搜索

x

留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

热传导系数跳跃的三维非Fourier温度场分布的奇摄动双参数解

包立平 李文彦 吴立群

引用本文:
Citation:

热传导系数跳跃的三维非Fourier温度场分布的奇摄动双参数解

包立平, 李文彦, 吴立群

Singularly perturbed solutions of a class of non-Fourier temperature field distribution

Bao Li-Ping, Li Wen-Yan, Wu Li-Qun
PDF
HTML
导出引用
  • 应用非Fourier热传导定律构建了温度场模型, 即一类在无界域上的三维奇摄动双曲抛物方程的初边值问题. 随着温度急剧变化, 热传导系数发生跳跃, 相应可以用非线性的具有间断系数的奇摄动双参数双曲方程表示. 通过奇摄动双参数展开方法, 得到了该问题的渐近解. 首先应用奇摄动方法得到该问题的展开式, 通过对解做出估计以及古典解的存在唯一性定理给出了内解和外解的存在性、唯一性. 其次, 由奇摄动理论, 得到该类奇摄动双曲方程进行了初始层矫正, 得到了解关于时间的导数的估计. 并且通过用Fourier 变换确定了热传导系数跳跃的位置表达式, 从而得到了解的形式渐近展开式. 最后通过余项估计, 得到了渐近解的一致有效性, 从而得到了热传导系数间断的温度场的分布.
    Thermoelastic coupling model excited by laser is of great significance in engineering. To study the thermoelastic coupling model, the distribution of temperature field must be determined firstly. Because the laser excitation time is short (usually femtosecond), the traditional Fourier heat conduction law is no longer suitable. Therefore, it is necessary to establish the distribution of temperature field by using the non-Fourier heat conduction law. Previous studies on the temperature field model mostly use numerical analysis and computer simulation to discuss its numerical solution, but few can directly solve the analytical solution of the model. Up to now, there are few reports about using singularly perturbed analysis method to solve the asymptotic solution of temperature field model and determine the jumping position of heat conductivity coefficient. In this paper, a temperature field model is constructed by using the non-Fourier heat conduction law, i.e. a class of singularly perturbed hyperbolic equations with small parameters in an unbounded domain. The nonlinear singularly perturbed two-parameter hyperbolic equations with discontinuous coefficients are obtained when the heat conduction coefficients jump due to sharp temperature changes. By using the singularly perturbed biparametric expansion method, the asymptotic solution of the problem is obtained. First, the expansion of the problem is obtained by using singularly perturbed method. The existence and uniqueness of the internal and external solutions are obtained by estimating the maximum modulus of the internal and external solutions and the maximum modulus estimates of the time derivatives, and the formal asymptotic expansion of the solutions is obtained. Secondly, the singularly perturbed hyperbolic equation is corrected by the singular perturbation theory, and the derivative of the solution is estimated. The position expression of the jump of the thermal conductivity coefficient is determined by the Fourier transform, and the seam method is used to connect the seams of the two sides of the jump position of the thermal conductivity coefficient, thus the form asymptotic expansion of the solution is obtained. Finally, the uniform validity of the asymptotic solution is obtained by estimating the residual term, and the distribution of the temperature field with discontinuous heat conduction coefficient is obtained. In this paper, we have synthetically applied the knowledge of ordinary differential equations, partial differential equations, mathematical and physical equations, nonlinear acoustics, mathematical analysis, singular perturbation theory and so on, which enriched the study of non-Fourier temperature field model.
      通信作者: 包立平, baolp@hdu.edu.cn
    • 基金项目: 国家自然科学基金(批准号: 51775154)和浙江省重点自然科学基金(批准号: LZ15E050004)资助的课题.
      Corresponding author: Bao Li-Ping, baolp@hdu.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No. 51775154) and the Natural Science Foundation of Zhejiang Province, China (Grant No. LZ15E050004).
    [1]

    关建飞, 沈中华, 许伯强 等 2005 光电子·激光 16 231Google Scholar

    Guan J F, Shen Z H, Xu B Q, et al. 2005 Photoelectronics and Laser 16 231Google Scholar

    [2]

    沈中华, 许伯强, 倪晓武 等 2004 中国激光 31 1275Google Scholar

    Shen Z H, Xu B Q, Ni X W, et al. 2004 China Laser 31 1275Google Scholar

    [3]

    Tzou D Y 1995 Int. J. Heat. Mass. Transf. 38 3231Google Scholar

    [4]

    Tzou D Y 1995 ASME J. Heat. Mass. Transf. 117 8Google Scholar

    [5]

    李金娥, 王保林, 常冬梅 2011 固体力学学报 s1 248

    Li J E, Wang B L, Chang D M 2011 J. Solid Mechanics s1 248

    [6]

    张浙, 刘登瀛 2000 力学进展 30 446Google Scholar

    Zhang Z, Liu D Y 2000 Progress in Mechanics 30 446Google Scholar

    [7]

    Liu Y, Li H, He S 2010 Numer. Math. A: J. Chin. Univ. 171 1

    [8]

    Amirov R 2014 Inter. Conference on Non. Differential and Difference Equations May 2 7

    [9]

