能源利用问题的代数显式解析解
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摘要
解析解有其不可代替的理论价值。流动与传热的各种基本方程的解析解，历史上对学科的发展曾起过非常关键的作用。由于它们严格表达了该方程在某一特定条件下的详尽准确情况，解析解还可以用来检查各种数值计算方法的准确度、收敛度与有效度，以及作为研究不同数值解法的基础，启发如何改进其差分格式、网格生成等。所以，即使对近年来迅速发展的计算流体力学与计算传热学，解析解也是非常有用的。而代数显式解析解尤其适合用于理论研究与作为数值计算的标准解。尽管如此，由于解析求解能源利用问题的偏微分方程在数学上较为困难，国际上的公开文献中仅有少量有关的代数显式解析解报道。本学位论文依托国家自然科学基金(No.50246003，No.50576097)和国家重大基础研究发展规划项目(No.G20000263)等科研任务，对能源利用的代数显式解析解问题进行了深入研究，涉及导热、对流换热、传质、非牛顿流体运动等领域。主要研究内容如下：
双曲型(热波)方程作为一种典型的非Fourier热传导模型得到了学术界的普遍关注。本研究对二维和三维双曲型热传导方程给出了一些含有任意函数的解析解。对于二维的情况，解中含有8个任意函数。对于三维的情况，解中含有无穷多个任意函数。初步讨论了这些解的边界和初始条件。通过对这些解的分析指出，当内热源以特定方式衰减时，无论其几何分布如何，它对于温度分布都不会有影响。这是由双曲型热传导方程导出的一个特有现象。
Chen-Holmes方程也许是现有最为完善的灌流组织传热模型。本研究导出了考虑非Fourier效应时Chen-Holmes生物传热模型特定热物性条件下的通解，以加深对该模型的了解，丰富生物传热学理论。由此通解可得到带有热波的解，热波解及其存在条件的生理学含义为，对于肿瘤、大脑等具有高血液灌注率的部位，由初始温度分布不均趋于温度均匀的过程中，考虑非Fourier效应时，组织内各点温度可能会在平衡温度附近上下振荡，而非直接过渡到平衡温度。
Pennes方程是目前应用最广泛的生物传热模型。本研究由Chen-Holmes方程通解出发，得到了考虑非Fourier效应时特定热物性条件下Pennes生物传热模型的通解，该解反映了对于任意的边界条件与初始条件，由Pennes方程所揭示的生物组织内的温度分布。
王补宣院士的多孔介质方程是一种富有代表性的生物传热模型。该方程的推导不涉及Darcy定律，也不限于牛顿流体，因而具有很好的通用性。本研究给出了非Fourier效应下该方程在一定内热源条件下的显式解析通解，以拓展对于生物传热这一高度复杂现象的认识。该解反映了对于任意的边界条件与初始条件，当总的内热产满足特定约束条件时，由王补宣方程所揭示的生物组织内的温度分布。当导热系数与体积比热为温度的函数时，非Fourier导热主控方程成为非线性偏微分方程，求得其解析解较为困难。根据作者的了解，国际上公开文献中极少有非线性非Fourier导热的显式解析解的报道。本研究对其导出了一些代数显式解析解，以发展相关理论并为数值计算提供标准解。
Brinkman模型是Darcy模型的一种改进模型，它可以反映一些各向异性与非Darcy效应(例如非滑移界面效应)。为更严格、准确地探讨Brinkman模型所反映的规律性，本研究推导并得出了该模型基本方程组的一些代数显式解析解。第一解的物理图景为在两块与y轴平行且相距δ的无限长固壁之间的、温度均匀的Brinkman流动。第二解代表多孔介质中一种在两块与x轴平行的无限长的可渗透壁之间的Brinkman自然对流。第三解可代表多孔介质中在两块平行于y轴的无限长固壁之间的Brinkman自然对流，且温度分布是线性的。
本研究给出了两套轴对称定常层流自然对流代数显式解析解以加强对这种流动的基础了解。第一解描述了一垂直无限长下移冷多孔介质圆管外半无限空间的有边界层假设的自然对流。第二解描述了两垂直同心圆管之间的自然对流。
本研究对对流进行了热力学上的严格讨论，说明了对流不是一种真正的传“热”方式，而主要是一种通过粒子运动输运内能的方式。基于对对流换热的讨论，从物理意义上阐释了场协同概念——当速度矢量处处与等温线垂直时，可获得最好的对流换热效果。为了进一步发展场协同理论，研究实现场协同的方法，导出了各种场协同解，包括具有热源的解与具有质量源的解以及边界协同解。
对流强化传热是当前学术界的研究热点之一。本研究对两平行可渗透壁之间的二维对流换热导出了一些解析解，并运用单相强化传热的统一理论——场协同原理进行了分析。这些结果有助于启发增强或削弱场协同程度的实用方法。本研究还讨论了一些因素对于换热强度、场协同度等的影响。分析表明对于这种流动场协同度可能会对不同壁面的换热条件具有不同的影响，此外局部场协同程度在一些情况下可能比整个流动区域的场协同程度更有意义。
就主控方程而言，非稳态对流比稳态对流问题更为复杂。一般而言，稳态对流问题可认为是非稳态问题的特例。本研究给出了一些在两平行壁面之间的非稳态二维对流换热的解析解，同时由这些解得出了一些有意义的结论。例如第一解表明非稳态对流换热在一定条件下可能退化为非稳态导热问题，此时流体运动对换热没有贡献。第三解指出非稳态流体运动可用于弱化换热。
本研究对考虑非Fourier效应和非Fick效应的热质耦合方程组导出了两组解析解。这些解有助于加深对多孔介质干燥中非Fourier非Fick超常传热传质过程的理解，同时可作为标准解校核数值计算结果，具有较好的参考价值。
双扩散对流的主控方程组相当复杂，为数学上三维的非线性偏微分方程组。本论文推导了双扩散对流的两套代数显式解析解。第一套解为无限长的柱形管中的双扩散对流；另一套解为无限长的具有多孔介质壁面的环管中的双扩散对流。它们对于传热传质理论具有重要的意义。
复杂流变液体种类繁多，一般以非牛顿本构方程加以描述。理解流变液体的各种流动现象有助于推动一些学科和产业的发展。本研究对环管中的Oldroyd-B型不可压非定常旋流推导出了两个代数显式解析解。尽管流变液体主控方程较为复杂，这些解析解却非常简单。此外，本文还导出了广义二阶流变流的一个定常解。
