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首页> 外文期刊>International Journal of Heat and Mass Transfer >On The Solution Of Damped Wave Conduction And Relaxation Equation In A Semi-infinite Medium Subject To Constant Wall Flux
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On The Solution Of Damped Wave Conduction And Relaxation Equation In A Semi-infinite Medium Subject To Constant Wall Flux

机译：定壁通量的半无限介质中阻尼波传导和松弛方程的解

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Eight reasons are given to seek a generalized Fourier's law of heat conduction and relaxation. Bounded solutions are obtained for the damped wave conduction and relaxation equation in one dimension in Cartesian coordinates for a semi-infinite medium subject to the constant wall flux boundary condition for the dimensionless heat flux and dimensionless temperature. Three different methods were employed. In the first approach the method of Laplace transforms was used. The solutions are domain restricted. Three regimes can be identified (a) zero transferring regime; (b) rising regime and (c) falling regime. In the second approach a generalized substitution is used to transformrnthe hyperbolic PDE into a parabolic PDE. The transform selected is one with spatiotemporal symme-rntry. The resulting parabolic PDE can be solved for using the Boltzmann transformation. In the thirdrnapproach the damping term was first removed from the governing equation. The resulting equationrnwas transformed into a Bessel differential equation using a spatiotemporal symmetric transformation variable. A approximate solution for the flux was obtained. The inertial regime, rising and falling regimes were identified in the solution. A Chebyshev polynomial approximation was used for the integrand with modified Bessel composite function in space and time. Telescoping power series leadsrnto more useful expression for transient heat flux. The temperature and heat flux solutions at the wavernfront were also developed. The solution for transient heat flux from the method of relativistic transformation is compared side by side with the solution for transient temperature from the method of Chebyshev economization. Both solutions are within 12% of each other. For conditions close to the wave front the solution from the Chebyshev economization is expected to be close to the exact solution and was found to be within 2% of the solution from the method of relativistic transformation. Far from the wave front, i.e., close to the surface the numerical error from the method of Chebyshev economization is expected to be significant and verified by a specific example. The solutions for dimensionless heat flux and dimensionless temperature is found to be continuous across the wave front without any singularities or jumps.
机译：给出了八个理由来寻求热传导和弛豫的广义傅立叶定律。对于无量纲热通量和无量纲温度的恒定壁通量边界条件，对于半无限介质，在笛卡尔坐标系中，在一维笛卡尔坐标中获得了衰减波传导和张弛方程的有界解。使用了三种不同的方法。在第一种方法中，使用拉普拉斯变换的方法。解决方案受域限制。可以确定三种制度（a）零转移制度； （b）上升政权和（c）下降政权。在第二种方法中，使用广义替换将双曲型PDE转换为抛物线型PDE。选择的变换是一种具有时空对称的变换。可以使用Boltzmann变换求解所得的抛物线形PDE。在第三种方法中，阻尼项首先从控制方程中删除。使用时空对称转换变量将所得方程转换为贝塞尔微分方程。获得了通量的近似解。在解决方案中确定了惯性态，上升态和下降态。 Chebyshev多项式逼近法用于具有改进的Bessel合成函数的时空积分。伸缩幂级数导致瞬态热通量的更有用表达。还开发了波前的温度和热通量解决方案。将相对论转换方法的瞬态热通量解与Chebyshev省煤法的瞬态温度解并列比较。两种解决方案之间的误差都在12％之内。对于接近波前的条件，预计切比雪夫节的解将接近于精确解，并且被发现在相对论变换法的解的2％之内。远离波前，即靠近表面，切比雪夫节约方法的数值误差预计会很大，并通过一个具体示例进行验证。发现无因次热通量和无因次温度的解决方案在整个波前都是连续的，没有任何奇异或跳跃。

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