Body semi-infinite

Several models have been proposed to predict adhesion force—the maximum force required to pull off the surfaces. Among these, the JKR theory is one receiving the greatest attention [2], which says that for an elastic spherical body in contact with a semi-infinite plane, the adhesion force can be estimated by... 

FIGURE 26.48 Calculated lines of equal stress for a line force made up of normal load and frictional force acting on the surface of a semi-infinite body. 

FIGURE 26.49 Calculated stored elastic energy and the horizontal stress component due to two line forces at an angle a to the plane of the rubber surface and a fixed distance x apart for different depths from the surface of a semi-infinite body. 

Classical heat transfer provides expressions for quantities such as view factors, radiation and temperature fields in semi—infinite bodies. The lining materials studied here were treated as semi-infinite bodies since the test duration is relatively short. 

The method can be demonstrated by considering diffusion into a semi-infinite body where the surface concentration c(x = 0, t) is fixed ... 

To estimate the time at which steady-state conditions are expected, the required penetration distance is set equal to the largest characteristic length over which diffusion can take place in the system. If L is the characteristic linear dimension of a body, steady state may be expected to apply at times r L2/Dmm, where Dmin is the smallest value of the diffusivity in the body. Of course, there are many physical situations where steady-state conditions will never arise, such as when the boundary conditions are time dependent or the system is infinite or semi-infinite. 

Maximum and minimum bounds for the growth of the vapor film at the surface of a rapidly heated plate in contact with a semi-infinite body of liquid initially at the saturation temperature have been deduced in a similar manner for arbitrary monotonic surface temperature or heat flux, following the method of Section II, D (H3). 

Figure 5.22 Schematic of a semi-infinite cooling body. Denoted are depths at which 50% and 1% of the temperature difference is felt.

For / < a, the 1/12 term dominates the interaction is like that of two semi-infinite bodies (a oo). For separation / comparable with or larger than thickness a, the finitude of thickness asserts itself. 

Begin with the interaction between two plane-parallel bodies separated by a distance Z across a medium m. Each of the bodies is semi-infinite, filling the space to the left or the right of the surface (see Fig. Ll.l). 

Of course, no body is infinitely or even semi-infinitely huge. The geometry here, chosen for mathematical convenience, in fact applies to cases in which the extent of the bodies is large compared with all other dimensions that are to be varied. Specifically, think of cases in which the depth away from the gap Z and the lateral extent of the interface enclos- Figure Ll.l... 

How strong are these long-range van der Waals forces between semi-infinite bodies A and B across a medium m or across a vacuum For the four materials whose dielectric responses are plotted in the preceding section, the corresponding Hamaker coefficients (with the neglect of retardation) make an instructive table. See Table LI. 3. 

Compared with the interaction of two semi-infinite bodies of hydrocarbon, the interaction between hydrocarbon-coated mica surfaces increases steadily versus separation whereas that between two finite slabs of hydrocarbon expectably decreases. Significant deviations are already clear at w/h = 1, when the varying separation w equals the constant layer thickness h. 

Table P.5. Multiply coated semi-infinite bodies A and B, small differences in s s, and // s Hamaker form...

Imagine the same two semi-infinite planar bodies for which the Lifshitz formulation was first carried out (see Fig. L2.9). 

NB Instead of using A and B to represent semi-infinite bodies, Level 3 derivations use L and R to orient "left" and "right" during equation solving. 

Follow an outside-in strategy. Consider a semi-infinite body with permittivity eout, coated with an inhomogeneous layer of constant thickness D and permittivity e(z), facing a medium em of variable thickness Z. Subscripts for a, b, L, and R will be added later (Fig. L3.14). 

What about the interaction of this coated semi-infinite body with another coated body, symmetric to it To avoid unnecessary mathematics, think of the mirror image of the problem just solved. Speak of variables za and Zb in Fig. L3.16 as increasing, respectively, left and right from the midpoint in variable region of thickness l. These za, Zb of convenience connect with the "real" z as za = -z, Zb = +z. By symmetry e(za) = e(Zb). 

Correlated charge fluctuations between anisotropic bodies acting across an anisotropic medium create torques as well as attractions or repulsions. The formulae derived here for semi-infinite media can also be specialized to express the torque and force between anisotropic small particles or between long rodlike molecules. (For example, Table C.4 and Subsection L2.3.G.)... 

FIG. 19.9 Schematic presentation of the progress of crystallisation into a semi-infinite body quenched at zero time at the plane x = 0 xc = distance at which crystallisation occurs, t = time. 