Analytical solutions have irreplaceable theoretical vaule. Many analytical solutions played key roles in the early development of fluid mechanics and heat conduction. Besides their theoretical meaning, analytical solutions can also be applied to check the accuracy, convergence and effectiveness of various numerical computation methods and to improve their differencing schemes, grid generation ways and so on. This is due to they can express the detailed analytical situation under certain initial and boundary conditions. The analytical solutions are therefore very useful even for the newly rapidly developing computational fluid dynamics and heat transfer. Especially, algebraically explicit analytical solutions are more suitable for theoretical research or to be used as benchmark solutions to check numerical calculations. Nevertheless, there have been few algebraically explicit analytical solutions reported in open literature up to now, due to mathematical difficulties. Supported by the National Natural Science Foundation of China (No.50246003, No.50576097) and the Major State Basic Research Development Program of China (No.G20000263), the dissertation researches algebraically explicit analytical solutions of energy utilization, concerning fields such as heat conduction, convection, mass transfer, non-Newtonian fluid flow and so on. The main contents are as follows:
Hyperbolic (thermal wave) equation has been widely concerned in the research community as a typical non-Fourier heat conduction model. Some analytical solutions with arbitrary functions for 2-D and 3-D hyperbolic heat conduction equation are presented. Especially, for 2-D case, 8 arbitrary functions are included in the final solution. For 3-D case, infinite arbitrary functions are included in the final solution. Some special boundary and initial conditions are discussed. It is pointed out that when the internal heat source attenuates with special manner, it would have no effect on the temperature distribution. This is a special feature of the hyperbolic heat conduction equation.
Chen-Holmes equation may be the optimum perfusion heat transfer model up to now. In order to further expand the understanding of this model and to enrich the theoretical research, an analytical general solution under certain kind of relationship between thermal parameters is presented for the non-Fourier Chen-Holmes model. It is possible to get special solutions with heat wave effect from it. The physiology meaning of the heat wave solutions is that for biotissue with high blood perfusion rate such as tumour and cerebrum, because of non-Fourier effect, the temperature may fluctuate around equilibrium point.
Pennes equation is the most widely used bioheat transfer model nowadays. An analytical general solution under certain kind of relationship between thermal parameters is presented for the non-Fourier Pennes model based on the abovementioned general solution of Chen-Holmes equation. The solution depicts the temperature distribution in biotissue medium for any boundary conditions and initial conditions discovered by the non-Fourier Pennes equation.
