{"id":71843,"date":"2016-06-07T08:10:24","date_gmt":"2016-06-07T00:10:24","guid":{"rendered":"http:\/\/highscope.ch.ntu.edu.tw\/wordpress\/?p=71843"},"modified":"2021-10-06T16:09:28","modified_gmt":"2021-10-06T08:09:28","slug":"%e9%80%a3%e7%ba%8c%e6%96%b9%e7%a8%8b%e5%bc%8f","status":"publish","type":"post","link":"http:\/\/localhost\/%e9%80%a3%e7%ba%8c%e6%96%b9%e7%a8%8b%e5%bc%8f\/","title":{"rendered":"\u9023\u7e8c\u65b9\u7a0b\u5f0f"},"content":{"rendered":"

\u9023\u7e8c\u65b9\u7a0b\u5f0f (Continuity Equation)<\/strong><\/span>
\n\u570b\u7acb\u81fa\u7063\u5927\u5b78\u5927\u6c23\u79d1\u5b78\u7cfb \u9673\u54c1\u5168<\/strong><\/span><\/p>\n

\u5728\u5f62\u5bb9\u7ba1\u4e2d\u4e00\u500b\u4e0d\u53ef\u58d3\u7e2e\u7684\u6d41\u9ad4\u4e4b\u904b\u52d5\u6642\uff0c\u6211\u5011\u5e38\u5e38\u6703\u4f7f\u7528\u5230\u4e00\u500b\u516c\u5f0f\uff1a<\/p>\n

$$A_1v_1=A_2v_2$$<\/p>\n

\"71843_p1\"<\/a>

\u5716\u4e00\u3001\u9023\u7e8c\u65b9\u7a0b\u5f0f\u793a\u610f\u5716\uff08\u672c\u6587\u4f5c\u8005\u9673\u54c1\u5168\u7e6a\u88fd\uff09<\/p><\/div>\n

\u6b64\u516c\u5f0f\u70ba\u9023\u7e8c\u65b9\u7a0b\u5f0f\u7684\u4e00\u500b\u7279\u4f8b\uff0c\u4ee3\u8868\u6d41\u9ad4\u7684\u8cea\u91cf\u5b88\u6046\u5f8b\u3002\u8981\u770b\u51fa\u9019\u4e00\u9ede\uff0c\u6211\u5011\u5728\u9019\u500b\u6d41\u7ba1\u53d6\u4e00\u500b\u5047\u60f3\u7684\u63a7\u5236\u9ad4\u7a4d (Control Volume)\uff0c\u5176\u4e2d $$A_1$$\u00a0\u548c $$A_2$$\u00a0\u5206\u5225\u662f\u5de6\u908a\u548c\u53f3\u908a\u6d41\u7ba1\u622a\u9762\u7a4d\uff0c\u800c $$v_1$$\u00a0\u548c $$v_2$$\u00a0\u5247\u662f\u6d41\u9ad4\u7684\u6d41\u9032\u548c\u6d41\u51fa\u901f\u7387\u3002\u5728\u55ae\u4f4d\u6642\u9593\u5167\uff0c\u6d41\u9032\u7684\u9ad4\u7a4d $$A_1v_1\\Delta t$$ \u6703\u7b49\u65bc\u6d41\u51fa\u7684\u9ad4\u7a4d $$A_2v_2\\Delta t$$\uff0c\u5982\u6b64\u4e00\u4f86\u6b64\u63a7\u5236\u9ad4\u7a4d\u5167\u7684\u6d41\u9ad4\u91cf\u624d\u6703\u7dad\u6301\u70ba\u5b9a\u503c \uff08\u7a69\u5b9a\u6d41\uff0cSteady flow\uff09\u3002<\/p>\n

\u5c0d\u65bc\u66f4\u4e00\u822c\u7684\u60c5\u6cc1\uff0c\u6211\u5011\u4e5f\u53ef\u4ee5\u6709\u66f4\u4e00\u822c\u6027 (general) \u7684\u9023\u7e8c\u65b9\u7a0b\u5f0f\u4f86\u5f62\u5bb9\u6d41\u9ad4\u3002\u6211\u5011\u53ef\u4ee5\u60f3\u50cf\u5728\u4e00\u500b\u4e09\u7dad\u7684\u6d41\u9ad4\u4e2d\u53d6\u4e00\u500b\u5c0f\u9ad4\u7a4d $$\\delta V$$ \u4f5c\u70ba\u63a7\u5236\u9ad4\u7a4d\uff0c\u800c\u9019\u500b\u9ad4\u7a4d\u5167\u542b\u6709\u7684\u6d41\u9ad4\u8cea\u91cf\u5247\u662f $$\\delta M=\\rho\\delta V$$\u3002\u56e0\u70ba\u63a7\u5236\u9ad4\u7a4d\u5167\u7684\u8cea\u91cf $$\\delta M$$ \u8981\u5b88\u6046\uff0c\u6240\u4ee5 $$\\frac{d(\\delta M)}{dt}=0$$\u3002\u5229\u7528\u9019\u500b\u7b49\u5f0f\uff0c\u6211\u5011\u53ef\u4ee5\u5c0e\u51fa\uff1a<\/p>\n

