Boussinesq theory

Modulus of elasticity of the subgrade for static load could be calculated with the Boussinesq theory in conjunction with the plate bearing test. The relationship, which can be... 

The Realizable k-e model ensures the positivity of normal stresses ( realizable ) by making the empirical constants of k-e turbulence model, C, a function of the mean flow (mean deformation) and the turbulence (/c, ) while the Boussinesq theory used in the standard and RNG k-e model allows for negative normal stresses. The Realizable k-e model is more accurate in the prediction of the spreading rate of both planar and round jets. Kang et al (2008) used a Realizable k-e model to simulate the hydrodynamics in the membrane filtration zone of pilot and full-scale MBRs. 

Johnson, K.L., Contact Mechanics. Cambridge University Press, Cambridge, 1985. Boussinesq, J., Theory of Elasticity. McGraw Hill, New York, 1945, p. 338. 

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... 

Later we shall include combustion and flame radiation effects, but we will still maintain all of assumptions 2 to 5 above. The top-hat profile and Boussinesq assumptions serve only to simplify our mathematics, while retaining the basic physics of the problem. However, since the theory can only be taken so far before experimental data must be relied on for its missing pieces, the degree of these simplifications should not reduce the generality of the results. We shall use the following conservation equations in control volume form for a fixed CV and for steady state conditions ... 

Boussinesq, J. (1903). Theorie Analytique de la Chaleur, 2. Paris Gauthier-Villars. 

It is correct to state that according to the Theory of Gases energy can be expressed as temperature. However, this is advantageous and reasonable only if the physical process is governed by molecular events. For macroscopic interrelations like the Boussinesq s problem, the molecular nature of the gas is irrelevant. Here, the microscopic parameters are replaced by mean values of the macroscopic ones, these appearing in measurable physical properties such as specific heat and heat conductivity. To equate energy with temperature as Riabouchinsky did, introduced irrelevant physics to the problem. See also the remarks of L.I. Sedov [48, p. 40+]. 

Before starting to discuss (116), we make an observation. The fast time evolution (116) is also observed in driven systems that cannot be described on the level Ath- For example, let us consider the Rayleigh-Benard system (i.e., a horizontal layer of a fluid heated from below). It is well established experimentally that this externally driven system does not reach thermodynamic equilibrium states but its behavior is well described on the level of fluid mechanics (by Boussinesq equations). This means that if we describe it on a more microscopic level, say the level of kinetic theory, then we shall observe the approach to the level of fluid mechanics. Consequently, the comments that we shall make below about (116) apply also to driven systems and to other types of systems that are prevented from reaching thermodynamical equilibrium states (as, e.g., glasses where internal constraints prevent the approach to Ath)-... 

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. 

In order to calculate wx/wT by solving the differential equation (3.149), the Reynolds stress w xw has to be known. The hypothesis introduced by Boussinesq (3.140) is unsuitable for this, as according to it, the Reynolds stress does not disappear at the wall. However, the condition w xw y = 0 at the wall is satisfied by Prandtl s mixing length theory, which will now be explained. In order to do this we will consider a fluid element in a turbulent boundary layer, at a distance y from the wall, Fig. 3.16. It has, at a distance y, the mean velocity wx(y) and... 

Boussinesq, J. M. Theorie analytique de la chaleur. 2nd ed. Paris Gauthier-Villars 1903... 

Prandtl s model derivation can then be briefly sketched, introducing the Boussinesq [19] [20] approximation for the turbulent viscosity. Starting out with the simple kinetic theory relation that the molecular viscosity equals the molecular velocity times the mean free path, an analogous relation can be formulated for the turbulent viscosity in terms of the turbulent mixing length and a suitable velocity scale, Ut Iv. ... 

Boussinesq J (1877) Essai sur la theorie des eaux courants. Mem pres par div savants a I Acad Sci, Paris 23(l) l-680... 

Boussinesq J (1897) Theorie de Pecoulement tourbillonnant et tumultueeux des liquids dans les lits rectilignes a grande section I-II. Gauthier-Villars, Paris... 

Any mobility of the surface decreases the velocity difference and the viscous stresses. The result is that the hydrodynamic resistance becomes smaller and the floating velocity of a bubble according to (8.6) increases by a factor of 3/2 as compared to Stokes Eq. (8.5). In early experiments, under the condition of Re < 1, it was found (Lebedev 1916) that small bubbles of a diameters less than 0.01 cm behave like rigid spheres since their velocity is described by Stokes formula (8.5). At the same time. Bond (1927) has found that drops of a sufficiently large size fall at velocities described by Eq. (8.6). To overcome contradictions with the Hadamard-Rybczynski theory, Boussinesq (1913) considered the hypothetical influence of the surfaee viscosity and derived the following relation. 

The concept of a retardation coefficient is associated with the idea of a uniform surface retardation, since in the absence of retardation the velocity distribution along the surface is also expressed by a sinusoidal relation. The considerations of surface viscosity in Boussinesq s theory and of the retarding effects of surfactants in Frumkin s theory result just in such angular dependence. Therefore, the discussion presented below can be carried out without predetermining the value of the retardation coefficient Xb ... 

According to the classical theories of Hadamard (H2), Rybczynski (R9), and Boussinesq (BIO), circulation sets in whenever a drop moves in a viscous fluid. Figure 6 shows Hadamard streamlines compared with the experimental ones obtained by Spelles (S18). Experiments (G8) indicated that circulation takes place readily, though not exclusively (G5), in the presence of a suitable driving force (temperature or concentration gradients) and only above a... 

Boussinesq s theory (Boussinesq 1885) was developed at the end of the 19th century. Even though Boussinesq s theory is not currently used in multi-layer pavement engineering, it was the basis for the development of all subsequent stress/strain distribution theories. Boussinesq s theory refers to only one layer of uniform and homogeneous material such as, by assumption, soil material-subgrade. 

Figure II.I Developing stresses in accordance with Boussinesq s theory, (a) Point load, (b) Uniformly distributed load.

Foster and Ahlvin (1954), using Boussinesq s theory, developed diagrams from which not only the stresses but also the elastic deflections caused by a circular, uniformly applied load can be calculated much easier, at any depth from the surface and distance from the axis of the loading area. 

Boussinesq s theory/analysis can also be used to derive the elastic deflection of a homogeneous material subjected to uniform circular loading. The elastic surface deflection, A, at the centre of the loading area under the influence of a uniformly distributed load can be calculated from the following equation ... 

Forty years after the development of Boussinesq s analysis, Westergaard (1926) developed his own analysis for the determination of stresses in a two-layer system, where the first layer consisted of a concrete slab. Even though this theory did not cover the case of flexible pavements, it triggered the development of analysis for flexible pavements with two layers. 

The above inconvenience can be overcome by using Odemark s equivalent thickness theory, which converts the two-layer system into a one-layer system, and then applying Boussinesq s analysis. 

Boussinesq, J.V. (1871) Theorie de I intumescence liquide appelec onde solitaire ou de translation se propageant dans un canal rectangulaire. C R. Hehd. Seances Acad. Sci. (Paris) 72 755-759. 

Boussinesq, J.V. (1872) Theorie des ondes et des remous qui se propagent le long d un canal rectangulaire horisontal en communiquant au liquid contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond../. Math. Pares Appl. 17 55-108. 

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