    Farrell P A, O’Riordan E, Shishkin G I 2005 Math. Computation 74 1759Google Scholar

    [10]

    Teixeira M A, Silva P R D 2012 Physica D: Non. Phenomena 241 1948Google Scholar

    [11]

    Teixeira M A 2012 Perturbation Theory for Non-smooth Systems (New York: Springer) pp32−49

    [12]

    林娟 2011 武汉大学学报(理学版) 57 109

    Lin J 2011 J. Wuhan University (Science Edition) 57 109

    [13]

    兴梅 2005 数学物理学报 25 685Google Scholar

    Xing M 2005 J. Math. Phys. 25 685Google Scholar

    [14]

    Tan Q J, Leng Z J 2008 Math. Research. Rev. (English Edition) 4 41

    [15]

    Llibre J, Silva P R D, Teixeira M A 2006 J. Dyn. Differ. Equ. 19 309

    [16]

    康连城 1989 数学年刊 A辑 (中文版) 13 529

    Kang L C 1989 Math. Annual Series A (Chinese Edition) 13 529

    [17]

    康连城 1992 应用数学和力学 13 135

    Kang L C 1992 Appl. Math. Mechanics 13 135

    [18]

    包立平 , 李文彦, 吴立群 2019 应用数学和力学 40 536

    Bao L P, Li W Y, Wu L Q 2019 Appl. Math. Mech. 40 536

    [19]

    Zhang J Q, Nie L R, Chen C Y, et al. 2016 AIP Advances 6 075212Google Scholar

    [20]

    Chen R, Nie L, Chen C 2018 Chaos 28 053115Google Scholar

    [21]

    Chen R, Nie L, Chen C, et al. 2017 J. Stat. Mech Theory. E 14 013201

    [22]

    Nie L, Yu L, Zheng Z, et al. 2013 Phys. Rev. 062142

    [23]

    伍卓群, 尹景学, 王春明 2003 椭圆与抛物型方程引论 (北京: 科学出版社) 第152−184页

    Wu Z Q, Yin J X, Wang C M 2003 Introduction to Elliptic and Parabolic Equations Fristplace (Beijing: Science Press) (in Chinese) pp152−184 (in Chinese)

  • [1]

    关建飞, 沈中华, 许伯强 等 2005 光电子·激光 16 231Google Scholar

    Guan J F, Shen Z H, Xu B Q, et al. 2005 Photoelectronics and Laser 16 231Google Scholar

    [2]

    沈中华, 许伯强, 倪晓武 等 2004 中国激光 31 1275Google Scholar

    Shen Z H, Xu B Q, Ni X W, et al. 2004 China Laser 31 1275Google Scholar

    [3]

    Tzou D Y 1995 Int. J. Heat. Mass. Transf. 38 3231Google Scholar

    [4]

    Tzou D Y 1995 ASME J. Heat. Mass. Transf. 117 8Google Scholar

    [5]

    李金娥, 王保林, 常冬梅 2011 固体力学学报 s1 248

    Li J E, Wang B L, Chang D M 2011 J. Solid Mechanics s1 248

    [6]

    张浙, 刘登瀛 2000 力学进展 30 446Google Scholar

    Zhang Z, Liu D Y 2000 Progress in Mechanics 30 446Google Scholar

    [7]

    Liu Y, Li H, He S 2010 Numer. Math. A: J. Chin. Univ. 171 1

    [8]

    Amirov R 2014 Inter. Conference on Non. Differential and Difference Equations May 2 7

    [9]

    Farrell P A, O’Riordan E, Shishkin G I 2005 Math. Computation 74 1759Google Scholar

    [10]

    Teixeira M A, Silva P R D 2012 Physica D: Non. Phenomena 241 1948Google Scholar

    [11]

    Teixeira M A 2012 Perturbation Theory for Non-smooth Systems (New York: Springer) pp32−49

    [12]

    林娟 2011 武汉大学学报(理学版) 57 109

    Lin J 2011 J. Wuhan University (Science Edition) 57 109

    [13]

    兴梅 2005 数学物理学报 25 685Google Scholar

    Xing M 2005 J. Math. Phys. 25 685Google Scholar

    [14]

    Tan Q J, Leng Z J 2008 Math. Research. Rev. (English Edition) 4 41

    [15]

    Llibre J, Silva P R D, Teixeira M A 2006 J. Dyn. Differ. Equ. 19 309

    [16]

    康连城 1989 数学年刊 A辑 (中文版) 13 529

    Kang L C 1989 Math. Annual Series A (Chinese Edition) 13 529

    [17]

    康连城 1992 应用数学和力学 13 135

    Kang L C 1992 Appl. Math. Mechanics 13 135

    [18]

    包立平 , 李文彦, 吴立群 2019 应用数学和力学 40 536

    Bao L P, Li W Y, Wu L Q 2019 Appl. Math. Mech. 40 536

    [19]

    Zhang J Q, Nie L R, Chen C Y, et al. 2016 AIP Advances 6 075212Google Scholar

    [20]

    Chen R, Nie L, Chen C 2018 Chaos 28 053115Google Scholar

    [21]