$$\\begin{array}{ll}\\displaystyle\\frac{1}{\\delta M}\\frac{d(\\delta M)}{dt}&\\displaystyle=\\frac{\\left[\\frac{\\rho d(\\delta V)}{dt}+\\frac{d\\rho}{dt}\\delta V\\right]}{\\rho\\delta V}\\\\&\\displaystyle=\\frac{1}{\\delta V}\\frac{d(\\delta V)}{dt}+\\frac{1}{\\rho}\\frac{d\\rho}{dt}=0\\end{array}$$<\/p>\n

\u5c0d\u65bc $$\\displaystyle\\frac{1}{\\delta V}\\frac{d(\\delta V)}{dt}$$\uff0c\u6211\u5011\u53ef\u4ee5\u5c07\u63a7\u5236\u9ad4\u7a4d\u5047\u60f3\u662f\u4e00\u500b\u5f88\u5c0f\u7684\u9577\u65b9\u9ad4\uff0c\u908a\u9577\u5206\u5225\u70ba $$\\delta x,\\delta y,\\delta z$$\uff08\u4e5f\u5c31\u662f\u5206\u5225\u70ba $$x$$\u3001$$y$$\u3001$$z$$ \u65b9\u5411\u7684\u5fae\u5c0f\u8b8a\u5316\uff09\u3002<\/p>\n

\"71843_p2\"<\/a>

\u5716\u4e8c\u3001\u6975\u5c0f\u7684\u9577\u65b9\u9ad4 (\u672c\u6587\u4f5c\u8005\u9673\u54c1\u5168\u7e6a\u88fd)<\/p><\/div>\n

$$\\begin{array}{ll}\\displaystyle\\frac{1}{\\delta V}\\frac{d(\\delta V)}{dt}&\\displaystyle=\\frac{1}{\\delta x\\delta y\\delta z}\\frac{d(\\delta x\\delta y\\delta z)}{dt}\\\\&\\displaystyle=\\frac{\\delta u\\delta y\\delta z+\\delta x\\delta v\\delta z+\\delta x\\delta y\\delta w}{\\delta x\\delta y\\delta z}\\\\&\\displaystyle=\\frac{\\delta u}{\\delta x}+\\frac{\\delta v}{\\delta y}+\\frac{\\delta w}{\\delta z}\\xrightarrow{\\delta x,\\delta y,\\delta z\\to 0}\\frac{\\partial u}{\\partial x}+\\frac{\\partial v}{\\partial y}+\\frac{\\partial w}{\\partial z}=\\nabla\\cdot\\vec{v}\\end{array}$$<\/p>\n

$$u,v,w$$ \u5206\u5225\u662f $$x,y,z$$ \u65b9\u5411\u7684\u901f\u5ea6\uff0c\u800c\u6211\u5011\u9019\u4e00\u9805\u6b63\u597d\u662f\u901f\u5ea6\u7684\u6563\u5ea6 (divergence)\u3002\u65bc\u662f\u6211\u5011\u5f97\u5230\u4e86\u4e00\u500b\u53ef\u4ee5\u5f62\u5bb9\u4e09\u7dad\u6d41\u9ad4\u7684\u9023\u7e8c\u65b9\u7a0b\u5f0f\uff1a<\/p>\n

$$\\displaystyle\\frac{1}{\\rho}\\frac{d\\rho}{dt}+\\nabla\\cdot\\vec{v}=0$$<\/p>\n

\u9019\u4e00\u689d\u9023\u7e8c\u65b9\u7a0b\u5f0f\u56e0\u70ba\u5305\u542b\u4e86 $$\\frac{1}{\\rho}\\frac{d\\rho}{dt}$$\uff0c\u56e0\u6b64\u662f\u53ef\u4ee5\u7528\u4f86\u5f62\u5bb9\u975e\u7a69\u5b9a\u7684\u6d41\u9ad4 (Non-steady flow)\u3002\u5982\u679c\u6d41\u9ad4\u901f\u5ea6\u7684\u6563\u5ea6\u5927\u65bc\u96f6\u7684\u8a71\uff0c\u5247\u4ee3\u8868\u9019\u500b\u63a7\u5236\u9ad4\u7a4d\u5167\u7684\u6d41\u9ad4\u6d41\u51fa\u7387\u5927\u65bc\u6d41\u5165\u7387\uff0c\u6240\u4ee5\u9019\u500b\u63a7\u5236\u9ad4\u7a4d\u5167\u7684\u5bc6\u5ea6\u6703\u96a8\u8457\u6642\u9593\u4e0b\u964d\u3002<\/p>\n