The porous medium Wang model is a representative bioheat transfer model. It is more general since its derivation has no relationship with Darcy Law and its application is not confined to Newtonian fluid. This dissertation presents an analytical general solution for the non-Fourier Wang equation with special internal heat production to further expand the understanding of the highly complex bioheat transfer phenomena. The solution depicts the temperature distribution in biotissue medium for any boundary conditions and initial conditions discovered by the non-Fourier Wang equation with special internal heat production.
When the thermal conductivity and volumetric specific heat are functions of temperature, the governing equation of non-Fourier heat conduction is nonlinear. Therefore it is difficult to find its analytical solutions. Several algebraically explicit analytical solutions of nonlinear non-Fourier heat conduction are derived, both to develop theory and to serve as benchmark solutions for numerical calculations.
Brinkman model is one of the improvements of Darcy model. It can reflect some anisotropic and non-Darcy effects (such as the no slip boundary effect). For the sake of more strictly and accurately studying the regularity reflected by Brinkman model, some algebraically explicit analytical solutions of the governing equation set are derived and presented. The physical feature of the first solution is a uniform temperature Brinkman flow in a channel between two infinite long solid walls parallel with y coordinate. The second solution represents a Brinkman natural convection flow in porous media between two infinite long permeable walls parallel to x coordinate. The third solution can represent a Brinkman natural convection in porous media between two infinite long solid walls parallel to y coordinate, and the temperature distribution is linear.
Two algebraically explicit analytical solutions of axisymmetric steady laminar natural convection are derived to develop the theoretical understanding. The first analytical solution represents the natural convection in a semi-infinite space with boundary suction along an infinitely long vertical cold moving down porous tube. The second solution describes the natural convection between two infinite concentric vertical solid tubes.
A thermodynamically strict discussion is given about the convection. It is pointed out that the convection is not a rigorous kind of "heat" transfer but mainly internal energy transfer by the movement of particles. Based on the discussion of convection, the concept of field synergy—the best convection "heat" transfer is the case where the velocity vectors are always perpendicular to the isothermal surfaces—is easy to understand. For further theoretically developing the field synergy principle and researching the artificial measures to accomplish field synergy, different kinds of algebraically explicit analytical solutions are derived and given, including solutions with heat source, with mass source, full field synergy solutions and boundary field synergy solutions.
Convective heat transfer enhancement is one of the research hot spots nowadays in the research community. Some analytical solutions for 2-D convective heat transfer between two parallel penetrable walls are presented and analysed using field synergy principle in this study. These results are valuable to inspire the methods to improve or weaken field synergy in practice. The influences of some factors on heat transfer and field synergy number are also discussed. It is demonstrated that the field synergy degree might have different influences on the heat transfer conditions of different walls for this kind of flow. It is also pointed out that the local field synergy degree might be more meaningful than the field synergy degree in the whole domain in some cases.
In regard to the governing equations, unsteady convection problems are more complex than steady ones. Generally speaking, steady convection problems can be regarded as simple special cases of unsteady ones. Some analytical solutions for unsteady 2-D convective heat transfer between two parallel walls are presented. And also some meaningful conclusions can be drawn form them. For example, the first solution demonstrates that the unsteady convective heat transfer could degenerate into an unsteady heat conduction problem under some conditions. Then the fluid flow would contribute nothing to the heat transfer. The third solution demonstrates that the unsteady fluid motion could be utilized to weaken the heat transfer.
Two algebraically explicit analytical solutions of the equation set describing non-Fourier and non-Fick heat and mass transfer in capillary porous media are reported. These analytical solutions are useful to deepen the understanding of non-Fourier and non-Fick heat and mass transfer in the rapid drying process of porous media. They can also be applied as benchmark solutions to check the numerical computation results. Therefore, these analytical solutions are of high value.
The governing equation set for the double diffusive convection is rather complicated. It is a nonlinear mathematical 3-D equation set. Two algebraically explicit analytical solutions for the double diffusive convection are derived. One is for the double diffusive convection in an infinite long cylindrical tube. Another one is for the double diffusive convection in an infinite long circular tube with porous wall. They are meaningful for the theory of heat and mass transfer.