    Chen R, Nie L, Chen C, et al. 2017 J. Stat. Mech Theory. E 14 013201

    [22]

    Nie L, Yu L, Zheng Z, et al. 2013 Phys. Rev. 062142

    [23]

    伍卓群, 尹景学, 王春明 2003 椭圆与抛物型方程引论 (北京: 科学出版社) 第152−184页

    Wu Z Q, Yin J X, Wang C M 2003 Introduction to Elliptic and Parabolic Equations Fristplace (Beijing: Science Press) (in Chinese) pp152−184 (in Chinese)

  • [1] 张荣培, 王迪, 蔚喜军, 温学兵. 基于广义交替数值通量的局部间断Galerkin方法求解二维波动方程. 物理学报, 2020, 69(2): 020202. doi: 10.7498/aps.69.20190613
    [2] 蒋涛, 陈振超, 任金莲, 李刚. 基于修正并行光滑粒子动力学方法三维变系数瞬态热传导问题的模拟. 物理学报, 2017, 66(13): 130201. doi: 10.7498/aps.66.130201
    [3] 王佐, 刘雁, 张家忠. 过渡区微尺度流动的有效黏性多松弛系数格子Boltzmann模拟. 物理学报, 2016, 65(1): 014703. doi: 10.7498/aps.65.014703
    [4] 胡金秀, 高效伟. 变系数瞬态热传导问题边界元格式的特征正交分解降阶方法. 物理学报, 2016, 65(1): 014701. doi: 10.7498/aps.65.014701
    [5] 赵国忠, 蔚喜军. 统一坐标系下二维气动方程组的Runge-Kutta间断Galerkin有限元方法. 物理学报, 2012, 61(11): 110208. doi: 10.7498/aps.61.110208
    [6] 龚建强, 梁昌洪. 基于TE10矩形波导的异向介质有效本构参数提取算法. 物理学报, 2011, 60(5): 059204. doi: 10.7498/aps.60.059204
    [7] 何兴道, 夏健, 史久林, 刘娟, 李淑静, 刘建安, 方伟. 水的衰减系数及有效增益长度对受激布里渊散射输出能量的影响. 物理学报, 2011, 60(5): 054207. doi: 10.7498/aps.60.054207
    [8] 蒋涛, 欧阳洁, 栗雪娟, 张林, 任金莲. 瞬态热传导问题的一阶对称SPH方法模拟. 物理学报, 2011, 60(9): 090206. doi: 10.7498/aps.60.090206
    [9] 杨娟, 卞保民, 闫振纲, 王春勇, 李振华. 典型随机信号特征参数统计分布的分形特性. 物理学报, 2011, 60(10): 100506. doi: 10.7498/aps.60.100506
    [10] 杨娟, 卞保民, 彭刚, 李振华. 随机信号双参数脉冲模型的分形特征. 物理学报, 2011, 60(1): 010508. doi: 10.7498/aps.60.010508
    [11] 徐慧, 崔麦玲, 马松山. 含有格点势的一维Fibonacci链热传导性质的研究. 物理学报, 2010, 59(10): 7266-7270. doi: 10.7498/aps.59.7266
    [12] 韩非, 崔俊芝, 于艳. 非一致随机分布复合材料力学参数预测的统计二阶双尺度方法. 物理学报, 2009, 58(13): 1-S7. doi: 10.7498/aps.58.1
    [13] 颜森林. 延时反馈半导体激光器双劈控制混沌方法研究. 物理学报, 2008, 57(5): 2827-2831. doi: 10.7498/aps.57.2827
    [14] 程荣军, 程玉民. 带源参数的热传导反问题的无网格方法. 物理学报, 2007, 56(10): 5569-5574. doi: 10.7498/aps.56.5569
    [15] 李新霞, 唐 翌. 阻尼作用下一维体系热传导性质的研究. 物理学报, 2006, 55(12): 6556-6561. doi: 10.7498/aps.55.6556
    [16] 蔡长英, 任中洲, 鞠国兴. 指数型变化有效质量的三维Schr?dinger方程的解析解. 物理学报, 2005, 54(6): 2528-2533. doi: 10.7498/aps.54.2528
    [17] 刘慕仁, 孔令江. 降低格子Boltzmann方程沾滞系数的有效方法. 物理学报, 1996, 45(3): 370-372. doi: 10.7498/aps.45.370
    [18] 麦振洪, 毛再先, 阳述林. 旋转圆盘下的流场解及熔硅中溶质有效分凝系数的计算. 物理学报, 1991, 40(6): 935-942. doi: 10.7498/aps.40.935
    [19] 武宇. 带圆孔的方柱中的热传导. 物理学报, 1963, 19(4): 268-270. doi: 10.7498/aps.19.268
    [20] 王竹溪. 物质内部有辐射的热传导问题. 物理学报, 1962, 18(1): 11-26. doi: 10.7498/aps.18.11
计量
  • 文章访问数:  5654
  • PDF下载量:  28
  • 被引次数: 0
出版历程
  • 收稿日期:  2019-01-25
  • 修回日期:  2019-09-11
  • 上网日期:  2019-10-01
  • 刊出日期:  2019-10-20

/

返回文章
返回