Semi-infinite Plate

Consider a semi-infinite plate at a uniform initial temperature 7b. From this condition, the ambient temperature is suddenly changed to a temperature Too. The heat transfer coefficient is h. 

The unsteady temperature of the plate corresponding to an infinite heat transfer coefficient is 

By referring to Eqs. (3.130) and (3.131), we may write the temperature of two-and three-dimensional comers in the form 

7 MIXED (DIFFERENTIAL-DIFFERENCE) FORMULATION. ANALOG SOLUTIONO 

As we have seen in the preceding sections, the solution of unsteady conduction problems is, in general, not mathematically simple, and one must usually resort to a number of solution methods to evaluate the unsteady temperature distribution. We have also learned how to obtain solutions by using the available charts for a class of analytical results. In Chapter 4 we will explore the use of numerical computations to evaluate multidimensional and unsteady conduction problems. These computations require approximate difference formulations to represent time and spatial derivatives. Actually there exists a third and hybrid (analog) method that allows us to evaluate the temperature distribution in a conduction problem by using a timewise differential and spacewise difference formulation. This method utilizes electrical circuits to represent unsteady conduction problems. The circuits are selected in such a way that the voltages (representing temperatures) obey the same differential equations as the temperature. 

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Figure 4 Hydrodynamic boundary layer development on the semi-infinite plate of Prandtl. <5D = laminar boundary layer, <5t = turbulent boundary layer, /vs = viscous turbulent sub-layer, <5ds = diffusive sub-layer (no eddies are present solute diffusion and mass transfer are controlled by molecular diffusion—the thickness is about 1/10 of <5vs)> B = point of laminar—turbulent transition. Source From Ref. 10.

K° consists of a combination of Prandtl s original proportionality constant used for the hydrodynamic boundary layer at a semi-infinitive plate, Ke, and a constant, K, characterizing a particular hydrodynamic system that is under consideration. The latter constant has to be determined experimentally. 

Transformations near the Edge of a Thin Semi-Infinite Plate... 

The van der Waals interaction energy between a sphere and a semi-infinite plate has been estimated by Hamaker (4)... 

Product solutions for temperatures In multidimensional systems (a) semi-infinite plate (b) infinite rectangular bar (c) semi-infinite rectangular bar (cf) rectangular parallelepiped (e) semi-infinite cylinder (0 short cylinder. 

One-Dimensional Conduction Semi-infinite Plate Consider a semi-infinite plate with an initial uniform temperature T,. Suppose that the temperature of the surface is suddenly raised to T that is, the heat-transfer coefficient is infinite. The unsteady temperature of the plate is... 

We consider the laminar two-dimensional motion of fluid past a hot semi-infinite plate, with the free stream velocity and temperature denoted by, Uoo and Too- We will focus our attention on the top of the plate, for which the temperature is T - that is greater than Too, while assuming the leading edge of the plate as the stagnation point. Governing equations are written in dimensional form (indicated by the quantities with asterisk), along with the Boussinesq approximation to represent the buoyancy effect,... 

The equations cited above are for an ideal semi-infinite plate, with no boundary effects. Application to real specimens requires calibration factors, so that the fracture toughness of Eq. (11-51), at the critical point is given by ... 

Consider a thin semi-infinite plate defined by y > 0 whose faces are insulated and whose edge y = 0 is kept at temperature zero except for the segment -1 < x < 1. The segment (-1 < x < 1) is kept at temperature unity. The steady-state temperature distribution T(x,y) is given by the Laplace equation... 

Reconsider a semi-infinite plate. Let the initial temperature of the plate be uniform, say To, and the surface temperature be suddenly changed to Tx. We wish to develop the integral formulation of this problem and its solution by approximate profiles. 

The steady-state flux of hydrogen through a containment structure can be estimated from Pick s First Law, assuming the structure can be modeled as a semi-infinite plate [15] ... 

Figure 3.2 Interaction between elements of volume dV containing qdV molecules, which on summation for all pairs oj volume elements gives the total-interaction free energy between the two bodies (a) particles oj arbitrary shape, (b) two parallel semi-infinite plates a distance H apart, (c) two spheres whose surfaces are a distance H apart.

J.H. De Boer and H.C. Hamaker computed the London interaction between a pair of semi-infinite plates and between spherical particles. They showed that, despite the relatively long-range attractive nature of the force between molecular pairs, the total attractive force between the bodies decays much less rapidly. 

For two parallel semi-infinite plates the attractive force between the plates was found, after integrating Eq. (8.1.18), to be... 

Cylindrical Indenters. The elastic stretching u produced by a rigid planar indenter in a semi-infinite plate was deduced by Streicher [15] ... 

A turbulence model for small polder additives In a boundary layer on a flat plate is proposed with an account being taken of optimum polymer concentration. Results of a numerical investigation of the solution flow on a flat semi-infinite plate are presented. These results are in a satisfactory 6ig-reement with experimental data. 

Figure 4.15-2. Steady-state heat conduction in two directions in a semi-infinite plate.

Temperatures in a Semi-Infinite Plate. A semi-infinite plate is similar to that in... 

As it turns out, this relation holds not only for the considered configuration with a crack in the center of an infinite plate, but also for finite plates and even for other locations of the crack, for example, for a semi-infinite plate with a notch at the edge. The critical value of the stress intensity factor Kic follows from a combination of Eqs. (8.55) and (8.50), as... 

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