There are many different kinds of rheological complicated fluids with non-Newtonian constitutive equations. The understanding of the flow phenomena of these fluids is helpful for many disciplines. Two algebraically explicit analytical solutions are derived for the incompressible unsteady rotational flow of a rheologic flow of Oldroyd-B type in an annular pipe. Even for complicated rheologic governing equation, the solutions are very simple. In addition, a similar solution is also derived for the generalized second grade rheologic fluid flow.
双曲型(热波)方程作为一种典型的非Fourier热传导模型得到了学术界的普遍关注。本研究对二维和三维双曲型热传导方程给出了一些含有任意函数的解析解。对于二维的情况，解中含有8个任意函数。对于三维的情况，解中含有无穷多个任意函数。初步讨论了这些解的边界和初始条件。通过对这些解的分析指出，当内热源以特定方式衰减时，无论其几何分布如何，它对于温度分布都不会有影响。这是由双曲型热传导方程导出的一个特有现象。
Chen-Holmes方程也许是现有最为完善的灌流组织传热模型。本研究导出了考虑非Fourier效应时Chen-Holmes生物传热模型特定热物性条件下的通解，以加深对该模型的了解，丰富生物传热学理论。由此通解可得到带有热波的解，热波解及其存在条件的生理学含义为，对于肿瘤、大脑等具有高血液灌注率的部位，由初始温度分布不均趋于温度均匀的过程中，考虑非Fourier效应时，组织内各点温度可能会在平衡温度附近上下振荡，而非直接过渡到平衡温度。
Pennes方程是目前应用最广泛的生物传热模型。本研究由Chen-Holmes方程通解出发，得到了考虑非Fourier效应时特定热物性条件下Pennes生物传热模型的通解，该解反映了对于任意的边界条件与初始条件，由Pennes方程所揭示的生物组织内的温度分布。
王补宣院士的多孔介质方程是一种富有代表性的生物传热模型。该方程的推导不涉及Darcy定律，也不限于牛顿流体，因而具有很好的通用性。本研究给出了非Fourier效应下该方程在一定内热源条件下的显式解析通解，以拓展对于生物传热这一高度复杂现象的认识。该解反映了对于任意的边界条件与初始条件，当总的内热产满足特定约束条件时，由王补宣方程所揭示的生物组织内的温度分布。当导热系数与体积比热为温度的函数时，非Fourier导热主控方程成为非线性偏微分方程，求得其解析解较为困难。根据作者的了解，国际上公开文献中极少有非线性非Fourier导热的显式解析解的报道。本研究对其导出了一些代数显式解析解，以发展相关理论并为数值计算提供标准解。
Brinkman模型是Darcy模型的一种改进模型，它可以反映一些各向异性与非Darcy效应(例如非滑移界面效应)。为更严格、准确地探讨Brinkman模型所反映的规律性，本研究推导并得出了该模型基本方程组的一些代数显式解析解。第一解的物理图景为在两块与y轴平行且相距δ的无限长固壁之间的、温度均匀的Brinkman流动。第二解代表多孔介质中一种在两块与x轴平行的无限长的可渗透壁之间的Brinkman自然对流。第三解可代表多孔介质中在两块平行于y轴的无限长固壁之间的Brinkman自然对流，且温度分布是线性的。
本研究给出了两套轴对称定常层流自然对流代数显式解析解以加强对这种流动的基础了解。第一解描述了一垂直无限长下移冷多孔介质圆管外半无限空间的有边界层假设的自然对流。第二解描述了两垂直同心圆管之间的自然对流。
本研究对对流进行了热力学上的严格讨论，说明了对流不是一种真正的传“热”方式，而主要是一种通过粒子运动输运内能的方式。基于对对流换热的讨论，从物理意义上阐释了场协同概念——当速度矢量处处与等温线垂直时，可获得最好的对流换热效果。为了进一步发展场协同理论，研究实现场协同的方法，导出了各种场协同解，包括具有热源的解与具有质量源的解以及边界协同解。
对流强化传热是当前学术界的研究热点之一。本研究对两平行可渗透壁之间的二维对流换热导出了一些解析解，并运用单相强化传热的统一理论——场协同原理进行了分析。这些结果有助于启发增强或削弱场协同程度的实用方法。本研究还讨论了一些因素对于换热强度、场协同度等的影响。分析表明对于这种流动场协同度可能会对不同壁面的换热条件具有不同的影响，此外局部场协同程度在一些情况下可能比整个流动区域的场协同程度更有意义。
就主控方程而言，非稳态对流比稳态对流问题更为复杂。一般而言，稳态对流问题可认为是非稳态问题的特例。本研究给出了一些在两平行壁面之间的非稳态二维对流换热的解析解，同时由这些解得出了一些有意义的结论。例如第一解表明非稳态对流换热在一定条件下可能退化为非稳态导热问题，此时流体运动对换热没有贡献。第三解指出非稳态流体运动可用于弱化换热。
本研究对考虑非Fourier效应和非Fick效应的热质耦合方程组导出了两组解析解。这些解有助于加深对多孔介质干燥中非Fourier非Fick超常传热传质过程的理解，同时可作为标准解校核数值计算结果，具有较好的参考价值。
双扩散对流的主控方程组相当复杂，为数学上三维的非线性偏微分方程组。本论文推导了双扩散对流的两套代数显式解析解。第一套解为无限长的柱形管中的双扩散对流；另一套解为无限长的具有多孔介质壁面的环管中的双扩散对流。它们对于传热传质理论具有重要的意义。
复杂流变液体种类繁多，一般以非牛顿本构方程加以描述。理解流变液体的各种流动现象有助于推动一些学科和产业的发展。本研究对环管中的Oldroyd-B型不可压非定常旋流推导出了两个代数显式解析解。尽管流变液体主控方程较为复杂，这些解析解却非常简单。此外，本文还导出了广义二阶流变流的一个定常解。
Analytical solutions have irreplaceable theoretical vaule. Many analytical solutions played key roles in the early development of fluid mechanics and heat conduction. Besides their theoretical meaning, analytical solutions can also be applied to check the accuracy, convergence and effectiveness of various numerical computation methods and to improve their differencing schemes, grid generation ways and so on. This is due to they can express the detailed analytical situation under certain initial and boundary conditions. The analytical