Singularly Perturbed Solution Of Non-Fourier Dual-Phase-Lag Model

Fourier law can accurately describe most of the heat conduction problems.However,due to its limitations（assuming that the heat conduction velocity is infinite）,Fourier law is no longer applicable under extreme conditions such as ultra-fast,which makes non-Fourier heat conduction phenomena found in many cases.Therefore,it is necessary to study non-Fourier thermal effects of various materials in practical engineering problem-s.Among the many non-Fourier heat conduction models,the dual-phase-lag heat con-duction model is the most comprehensive,covering the physical response from micro to macro levels,with universality.Most of the existing researches on non-Fourier effect are on the unbounded region or paraboloid,and most of the solutions of non-Fourier temper-ature field models use numerical analysis methods.Therefore,there is still a lot of space for exploration and development to solve the model by using the singular perturbation method.In this dissertation,we construct the temperature field models of general mate-rials,one and three-dimensional laminate materials,and phase change materials by using the dual-phase-lag heat conduction law.We obtain the definite solutions of the model-s and use the singularly perturbed method to find their formal asymptotic solutions and estimate the remainder of the asymptotic solutions.The main contents are as follows:1、Singularly perturbed solutions of the dual-phase-lag model for general materials.Based on the dual-phase-lag conduction model,we obtain the definite solution problem of the third-order mixed equation with mixed non-homogeneous boundary value conditions.First,the parabolic equations are obtained by regular perturbation expansion,and the outer solutions are obtained by the separation of variables.Then,the initial layer and corner layer are corrected,and Laplace transformation method is used to solve the problem,so the asymptotic solution expansion is obtained.Finally,the remainder of the solution is estimated,and the consistent validity is established.2、Singularly perturbed solution of dual-phase-lag model for one-dimensional lami-nates.Based on non-Fourier heat conduction law,we build the temperature field model of laminates.The model is a nonlinear mixed equation boundary value problem with discon-tinuous coefficient.The nonlinear discontinuous equation problem is transformed into a series of linear parabolic equation problems by singular perturbation method.The formal external solutions are obtained by eigenfunction method combined with the corresponding boundary conditions.Secondly,the internal solutions are obtained by Laplace transform method after correcting the initial moment,corner points and discontinuity.Then,ac-cording to the properties of continuous and smooth temperature,we get the expression of temperature field at the lamination.Finally,the consistent validity of the asymptotic solution in the sense of L₂is obtained by the remainder estimation.3、Singularly perturbed solution of non-Fourier dual-phase-lag model for three-dimensional laminates.The third-order nonlinear mixed equation with discontinuity co-efficient in three dimensions is discussed.In this case,the position where the heat conduc-tion coefficient jumps are changed from point to surface.Firstly,the nonlinear discontin-uous equation is transformed into a series of linear equations by the singularly perturbed method,and the parabolic equations satisfied by the external solutions are obtained.The existence and uniqueness of the external solutions are proved by the energy method.Sec-ondly,a series of second-order mixed equations are obtained by correcting the initial layer and the corner layer,which are solved by Laplace transform method.Then the tempera-ture field expression of the laminar surface is determined by joint of the two parts.Finally,the Gronwall inequality and Gauss formula are used to prove the uniform validity of the solution.4、Singularly perturbed solution of one-dimensional dual-phase-lag heat conduc-tion model with discontinuous heat conduction coefficient.Based on the non-Fourier dual-phase-lag heat conduction law,the temperature field model of material phase tran-sition（the process of a substance from a solid or liquid state to a liquid or vapor state）is constructed.A class of nonlinear singular perturbed equations in bounded domain is obtained because of the discontinuity of heat conduction coefficient due to the change of material morphology.Firstly,we obtain the parabolic equations satisfied by the outer solutions by regular expansion and prove the existence and uniqueness of the outer so-lutions in the irregular bounded domain.To eliminate the errors caused by perturbation expansion,the initial layer and corner layer are corrected,and the internal solutions are obtained by the Laplace transform method.Secondly,we use the polishing method and Taylor expansion to determine the melting position of the material.Finally,we show the uniform validity of the asymptotic solution.5、Singularly perturbed solution of a three-dimensional dual-phase-lag heat con-duction model with discontinuous heat conduction coefficient.In this paper,the bound-ary value problem of three-dimensional nonlinear mixed equation with discontinuity of heat conduction coefficient in a bounded and irregular region is studied.The singular perturbation method is applied to transform the nonlinear discontinuous equation into a series of linear equation problems,and the equations satisfied by the external solutions are obtained,and their existence and uniqueness are proved.Secondly,the method of Laplace transformation is used to solve the problems by correcting the initial moment,angular region and discontinuity.Then,according to the equation conditions satisfied by the undetermined function,the expression of the material melting position is determined.In order to ensure the accuracy of the asymptotic solution,we estimate the remainder.