On the Stefan Problem With Nonlinear Thermal Conductivity

The solution is obtained and validated by an existence and uniqueness theorem for the following nonlinear boundary value problem \[ \frac{d}{dx}(1+\delta y+\gamma y^{2})^{n}\frac{dy}{dx}]+2x\frac{dy}{dx}=0,\,\,\,x>0,\,\,y(0)=0,\,\,\,y(\infty)=1, \] which was proposed in 1974 by [1] to represent a Stefan problem with a nonlinear temperature-dependent thermal conductivity on the semi-infinite line (0;1). The modified error function of two parameters $\varphi_{\delta,\gamma}$ is introduced to represent the solution of the problem above, and some properties of the function are established. This generalizes the results obtained in [3, 4].

On the New Nonlinear Properties of the Nonlinear Heat Conductivity Problem

In this article, we discuss one problem of nonlinear thermal conductivity with double nonlinearity; an exact analytical solution has been found for it, the analysis of which allows revealing a number of characteristic features of thermal processes in nonlinear media. The following nonlinear effects have been established: the inertial effect, the finite propagation velocity of thermal disturbances, the spatial localization of heat, and the effect of the finite time of the existence of a thermal structure in an absorption medium.

Adaptive Thermal Conductivity Metamaterials: Enabling Active and Passive Thermal Control

The novel adaptive thermal metamaterial developed in this paper provides a unique thermal management capability that can address the needs of future spacecraft. While advances in metamaterials have provided the ability to generate materials with a broad range of material properties, relatively little advancement has been made in the development of adaptive metamaterials. This metamaterial concept enables the development of materials with a highly nonlinear thermal conductivity as a function of temperature. Through enabling active or passive control of the metamaterials bulk effective thermal conductivity, this metamaterial that can improve the spacecraft's thermal management systems performance. This variable thermal conductivity is achieved through induced contact that results in changes in the F path length and the conductive path area. The contact can be generated internally using thermal strain from shape memory alloys, bimetal springs, and mismatches in coefficient of thermal expansion (CTE) or it can be generated externally using applied mechanical loading. The metamaterial can actively control the temperature of an interface by dynamically changing the bulk thermal conductivity controlling the instantaneous heat flux through the metamaterial. The design of thermal stability regions (regions of constant thermal conductivity versus temperature) into the nonlinear thermal conductivity as a function of temperature can provide passive thermal control. While this concept can be used in a wide range of applications, this paper focuses on the development of a metamaterial that achieves highly nonlinear thermal conductivity as a function of temperature to enable passive thermal control of spacecraft systems on orbit.

Nonlinear thermal conductivity in gases

The nonlinear diffusion equation corresponds to the diffusion processes which can occur with a finite velocity. A.J. Janavičius proposed nonlinear equation which describes more exactly the diffusion of impurities in Si crystals in many interesting practical applications. The heat transfer in gases is also based on diffusion of gas molecules from hot regions to the coldest ones with a finite velocity by random Brownian motions. In this case the heat transfer can be considered using similar nonlinear thermal diffusivity equation. The approximate analytical solution of this nonlinear equation can be used for the experimental analysis of thermal conductivity coefficients using temperature profiles dependence on different temperatures and pressures in gases.