solutions are therefore very useful even for the newly rapidly developing computational fluid dynamics and heat transfer. Especially, algebraically explicit analytical solutions are more suitable for theoretical research or to be used as benchmark solutions to check numerical calculations. Nevertheless, there have been few algebraically explicit analytical solutions reported in open literature up to now, due to mathematical difficulties. Supported by the National Natural Science Foundation of China (No.50246003, No.50576097) and the Major State Basic Research Development Program of China (No.G20000263), the dissertation researches algebraically explicit analytical solutions of energy utilization, concerning fields such as heat conduction, convection, mass transfer, non-Newtonian fluid flow and so on. The main contents are as follows:
Hyperbolic (thermal wave) equation has been widely concerned in the research community as a typical non-Fourier heat conduction model. Some analytical solutions with arbitrary functions for 2-D and 3-D hyperbolic heat conduction equation are presented. Especially, for 2-D case, 8 arbitrary functions are included in the final solution. For 3-D case, infinite arbitrary functions are included in the final solution. Some special boundary and initial conditions are discussed. It is pointed out that when the internal heat source attenuates with special manner, it would have no effect on the temperature distribution. This is a special feature of the hyperbolic heat conduction equation.
Chen-Holmes equation may be the optimum perfusion heat transfer model up to now. In order to further expand the understanding of this model and to enrich the theoretical research, an analytical general solution under certain kind of relationship between thermal parameters is presented for the non-Fourier Chen-Holmes model. It is possible to get special solutions with heat wave effect from it. The physiology meaning of the heat wave solutions is that for biotissue with high blood perfusion rate such as tumour and cerebrum, because of non-Fourier effect, the temperature may fluctuate around equilibrium point.
Pennes equation is the most widely used bioheat transfer model nowadays. An analytical general solution under certain kind of relationship between thermal parameters is presented for the non-Fourier Pennes model based on the abovementioned general solution of Chen-Holmes equation. The solution depicts the temperature distribution in biotissue medium for any boundary conditions and initial conditions discovered by the non-Fourier Pennes equation.
The porous medium Wang model is a representative bioheat transfer model. It is more general since its derivation has no relationship with Darcy Law and its application is not confined to Newtonian fluid. This dissertation presents an analytical general solution for the non-Fourier Wang equation with special internal heat production to further expand the understanding of the highly complex bioheat transfer phenomena. The solution depicts the temperature distribution in biotissue medium for any boundary conditions and initial conditions discovered by the non-Fourier Wang equation with special internal heat production.
