本文旨在探討冪次率流體在垂直非等溫平板上之膜狀凝結熱傳問題，分析了流場、溫度場、凝結液流量、表面摩擦係數、在地紐塞數等等重要的物理現象在給定不同流動行為指數、無因次液膜厚度、修正普朗特數、平板變溫係數之下的變化，更延伸探討了相似轉換後常出現的多重解現象。本文利用相似轉換使非線性偏微分統制方程式轉換為非線性常微分方程式，使原先與x、y相關聯的方程式轉化為只與η相關，大幅簡化了方程式的複雜性。
數值方法採用LADM求解，結合了拉普拉斯轉換後的LADM具有更快的速度，可以廣泛的應用在各類問題上，所得之解為一截斷級數解，在結果不易收斂時常與帕德近似搭配使用。
研究顯示相較於膨脹性流體與牛頓流體，擬塑性流體擁有較薄的速度及溫度邊界層，在本文薄膜凝結的問題中擁有較高的熱傳效率。修正普朗特數的增加會加快熱傳遞的速度，溫度變界層變薄，使凝結的效率增加。本文延伸討論了非等溫平板的薄膜凝結現象，並以平板變溫參數r來控制平板溫度的變化速度，結果顯示隨著r的增加會使溫度的下降速度更快，溫度邊界層變薄，凝結效率增加。由結果可知凝結液膜總流量會隨著液膜厚度μ_δ與流動行為指數n的增加而增加，因為膨脹性流體具有較厚的邊界層。而表面摩擦係數會隨著液膜厚度的增加而增加，且擬塑性流體擁有略高的表面摩擦係數。隨著修正普朗特數的增加，熱傳逐漸由對流主導，使溫度下降更快速，局部紐賽數較高。本文模型中主要熱阻為凝結液膜，故較薄的液膜擁有較高的熱傳效率，紐塞數較高，且紐塞數會隨著液膜厚度的增長而降低。除此之外，本文也驗證了LADM適用於擁有多重解的問題，儘管其中只有一組解符合物理意義。

This thesis aims to investigate the effects of non-Newtonian fluids film condensation on a non-isothermal vertical plate and analyze some critical physical phenomena such as flow field, temperature field, total mass flow, skin-friction coefficient, and local Nusselt number. The influence parameters included the flow behavior index n, non-dimensional film thickness η_δ, generalized Prandtl number Npr, and plate temperature index r. Moreover, the multiple solutions phenomenon frequently occurs after similarity transformation is demonstrated here. The similarity transformation is used in this thesis to convert the non-linear partial differential equations into non-linear ordinary differential equations. Then, the original equations related to x and y are transformed to only correlate with η, which significantly simplifies the governing equations.
The results are obtained by the Laplace Adomian decomposition method (LADM). The Laplace transform replaces the multiple integral calculations and yields a faster convergence speed than ADM. Consequently, LADM can be widely applied to various problems. The solution obtained by LADM or ADM is in series form, usually coupled with the Padé approximant to improve the convergence speed of the results.
The analysis shows that the pseudo-plastic fluid has a thinner velocity and thermal boundary layer thickness compared with the dilatant and Newtonian fluids, which means the pseudo-plastic fluid has a higher heat transfer efficiency in this thesis. The increase in the generalized Prandtl number will cause a faster heat transfer rate, leading to a thinner thermal boundary layer and higher condensation efficiency. Furthermore, the non-isothermal plate is considered in this condensation model and uses a plate temperature index r to control the change speed of plate temperature. The investigation reveals that the temperature will drop faster as r increases, indicating the thin temperature boundary layer and the high heat transfer efficiency. Subsequently, some physical quantities are also discussed here; it can be seen that the total condensate mass flow will increase with the increase of the film thickness η_δ and the flow behavior index n. The skin-friction coefficient will increase with the film thickness η_δ and the pseudo-plastic fluid has a slightly higher skin-friction coefficient. The higher generalized Prandtl number represents that heat transfer is gradually dominated by convection, which results in a faster temperature drop and a higher local Nusselt number. The primary thermal resistance of this condensation model is the film generated on the wall surface; hence, the thinner liquid film thickness will cause a higher heat transfer efficiency and a higher local Nusselt number. Apart from the preceding aspects, the applicability of LADM in solving a problem with multiple solutions is demonstrated, though only one of the solutions agrees with the physical significance.

[1] Nusselt, W., Die oberflachenkondensation des wasserdamphes. VDI-Zs, 1916. 60: p. 541.
[2] Rohsenow, W.M., Heat transfer and temperature distribution in laminar-film condensation. Transactions of the American Society of Mechanical Engineers, 1956. 78(8): p. 1645-1647.
[3] Sparrow, E.M. and J.L. Gregg, A boundary-layer treatment of laminar-film condensation. Journal of Heat Transfer, 1959. 81(1): p. 13-18.
[4] Chen, M.M., An analytical study of laminar film condensation: part 1—flat plates. 1961.
[5] Koh, J., E.M. Sparrow, and J. Hartnett, The two phase boundary layer in laminar film condensation. International Journal of Heat and Mass Transfer, 1961. 2(1-2): p. 69-82.
[6] Rose, J., Fundamentals of condensation heat transfer: laminar film condensation. JSME international journal. Ser. 2, engineering, heat transfer, power, combustion, thermophysical properties, 1988. 31(3): p. 357-375.
[7] Levy, S., Heat transfer to constant-property laminar boundary-layer flows with power-function free-stream velocity and wall-temperature variation. Journal of the Aeronautical Sciences, 1952. 19(5): p. 341-348.