We start this chapter with the analysis of lumped systems in which the temperature of a body varies with time but remains uniform throughout at any time. Then we consider the variation of temperature with time as well as position for one-dimensional heat conduction problems such as those associated with a large plane wall, a long cylinder, a sphere, and a semi infinite medium using transient temperature charts and analytical solutions. Finally, we consider transient heat conduction in multidimensional systems by utilizing the product solution. 

A semi-infinite solid is an idealized body that has a single plane surface and e.xiends to infinity in all directions, as shown in Figure 4—24. This idealized body is used to indicate that the temperature change in the part of the body in... 

For short periods of time, most bodies can be modeled as semi-infinite solids since heat does not have sufficient time to penetrate deep into the body, and the thickness of the body does not enter into the heat transfer analysis. A steel piece of any shape, for example, can be treated as a. semi-infinite solid when it is quenched rapidly to harden its surface. A body whose surface is heated by a laser pulse can be treated the same why. 

If the bodies are of different materials, they still achieve a temperature equality, but the surface temperature T, in this case will be different than the arithmetic average. Noting that both bodies can be treated as semi-infinite solids with the same specified surface temperature, the energy balance on the contact surface gives, from Eq. 4- 15,... 

When the lumped system analysis is not applicable, the variation of temperature with position as well as time can be determined using the transient temperaiure charts given in Figs, 4-15,4-16, 4 17, and 4-29 for a large plane wall, a long cylinder, a sphere, and a semi-infinite medium, respectively. These charts are applicable for one-dimensional heal transfer in those geometries. Therefore, their use is limited to situations in which the body is initially at a uniform temperature, all surfaces are subjected to the same thermal conditions, and the body docs not involve any heat geiieiation. Tliese charts can also be used to determine the total heat transfer from the body up to a specified lime I. 

C What is a semi-infinite medium Give examples of solid bodies that can he treated as semi-infinite mediums for heat transfer purposes. 

Lewandowska, M. (2001) Hyperbolic Heat Conduction in the Semi-Infinite Body with a Time-Dependent Laser Heat Source,J. Heat Mass Transfer, Vol. 37, pp.333-342. 

Blackwell, B. F. (1990) Temperature Profile in Semi-Infinite Body with Exponential Sources and Convective Boundary Condition, Journal of Heat Transfer, Vol. 112, pp. 567-571. 

Henriques treated the skin as a semi-infinite body in which all skin layers have the same thermal properties, and the total skin thickness is far greater than that heated by the thermal source. Based on this assumption, he obtained the following formula for the temperature of the basal epidermal layer as a function of time ... 

Fig. 2.21 Heating of a semi-infinite body with different boundary conditions at the surface (x = 0). a jump in the surface temperature to s, b constant heat flux qg, c heat transfer from a fluid at = s...

Two semi-infinite bodies in contact with each other... 

We will consider the two semi-infinite bodies shown in Fig. 2.24, which have different, but constant, initial temperatures d01 and d02- Their material properties Ax, a1 and A2, a2 are also different. At time t = 0 both bodies are brought into (thermal) contact with each other along the plane indicated by x = 0. After a very short period of time an average temperature is reached along the plane. Heat flows from body 1 with the higher initial temperature to body 2 which has a lower temperature. The transient conduction process described here serves as a model for the description of short-time contact between two (finite) bodies at different temperatures. Examples of this include the touching of different objects with a hand or foot and the short-time interaction of a heated metal body with a cooled object in reforming processes. 

Fig. 2.24 Temperature pattern in two semi-infinite bodies with initial temperatures oi and 02 in contact with each other along the plane x = 0...

This solution allows for a clear interpretation. The terms in the curly brackets correspond to the temperature pattern in a semi-infinite solid. Its surface lies at r+ = — 1 for the first term, and it stretches itself out in the positive redirection. The second term belongs to a semi-infinite body which stretches out from its surface at r+ = +1 in the negative redirection. The subtraction of these two temperature profiles from the initial temperature d+ = 1 yields the symmetrical temperature profile for the plate shown in Fig. 2.32, which satisfies the condition di)+/dr+ = 0 at r+ = 0. 

In a semi-infinite body (x > 0), a spatially constant, but time dependent power density is... 

Here Wo represents the power density at time to- By using (2.224), a heat release is modelled as it occurs, for example when concrete sets. Then we have an initial large release of heat which rapidly decreases with advancing time. The semi-infinite body initially has the constant (over) temperature... 

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