixing (Figures 1 -10) is based on the solution of the equations governing convection in the Earth s mantle, assuming that the mantle can be described as an anelastic and weakly compressible fluid at infinite Prandtl number. Under the extended Boussinesq approximation, we can write the equation of motion as... 

Contact problems have their origins in the works of Hertz (1881) and Boussinesq (1885) on elastic materials. Indentation problems are an important subset of contact problems (17,18). The assessment of mechanical properties of materials by means of indentation experiments is an important issue in polymer physics. One of the simplest pieces of equipment used in the experiments is the scleroscope, in which a rigid metallic ball indents the surface of the material. To gain some insight into this problem, we consider the simple case of a flat circular cylindrical indentor, which presents a relatively simple solution. This problem is also interesting from the point of view of soil mechanics, particularly in the theory of the safety of foundations. In fact, the impacting cylinder can be considered to represent a circular pillar and the viscoelastic medium the solid upon which it rests. 

Circulation models are based on the equations of motion of the geophysical fluid dynamics and on the thermodynamics of seawater. The model area is divided into finite size grid cells. The state of the ocean is described by the velocity, temperature, and salinity in each grid cell, and its time evolution can be computed from the three-dimensional model equations. To reduce the computational demands, the model ocean is usually incompressible and the vertical acceleration is neglected, the latter assumption is known as hydrostatic approximation. This removes sound waves in the ocean from the model solution. In the horizontal equations, the Boussinesq approximation is applied and small density changes are ignored except in the horizontal pressure gradient terms. This implies that such models conserve... 

Even with these simplifications, however, it is rarely possible to obtain analytic solutions for fluid mechanics or heat transfer problems. The Navier Stokes equation for an isothermal fluid is still nonlinear, as can be seen by examination of either (2 89) or (2 91). The Bousi-nesq equations involve a coupling between u and 6, introducing additional nonlinearities. It will be noted, however, that, provided the density can be taken as constant in the body-force term (thus neglecting any natural convection), the fluid mechanics problem is decoupled from the thermal problem in the sense that the equations of motion, (2 89) or (2-91), and continuity, (2-20), do not involve the temperature 0. The thermal energy equation, (2-93), is actually a linear equation in the unknown 6, once the Boussinesq approximation has been introduced. In that case, the only nonlinear term is dissipation, but this involves the product E E and can be treated simply as a source term that will be known once Eqs. (2-89) or (2 91) and (2 20) have been solved to determine the velocity. In spite of being linear, however, the velocity u appears as a coefficient (in the convective derivative term). Even when the form of u is known (either exactly or approximately), it is normally quite a complicated function, and this makes it extremely difficult to obtain analytic solutions for 0 even though the governing equation is linear. 

Of course, (12-164) is still exact, and the system of equations (12-164), (12-160), and (12-161) is no easier to solve than the original system of equations. To produce a tractable problem for analytic solution, it is necessary to introduce the so-called Boussinesq approximation, which has been used for many of the existing analyses of natural and mixed convection problems. The essence of this approximation is the assumption that the temperature variations in the fluid are small enough that the material properties p, p, k, and Cp can be approximated by their values at the ambient temperature Tq, except in the body-force term in (12-164), where the approximation p = po would mean that the fluid remains motionless. 

Equations (12—168)—(12 170) are known as the Boussinesq equations of motion and will form the basis for the natural convection stability analyses in this chapter. In fact, the Boussinesq approximation has been used in much of the published theoretical work on natural convection flows. Although one should expect quantitative deviations from the Boussinesq predictions for systems in which the temperature differences are large (greater than 10°C-20°C), it is likely that the Boussinesq equations remain qualitatively useful over a considerably larger range of temperature differences. In any case, although the Boussinesq equations represent a very substantial simplification of the exact equations, the essential property of coupling between the thermal and velocity fields is preserved, and, even in the Boussinesq approximation, the solution of natural convection problems is more complicated than the forced convection heat transfer problems that we encountered earlier. 

We have already noted that the general class of flows driven by buoyancy forces that are created because the density is nonuniform is known as natural convection. If we examine the Boussinesq approximation of the Navier-Stokes equations, (12-170), we can see that there are actually two types of natural convection problems. In the first, we assume that a fluid of ambient temperature 71, is heated at a bounding surface to a higher temperature I. This will produce a nonuniform temperature distribution in the contiguous fluid, and thus a nonuniform density distribution too. Let us suppose that the heated surface is everywhere horizontal. Then there is a steady-state solution of (12-170) with u = 0, and the body-force terms balanced by a modification to the hydrostatic pressure distribution, such that... 

In 1960, Stuart and Watson (cf. S12) examined the same problem using the time-dependent Boussinesq equations and obtained a solution which converged to that of Malkus and Veronis as time approached infinity, thus demonstrating that under unstable conditions a differential disturbance can indeed lead to finite amplitude steady convection. 

In this case, the flow cross section is 2a in the x direction by 2b in the y direction. A series solution has been obtained by Boussinesq (1868) in the following form ... 

Thus, in order to solve the hydrodynamic problem of liquid motion in view of the change of 2 at the interface, we should first And out the distribution of substance concentration, temperature and electric charge over the surface. These distributions, in turn, are influenced by the distribution of hydrodynamic parameters. Therefore the solution of this problem requires utilization of conservation laws - the equations of mass, momentum, energy, and electric charge conservation with the appropriate boundary conditions that represent the balance of forces at the interface the equality of tangential forces and the jump in normal forces which equals the capillary pressure. In the case of Boussinesq model, it is necessary to know the surface viscosity of the layer. From now on, we are going to neglect the surface viscosity. 

The tidal current fields at the project sea area are simulated by the three-dimensional numerical model (MIKE3 FM) which developed by Danish Hydraulics Research Institute. The model is based on the solution of three-dimensional incompressible Reynolds Navier-Stokes equations, subject to the assumption of Boussinesq and hydrostatic pressure. 

The solution to this equation is more difficult than the solution to Eq. 7.194. The case of pure pressure flow was first solved by Boussinesq [130] in 1868. The solution to the combined drag and pressure flow was first published in 1922 [98] the authorship of this publication remains a question. Since the 1922 publication, numerous workers have presented solutions to this problem. Meskat [131] reviewed various solutions and demonstrated that they were equivalent. The velocity profile resulting from the drag flow can be written as ... 

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