\u9019\u500b\u516c\u5f0f\u9084\u53ef\u4ee5\u6709\u4e00\u500b\u8b8a\u5f62\uff1a<\/p>\n

$$\\displaystyle\\frac{1}{\\rho}\\frac{d\\rho}{dt}+\\nabla\\cdot\\vec{v}=\\frac{1}{\\rho}\\left(\\frac{\\partial \\rho}{\\partial t}+v\\cdot\\nabla\\rho\\right)+\\nabla\\cdot\\vec{v}=0$$<\/p>\n

$$\\displaystyle\\rightarrow \\frac{\\partial \\rho}{\\partial t}+v\\cdot\\nabla\\rho+\\rho\\nabla\\cdot\\vec{v}=\\frac{\\partial \\rho}{\\partial t}+\\nabla\\cdot(\\rho\\vec{v})=0$$<\/p>\n

\u5c0d\u65bc\u4e0d\u53ef\u58d3\u7e2e\u7684\u6d41\u9ad4\uff0c$$\\frac{\\partial \\rho}{\\partial t}=0$$\uff0c\u56e0\u6b64 $$\\nabla\\cdot(\\rho\\vec{v})=0$$\u3002\u5728\u96fb\u78c1\u5b78\u4e2d\uff0c\u6211\u5011\u5e38\u5e38\u6703\u628a $$\\rho\\vec{v}$$ \u5beb\u6210 $$\\vec{J}$$\uff0c\u4e5f\u5c31\u662f\u6240\u8b02\u7684\u96fb\u6d41\u5bc6\u5ea6\u3002<\/p>\n

\u88dc\u5145\uff1aDivergence theorem<\/p>\n

\u5c0d\u65bc $$\\nabla\\cdot(\\rho\\vec{v})$$\uff0c\u6211\u5011\u53ef\u4ee5\u5229\u7528 divergence theorem \u5beb\u6210<\/p>\n

$$\\displaystyle\\nabla\\cdot(\\rho\\vec{v})=\\lim_{V\\to 0}\\int\\nabla\\cdot(\\rho\\vec{v})\\frac{dV}{V}=\\lim_{A\\to 0}\\oint(\\rho\\vec{v})\\cdot d\\vec{A}$$<\/p>\n

\u6545\u53ef\u4ee5\u60f3\u50cf\u6210\u662f\u5728\u4e00\u500b\u9ad4\u7a4d\u975e\u5e38\u5c0f\u7684\u63a7\u5236\u9ad4\u7a4d\u4e2d\uff0c\u770b\u770b\u6d41\u9ad4\u5bc6\u5ea6\u6d41\u5c0d\u65bc\u63a7\u5236\u9762\u7a4d\uff08Control surface\uff0c\u63a7\u5236\u9ad4\u7a4d\u7684\u8868\u9762\uff09\u7684\u7e3d\u901a\u91cf\u70ba\u4f55\uff08\u901a\u5e38\u6703\u9078\u64c7\u5411\u5916\u70ba\u6b63\uff09\u3002<\/p>\n

\u66f4\u8a73\u7d30\u7684\u4ecb\u7d39\u8acb\u898b\u300a\u6563\u5ea6\u548c\u65cb\u5ea6 (Divergence and Curl)\u300b\u3002<\/p>\n

\n

\u53c3\u8003\u6587\u737b<\/strong><\/p>\n

    \n
  1. \u694a\u660e\u4ec1\u3002Chapter 3, Kinematics of fluid flow. \u6d41\u9ad4\u529b\u5b78\u4e0a\u8ab2\u8b1b\u7fa9\u3002<\/li>\n
  2. Continuity and Conservation of Matter — CIVE1400: Fluid Mechanics, University of Leeds. http:\/\/www.efm.leeds.ac.uk\/CIVE\/CIVE1400\/Section3\/continuity.htm<\/li>\n<\/ol>\n<\/div>","protected":false},"excerpt":{"rendered":"

    \u9023\u7e8c\u65b9\u7a0b\u5f0f (Continuity Equation) \u570b\u7acb\u81fa\u7063\u5927\u5b78\u5927\u6c23\u79d1\u5b78\u7cfb \u9673\u54c1\u5168 \u5728\u5f62\u5bb9\u7ba1\u4e2d\u4e00\u500b\u4e0d\u53ef\u58d3\u7e2e…<\/p>\n","protected":false},"author":50,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[14,15,107],"tags":[10143,10141,10145,10142,10140,10144],"class_list":["post-71843","post","type-post","status-publish","format-standard","hentry","category-physics02","category-physics02-01","category-physics00","tag-control-surface","tag-control-volume","tag-steady-flow","tag-10142","tag-10140","tag-10144","loop-entry","cat-14","cat-15","cat-107","no-thumbnail"],"views":9570,"_links":{"self":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/71843","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/users\/50"}],"replies":[{"embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/comments?post=71843"}],"version-history":[{"count":1,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/71843\/revisions"}],"predecessor-version":[{"id":86320,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/71843\/revisions\/86320"}],"wp:attachment":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/media?parent=71843"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/categories?post=71843"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/tags?post=71843"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}