When the thermal conductivity and volumetric specific heat are functions of temperature, the governing equation of non-Fourier heat conduction is nonlinear. Therefore it is difficult to find its analytical solutions. Several algebraically explicit analytical solutions of nonlinear non-Fourier heat conduction are derived, both to develop theory and to serve as benchmark solutions for numerical calculations.
Brinkman model is one of the improvements of Darcy model. It can reflect some anisotropic and non-Darcy effects (such as the no slip boundary effect). For the sake of more strictly and accurately studying the regularity reflected by Brinkman model, some algebraically explicit analytical solutions of the governing equation set are derived and presented. The physical feature of the first solution is a uniform temperature Brinkman flow in a channel between two infinite long solid walls parallel with y coordinate. The second solution represents a Brinkman natural convection flow in porous media between two infinite long permeable walls parallel to x coordinate. The third solution can represent a Brinkman natural convection in porous media between two infinite long solid walls parallel to y coordinate, and the temperature distribution is linear.
Two algebraically explicit analytical solutions of axisymmetric steady laminar natural convection are derived to develop the theoretical understanding. The first analytical solution represents the natural convection in a semi-infinite space with boundary suction along an infinitely long vertical cold moving down porous tube. The second solution describes the natural convection between two infinite concentric vertical solid tubes.
A thermodynamically strict discussion is given about the convection. It is pointed out that the convection is not a rigorous kind of "heat" transfer but mainly internal energy transfer by the movement of particles. Based on the discussion of convection, the concept of field synergy—the best convection "heat" transfer is the case where the velocity vectors are always perpendicular to the isothermal surfaces—is easy to understand. For further theoretically developing the field synergy principle and researching the artificial measures to accomplish field synergy, different kinds of algebraically explicit analytical solutions are derived and given, including solutions with heat source, with mass source, full field synergy solutions and boundary field synergy solutions.
Convective heat transfer enhancement is one of the research hot spots nowadays in the research community. Some analytical solutions for 2-D convective heat transfer between two parallel penetrable walls are presented and analysed using field synergy principle in this study. These results are valuable to inspire the methods to improve or weaken field synergy in practice. The influences of some factors on heat transfer and field synergy number are also discussed. It is demonstrated that the field synergy degree might have different influences on the heat transfer conditions of different walls for this kind of flow. It is also pointed out that the local field synergy degree might be more meaningful than the field synergy degree in the whole domain in some cases.
In regard to the governing equations, unsteady convection problems are more complex than steady ones. Generally speaking, steady convection problems can be regarded as simple special cases of unsteady ones. Some analytical solutions for unsteady 2-D convective heat transfer between two parallel walls are presented. And also some meaningful conclusions can be drawn form them. For example, the first solution demonstrates that the unsteady convective heat transfer could degenerate into an unsteady heat conduction problem under some conditions. Then the fluid flow would contribute nothing to the heat transfer. The third solution demonstrates that the unsteady fluid motion could be utilized to weaken the heat transfer.
Two algebraically explicit analytical solutions of the equation set describing non-Fourier and non-Fick heat and mass transfer in capillary porous media are reported. These analytical solutions are useful to deepen the understanding of non-Fourier and non-Fick heat and mass transfer in the rapid drying process of porous media. They can also be applied as benchmark solutions to check the numerical computation results. Therefore, these analytical solutions are of high value.
The governing equation set for the double diffusive convection is rather complicated. It is a nonlinear mathematical 3-D equation set. Two algebraically explicit analytical solutions for the double diffusive convection are derived. One is for the double diffusive convection in an infinite long cylindrical tube. Another one is for the double diffusive convection in an infinite long circular tube with porous wall. They are meaningful for the theory of heat and mass transfer.
There are many different kinds of rheological complicated fluids with non-Newtonian constitutive equations. The understanding of the flow phenomena of these fluids is helpful for many disciplines. Two algebraically explicit analytical solutions are derived for the incompressible unsteady rotational flow of a rheologic flow of Oldroyd-B type in an annular pipe. Even for complicated rheologic governing equation, the solutions are very simple. In addition, a similar solution is also derived for the generalized second grade rheologic fluid flow.
引文
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6 Kosugi T, Pak PS. Economic evaluation of solar thermal hybrid H_2O turbine power generation systems. Int. J. Energy 2003; 28: 185-198.
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8 Cai R, Jiang L. Analysis of the recuperative gas turbine cycle with a recuperator located between turbines. Applied Thermal Engineering 2006; 26: 89-96.
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