[8] Sparrow, E. and J. Gregg, Similar solutions for free convection from a nonisothermal vertical plate. Transactions of the American Society of Mechanical Engineers, 1958. 80(2): p. 379-386.
[9] Nagendra, H. and M. Tirunarayanan, Laminar film condensation from nonisothermal vertical flat plates. Chemical Engineering Science, 1970. 25(6): p. 1073-1079.
[10] Schowalter, W.R., The application of boundary‐layer theory to power‐law pseudoplastic : Similar solutions. AIChE Journal, 1960. 6(1): p. 24-28.
[11] Acrivos, A., M. Shah, and E. Petersen, Momentum and heat transfer in laminar boundary‐layer flows of non‐Newtonian past external surfaces. AIChE Journal, 1960. 6(2): p. 312-317.
[12] Astarita, G., G. Marrucci, and G. Palumbo, Non-Newtonian gravity flow along inclined plane surfaces. Industrial & Engineering Chemistry Fundamentals, 1964. 3(4): p. 333-339.
[13] Therien, N., B. Coupal, and J.L. Corneille, Vérification expérimentale de l'épaisseur du film pour des liquides non‐newtoniens s' écoulant par gravité sur un plan incliné. The Canadian Journal of Chemical Engineering, 1970. 48(1): p. 17-20.
[14] Yang, T. and D. Yarbrough, A numerical study of the laminar flow of non-Newtonian along a vertical wall. 1973.
[15] Yang, T. and D. Yarbrough, Laminar flow of non-Newtonian liquid films inside a vertical pipe. Rheologica Acta, 1980. 19(4): p. 432-436.
[16] Narayana Murthy, V. and P. Sarma, A note on hydrodynamic entrance lengths of non-Newtonian laminar falling films. Chem. Eng. Sci., 1977. 32: p. 566-567.
[17] Narayana Murthy, V. and P. Sarma, Dynamics of developing laminar non-Newtonian falling liquid films with free surface. 1978.
[18] Andersson, H.I. and F. Irgens, Gravity-driven laminar film flow of power-law along vertical walls. Journal of Non-Newtonian Fluid Mechanics, 1988. 27(2): p. 153-172.
[19] De-Yi, S. and W. Bu-Xuan, Effect of variable thermophysical properties on laminar free convection of gas. International journal of heat and mass transfer, 1990. 33(7): p. 1387-1395.
[20] Shang, D.-Y. and T. Adamek, Study on laminar film condensation of saturated steam on a vertical flat plate for consideration of various physical factors including variable thermophysical properties. Heat and Mass Transfer, 1994. 30(2): p. 89-100.
[21] Andersson, H.I. and D.-Y. Shang, An extended study of the hydrodynamics of gravity-driven film flow of power-law . Fluid dynamics research, 1998. 22(6): p. 345.
[22] Si, X., et al., Laminar film condensation of pseudo-plastic fluid non-Newtonian fluid with variable thermal conductivity on an isothermal vertical plate. International Journal of Heat and Mass Transfer, 2016. 92: p. 979-986.
[23] Adomian, G., An analytical solution of the stochastic Navier-Stokes system. Foundations of Physics, 1991. 21(7): p. 831-843.
[24] Adomian, G., Solving frontier problems of physics: the decomposition method. Vol. 60. 1993: Springer Science & Business Media.
[25] Adomian, G., R. Rach, and R. Meyers, An efficient methodology for the physical sciences. Kybernetes, 1991.
[26] Adomian, G. and R. Rach, Nonlinear differential equations with negative power nonlinearities. Journal of mathematical analysis and applications, 1985. 112(2): p. 497-501.
[27] Adomian, G. and R. Rach, Solving nonlinear differential equations with decimal power nonlinearities. Journal of mathematical analysis and applications, 1986. 114(2): p. 423-425.
[28] Adomian, G., Nonlinear stochastic systems theory and applications to physics. Vol. 46. 1988: Springer Science & Business Media.
[29] Wazwaz, A.-M., A new algorithm for solving differential equations of Lane–Emden type. Applied mathematics and computation, 2001. 118(2-3): p. 287-310.
[30] Wazwaz, A.-M. and S.M. El-Sayed, A new modification of the Adomian decomposition method for linear and nonlinear operators. Applied Mathematics and computation, 2001. 122(3): p. 393-405.
[31] Yang, Y.-T. and S.-K. Chien, A double decomposition method for solving the periodic base temperature in convective longitudinal fins. Energy Conversion and Management, 2008. 49(10): p. 2910-2916.
[32] Khuri, S.A., A Laplace decomposition algorithm applied to a class of nonlinear differential equations. Journal of Applied Mathematics, 2001. 1(4): p. 141-155.
[33] Tsai, P.Y. and Chen C.K., Free vibration of the nonlinear pendulum using hybrid Laplace Adomian decomposition method. International journal for numerical methods in biomedical engineering, 2011. 27(2): p. 262-272.
[34] 陳柏維, Laplace Adomian 混合分解法應用於邊界層流流經楔型板之研究. 2011.
[35] 吳昱成, Laplace/Adomian 混合解法應用於非牛頓冪次律流體流經楔型板之邊界層流受磁場影響之熱流特性研究. 2019.
[36] Tamm, G., et al., On the similarity solution for condensation heat transfer. Journal of heat transfer, 2009. 131(11).
[37] Xu, H., X.-C. You, and I. Pop, Analytical approximation for laminar film condensation of saturated stream on an isothermal vertical plate. Applied mathematical modelling, 2008. 32(5): p. 738-748.
[38] Durlofsky, L. and J. Brady, The spatial stability of a class of similarity solutions. The Physics of , 1984. 27(5): p. 1068-1076.