{"id":209,"date":"2013-06-18T23:28:12","date_gmt":"2013-06-18T15:28:12","guid":{"rendered":"http:\/\/lttt.blog.ustc.edu.cn\/?p=209"},"modified":"2013-06-18T23:28:12","modified_gmt":"2013-06-18T15:28:12","slug":"navier-stokes-equations-2","status":"publish","type":"post","link":"https:\/\/lttt.vanabel.cn\/?p=209","title":{"rendered":"Navier\u2013Stokes equations"},"content":{"rendered":"<p>Lecture notes taken from <a title=\"Changyou Wang's homepage\" href=\"http:\/\/www.ms.uky.edu\/~cywang\/\" target=\"_blank\">C.Y. Wang&#8217;<\/a>&#8216;s short course.<\/p>\n<p>Please note that the complete notes is now available at Changyou Wang&#8217;s <a href=\"http:\/\/www.ms.uky.edu\/~cywang\/BNU_NSE_Note.pdf\" target=\"_blank\">homapage<\/a>.<\/p>\n<h3>Reference<\/h3>\n<ol>\n<li>Temam, Roger. <em>Navier-Stokes equations: theory and numerical analysis<\/em>. Vol. 343. Oxford University Press, 2001.<\/li>\n<li>Majda, A. J., A. L. Bertozzi, and A. Ogawa. &#8220;Vorticity and Incompressible Flow. Cambridge Texts in Applied Mathematics.&#8221; <em>Applied Mechanics Reviews<\/em> 55 (2002): 77.<\/li>\n<\/ol>\n<p><!--more--><\/p>\n<h3>Table of Contents<\/h3>\n<div class=\"toc\">\n<ul>\n<li><a href=\"#reference\">Reference<\/a>\n<ul>\n<li><a href=\"#flow-map\">Flow map<\/a><\/li>\n<li><a href=\"#stead-fluid-state\">Stead fluid state<\/a><\/li>\n<li><a href=\"#strain-rate\">Strain rate<\/a><\/li>\n<li><a href=\"#conservation-of-mass\">Conservation of Mass<\/a><\/li>\n<li><a href=\"#homogenous-nse\">Homogenous NSE<\/a><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<h4 id=\"flow-map\">Flow map<\/h4>\n<p>Let $\\Omega\\subset \\R^n$ be a fluid, $n=2,3$. $\\rho$ is the density of fluid, $u:\\omega\\to\\R^n$ is the velocity field of fluid. The flow map is defined as:<br \/>\n\\begin{align*}<br \/>\nx:\\Omega\\times[0,+\\infty)&amp;\\to\\R^n\\\\<br \/>\n(\\alpha,t)&amp;\\mapsto x(\\alpha,t),<br \/>\n\\end{align*}<br \/>\nwhere $\\alpha$ is called the <strong>Lagrangian coordinates<\/strong> and $x(\\alpha,t)$ is called the <strong>Eulerian coordinates<\/strong>. We must have the following equation:<br \/>\n$$\\begin{cases}<br \/>\n\\frac{d x(\\alpha,t)}{d t}=u(x(\\alpha,t),t)\\<br \/>\nx(\\alpha,0)=\\alpha.<br \/>\n\\end{cases}$$<br \/>\nWhat&#8217;s we concern is the acceleration<br \/>\n\\begin{align*}<br \/>\na&amp;=\\frac{d^2}{d t^2}x(\\alpha,t)=\\frac{d}{dt}u(x,t)=u_t+\\frac{\\partial u}{\\partial x_i}\\frac{d x^i}{d t}\\\\<br \/>\n&amp;=u_t+\\frac{\\partial u}{\\partial x_i}u^i=u_t+(u\\cdot\\nabla)u,<br \/>\n\\end{align*}<br \/>\n$\\frac{D}{D t}:=\\frac{\\partial }{\\partial t}+(u\\cdot\\nabla)$ is called <strong>material derivative<\/strong>.<\/p>\n<p>By the <strong>Newtonian second law<\/strong>, we have $Ma=F$, that is $\\rho\\frac{Du}{Dt}=F$, where $F=f+T$, where $f$ is the external <strong>body force<\/strong> and $T$ is the <strong>friction force<\/strong>.<\/p>\n<p>\\begin{defn}[Incompressibility]<br \/>\nIf $x(\\alpha, t):\\Omega\\to\\Omega_t$ is volume preserving diffeomorphism, i.e., $\\forall O\\subset\\Omega$, we have $\\vol(O)=\\mathrm{Vol}(O_t)$, where $O_t=x(O,t)$.