The Alekseevskii–Tate model is the most successful semi-hydrodynamic model applied to long-rod penetration into semi-infinite targets. However, due to the nonlinear nature of the equations, the rod(tail) velocity, pe...The Alekseevskii–Tate model is the most successful semi-hydrodynamic model applied to long-rod penetration into semi-infinite targets. However, due to the nonlinear nature of the equations, the rod(tail) velocity, penetration velocity, rod length, and penetration depth were obtained implicitly as a function of time and solved numerically By employing a linear approximation to the logarithmic relative rod length, we obtain two sets of explicit approximate algebraic solutions based on the implicit theoretica solution deduced from primitive equations. It is very convenient in the theoretical prediction of the Alekseevskii–Tate model to apply these simple algebraic solutions. In particular, approximate solution 1 shows good agreement with the theoretical(exact) solution, and the first-order perturbation solution obtained by Walters et al.(Int. J. Impac Eng. 33:837–846, 2006) can be deemed as a special form of approximate solution 1 in high-speed penetration. Meanwhile, with constant tail velocity and penetration velocity approximate solution 2 has very simple expressions, which is applicable for the qualitative analysis of long-rod penetration. Differences among these two approximate solutions and the theoretical(exact) solution and their respective scopes of application have been discussed, and the inferences with clear physical basis have been drawn. In addition, these two solutions and the first-order perturbation solution are applied to two cases with different initial impact velocity and different penetrator/target combinations to compare with the theoretical(exact) solution. Approximate solution 1 is much closer to the theoretical solution of the Alekseevskii–Tate model than the first-order perturbation solution in both cases, whilst approximate solution 2 brings us a more intuitive understanding of quasi-steady-state penetration.展开更多
WT5”BZ]In this paper, the flow in a rotating curved annular pipe is examined by a perturbation method. A second order perturbation solution is presented. The characteristics of the secondary flow and the axial flow a...WT5”BZ]In this paper, the flow in a rotating curved annular pipe is examined by a perturbation method. A second order perturbation solution is presented. The characteristics of the secondary flow and the axial flow are studied in detail. The study indicates that the loops of the secondary flow are more complex than those in a curved annular pipe without rotation and its numbers depend on the ratio of the Coriolis force to centrifugal force F. As F≈-1, the secondary flow has eight loops and its intensity reaches the minimum value, and the distribution of the axial flow is like that of the Poiseuille flow. The position of the maximum axial velocity is pushed to either outer bend or inner bend, which is also determined by F. [WT5”HZ]展开更多
This paper deals with how the purely mathematical approach can be used to solve transient-state instability problems of dissolution-timescale reactive infiltration(DTRI) in fluid-saturated porous rocks. Three key step...This paper deals with how the purely mathematical approach can be used to solve transient-state instability problems of dissolution-timescale reactive infiltration(DTRI) in fluid-saturated porous rocks. Three key steps involved in such an approach are:(1) to mathematically derive an analytical solution(known as the base solution or conventional solution) for a quasi-steady state problem of the dissolution timescale, which is viewed as a frozen state of the original transient-state instability problem;(2)to mathematically deduce a group of first-order perturbation partial-differential equations(PDEs) for the quasi-steady state problem;(3) to mathematically derive an analytical solution(known as the perturbation solution or unconventional solution) for this group of first-order perturbation PDEs. Because of difficulty in mathematically solving a transient-state instability problem of DTRI in general cases, only a special case, in which some nonlinear coupling between governing PDEs of the problem can be decoupled, is considered to illustrate these three key steps in this study. The related theoretical results demonstrated that the transient chemical dissolution front can become unstable in the DTRI system of large Zh numbers when the long wavelength perturbations are applied to the system. This new finding may lay the theoretical foundation for developing innovative technique to exploit shale gas resources in the deep Earth.展开更多
基金supported by the National Outstanding Young Scientist Foundation of China (Grant 11225213)the Key Subject "Computational Solid Mechanics" of China Academy of Engineering Physics
文摘The Alekseevskii–Tate model is the most successful semi-hydrodynamic model applied to long-rod penetration into semi-infinite targets. However, due to the nonlinear nature of the equations, the rod(tail) velocity, penetration velocity, rod length, and penetration depth were obtained implicitly as a function of time and solved numerically By employing a linear approximation to the logarithmic relative rod length, we obtain two sets of explicit approximate algebraic solutions based on the implicit theoretica solution deduced from primitive equations. It is very convenient in the theoretical prediction of the Alekseevskii–Tate model to apply these simple algebraic solutions. In particular, approximate solution 1 shows good agreement with the theoretical(exact) solution, and the first-order perturbation solution obtained by Walters et al.(Int. J. Impac Eng. 33:837–846, 2006) can be deemed as a special form of approximate solution 1 in high-speed penetration. Meanwhile, with constant tail velocity and penetration velocity approximate solution 2 has very simple expressions, which is applicable for the qualitative analysis of long-rod penetration. Differences among these two approximate solutions and the theoretical(exact) solution and their respective scopes of application have been discussed, and the inferences with clear physical basis have been drawn. In addition, these two solutions and the first-order perturbation solution are applied to two cases with different initial impact velocity and different penetrator/target combinations to compare with the theoretical(exact) solution. Approximate solution 1 is much closer to the theoretical solution of the Alekseevskii–Tate model than the first-order perturbation solution in both cases, whilst approximate solution 2 brings us a more intuitive understanding of quasi-steady-state penetration.
文摘WT5”BZ]In this paper, the flow in a rotating curved annular pipe is examined by a perturbation method. A second order perturbation solution is presented. The characteristics of the secondary flow and the axial flow are studied in detail. The study indicates that the loops of the secondary flow are more complex than those in a curved annular pipe without rotation and its numbers depend on the ratio of the Coriolis force to centrifugal force F. As F≈-1, the secondary flow has eight loops and its intensity reaches the minimum value, and the distribution of the axial flow is like that of the Poiseuille flow. The position of the maximum axial velocity is pushed to either outer bend or inner bend, which is also determined by F. [WT5”HZ]
基金supported by the National Natural Science Foundation of China(Grant No.11272359)。
文摘This paper deals with how the purely mathematical approach can be used to solve transient-state instability problems of dissolution-timescale reactive infiltration(DTRI) in fluid-saturated porous rocks. Three key steps involved in such an approach are:(1) to mathematically derive an analytical solution(known as the base solution or conventional solution) for a quasi-steady state problem of the dissolution timescale, which is viewed as a frozen state of the original transient-state instability problem;(2)to mathematically deduce a group of first-order perturbation partial-differential equations(PDEs) for the quasi-steady state problem;(3) to mathematically derive an analytical solution(known as the perturbation solution or unconventional solution) for this group of first-order perturbation PDEs. Because of difficulty in mathematically solving a transient-state instability problem of DTRI in general cases, only a special case, in which some nonlinear coupling between governing PDEs of the problem can be decoupled, is considered to illustrate these three key steps in this study. The related theoretical results demonstrated that the transient chemical dissolution front can become unstable in the DTRI system of large Zh numbers when the long wavelength perturbations are applied to the system. This new finding may lay the theoretical foundation for developing innovative technique to exploit shale gas resources in the deep Earth.
