Partial differential equations unsteady heat transfer

Mathematical modeling of mass or heat transfer in solids involves Pick s law of mass transfer or Fourier s law of heat conduction. Engineers are interested in the distribution of heat or concentration across the slab, or the material in which the experiment is performed. This process is represented by parabolic partial differential equations (unsteady state) or elliptic partial differential equations. When the length of the domain is large, it is reasonable to consider the domain as semi-infinite which simplifies the problem and helps in obtaining analytical solutions. These partial differential equations are governed by the initial condition and the boundary condition at x = 0. The dependent variable has to be finite at distances far (x = ) from the origin. Both parabolic and elliptic partial... 

In order to simulate fluid flow, heat transfer, and other related physical phenomena over various length scales, it is necessary to describe the associated physics in mathematical terms. Nearly all the physical phenomena of interest to the fluid dynamics research community are governed by the principles of continuum conservation and are expressed in terms of first- or second-order partial differential equations that mathematically represent these principles (within the restrictions of a continuum-based firamework). However, in case the requirements of continuum hypothesis are violated altogether for certain physical problems (for instance, in case of high Knudsen number rarefied gas flows), alternative formulations in terms of the particle-based statistical tools or the atomistic simulation techniques need to be resorted to. In this entry, we shall only focus our attention to situations in which the governing differential equations physically originate out of continuum conservation requirements and can be expressed in the form of a general differential equation that incorporates the unsteady term, the advection term, the diffusion term, and the source term to be elucidated as follows. 

In Chapter 5 the conservation-of-energy equations (2.7-2) and (4.1-3) will be used again when the rate of accumulation is not zero and unsteady-state heat transfer occurs. The mechanistic expression for Fourier s law in the form of a partial differential equation will be used where temperature at various points and the rate of heat transfer change with time. In Section 5.6 a general differential equation of energy change will be derived and integrated for various specific cases to determine the temperature profile and heat flux. 

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