The heat conduction equation

The basis of the solution of complex heat conduction problems, which go beyond the simple case of steady-state, one-dimensional conduction first mentioned in section 1.1.2, is the differential equation for the temperature field in a quiescent medium. It is known as the equation of conduction of heat or the heat conduction equation. In the following section we will explain how it is derived taking into account the temperature dependence of the material properties and the influence of heat sources. The assumption of constant material properties leads to linear partial differential equations, which will be obtained for different geometries. After an extensive discussion of the boundary conditions, which have to be set and fulfilled in order to solve the heat conduction equation, we will investigate the possibilities for solving the equation with material properties that change with temperature. In the last section we will turn our attention to dimensional analysis or similarity theory, which leads to the definition of the dimensionless numbers relevant for heat conduction. 

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The heat conduction equation in terms of these variables has the components... 

Grady and Asay [49] estimate the actual local heating that may occur in shocked 6061-T6 Al. In the work of Hayes and Grady [50], slip planes are assumed to be separated by the characteristic distance d. Plastic deformation in the shock front is assumed to dissipate heat (per unit area) at a constant rate S.QdJt, where AQ is the dissipative component of internal energy change and is the shock risetime. The local slip-band temperature behind the shock front, 7), is obtained as a solution to the heat conduction equation with y as the thermal diffusivity... 

From the heat conduction equation in the presence of adiabatic compression [51]... 

A typical method for thermal analysis is to solve the energy equation in hydrodynamic films and the heat conduction equation in solids, simultaneously, along with the other governing equations. To apply this method to mixed lubrication, however, one has to deal with several problems. In addition to the great computational work required, the discontinuity of the hydrodynamic films due to asperity contacts presents a major difficulty to the application. As an alternative, the method of moving point heat source integration has been introduced to conduct thermal analysis in mixed lubrication. 

Example 4. The first boundary-value problem for the heat conduction equation ... 

By having recourse to the heat conduction equation for a = 0 we establish a precise relationship... 

In the preceding section the Dirichlet difference problem was set up in the form (1) 3 Consider as one possible example the so-called scheme with weights for the heat conduction equation... 

In this chapter difference schemes for the simplest time-dependent equations are studied, namely, for the heat conduction equation with one or more spatial variables, the one-dimensional transfer equation and the equation of vibrations of a string. Two-layer and three-layer schemes are designed for the first, second and third boundary-value problems. Stability is investigated by different methods such as the method of separation of variables and the method of energy inequalities as well as by means of the maximum principle. Asymptotic stability of difference schemes is discovered for the heat conduction equation in ascertaining the viability of difference approximations. Finally, stability theory is being used, increasingly, to help us understand a variety of phenomena, so it seems worthwhile to discuss it in full details. 

The scheme of second-order accuracy (unconditionally stable in the asymptotic sense). Before taking up the general case, our starting point is the existing scheme of order 2 for the heat conduction equation possessing the unconditional asymptotic stability and having the form... 

SCHEMES FOR THE HEAT CONDUCTION EQUATION WITH SEVERAL SPATIAL VARIABLES... 

The explicit difference scheme. The schemes considered in Section 1 may be generalized to the case of the heat conduction equation with several spatial variables. 

By analogy with the heat conduction equation we employ the method of separation of variables, in the framework of which a solution of this problem is sought as the series... 

For the heat conduction equation the difference scheme is suggested ... 

For the heat conduction equation with a variable coefficient k x) we might have... 

Example 1 The explicit three-layer Du Fort Frankel scheme for the heat conduction equation from Section 3.7 belongs to the family... 

It is not difficult to write down an explicit stable scheme for the heat conduction equation with variable coefficients... 

The origiiral problem. We begin by placing the first boundary-value problem for for the heat conduction equation in which it is required to find a continuous in the rectangle Dt = 0[Pg.459]

One way of covering this for the heat conduction equation is to construct a homogeneous conservative scheme by means of the integro-interpo-lation method. To make our exposition more transparent, we may assume that the coefficient of heat conductivity k = k(x) is independent of t. The general case k = k(x,t) will appear on this basis in Section 8 without any difficulties. 

A concentrated heat capacity. We now consider the boundary-value problem for the heat conduction equation with some unusual condition placing the concentrated heat capacity Co on the boundary, say at a single point X = 0. The traditional way of covering this is to impose at the point a = 0 an unusual boundary condition such as... 

Cylindrically symmetric and spherically symmetric heat conduction problems. In explorations of many physical processes such as diffusion or heat conduction it may happen that the shape of available bodies is cylindrical. In this view, it seems reasonable to introduce a cylindrical system of coordinates (r, ip, z) and write down the heat conduction equation with respect to these variables (here x = r) ... 

In the case of a spherical symmetry the heat conduction equation acquires the form... 

After that, Newton s method of iterations applies equally well to either of these groups independently. By analogy with the isotermic case the first group of equations is to be solved with a prescribed temperature, while the second one needs the assigned values of rj and v. The essence of the matter in the last case is that the origin of the heat conduction equation is stipulated by the available sources of a dynamical nature. 

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