<br \/>\n\\end{defn}<\/p>\n<p>\\begin{prop}<br \/>\n$x(\\alpha,t)$ is incompressible iff $\\mathrm{div} u=\\sum_{i=1}^n\\frac{\\partial u^i}{\\partial x_i}=0$.<br \/>\n\\end{prop}<br \/>\n\\begin{proof}<br \/>\nIn fact, by the change of variable formula for integrals, we have $$\\vol(O_t)=\\int_O\\det\\left(\\frac{\\pt x}{\\pt \\alpha}\\right)\\rd \\alpha,$$<br \/>\nthus<br \/>\n$$0=\\frac{\\rd \\vol(O_t)}{\\rd t}=\\frac{\\rd }{\\rd t}\\int_O\\det \\left(\\frac{\\pt x}{\\pt \\alpha}\\right)\\rd \\alpha=\\int_O\\frac{\\rd }{\\rd t}\\det\\left(\\frac{\\pt x}{\\pt \\alpha}\\right)\\rd \\alpha.$$<br \/>\nIf we denote $A=(a_{ij})$, where $a_{ij}=\\frac{\\pt x^i(\\alpha,t)}{\\pt \\alpha^j}$, and the cofactor of $a_{ij}$ is $A_{ij}$, then apply the ralation $A_{ij}a_{jk}=\\delta_{ik}$, we obtain<br \/>\n\\begin{align*}<br \/>\n\\frac{\\rd}{\\rd t}\\det\\left(\\frac{\\pt x}{\\pt \\alpha}\\right)<br \/>\n&amp;=A_{ij}\\frac{\\rd }{\\rd t}\\left(\\frac{\\pt x^i}{\\pt \\alpha^j}\\right)<br \/>\n=A_{ij}\\frac{\\pt}{\\pt\\alpha^j}\\left(u^i(x,t)\\right)\\\\<br \/>\n&amp;=A_{ij}\\frac{\\pt u^i}{\\pt x^k}\\frac{\\pt x^k}{\\pt \\alpha^j}<br \/>\n=\\sum_i\\frac{\\pt u^i}{\\pt x^i}.<br \/>\n\\end{align*}<br \/>\nTherefore,<br \/>\n$$<br \/>\n0=\\int_O\\sum_i\\frac{\\pt u^i}{\\pt x^i}\\rd\\alpha,\\quad\\forall O\\subset\\Omega,<br \/>\n$$<br \/>\ni.e., $x$ is incompressible iff $\\div u=0$.<br \/>\n\\end{proof}<\/p>\n<h4 id=\"stead-fluid-state\">Stead fluid state<\/h4>\n<p>Since in this case the fluid is force balance, we have<br \/>\n$$<br \/>\n\\int_O f\\rd v+\\int_{\\partial O}\\tau\\cdot \\nu\\rd \\sigma=0,<br \/>\n$$<br \/>\nwhere $f$ is the body force and $\\tau$ is the <strong>cauchy stress tensor<\/strong>, it is a matrix of $n\\times n$, and $\\rd v$ is the volume elements, $\\rd\\sigma$ is the area elements, $\\nu$ is the unit outer normal vector field.<\/p>\n<p>By the divergence theorem (view $\\tau$ as a vector with three components, each component is a row of $\\tau$.) we have the vector styled equation:<br \/>\n$$<br \/>\nf+\\div \\tau=0.<br \/>\n$$<br \/>\nUsually, $\\tau=-pI_n+\\sigma$, where $p$ is the <strong>pressure<\/strong>, and $I_n$ is the identity matrix and $\\sigma$ is the <strong>viscous stress<\/strong>.<\/p>\n<h4 id=\"strain-rate\">Strain rate<\/h4>\n<p>The differential of $u:\\Omega\\to\\R^3$ will be denoted as $Du$, set $Du=\\dd (u)+\\Omega(u)$, where $\\dd(u)=\\frac{1}{2}\\left[Du+(Du)^T\\right]$ is the rate-of-strain tensor and $\\Omega(u)=\\frac{1}{2}\\left[Du-(Du)^T\\right]$ is the rate of expansion of the flow. If $\\sigma$ is linearly depended on $\\dd(u)$, we say in this case the fluid is <strong>simple fluid<\/strong>, denoted as $\\sigma=L(\\dd(u))$. Note that, for the simple fluid, we have<br \/>\n$$<br \/>\nL(Q^T\\dd(u)Q)=Q^TL(\\dd(u))Q,\\quad \\forall Q\\in O(n),<br \/>\n$$<br \/>\nThus,<br \/>\n$$<br \/>\n\\sigma=2\\mu\\dd(u)+\\lambda\\tr(\\dd(u))I_n<br \/>\n=2\\mu\\dd(u)+\\lambda(\\div u)I_n,<br \/>\n$$<br \/>\nwhere $\\mu$ is called <strong>shear viscosity<\/strong>.<\/p>\n<p>Now, $\\tau=-pI_n+\\sigma=-pI_n+2\\mu\\dd(u)+\\lambda(\\div u)I_n$, from which we obtain, in the steady case:<br \/>\n$$<br \/>\n0=f+\\div(-pI_n+2\\mu\\dd(u)+\\lambda(\\div u)I_n),<br \/>\n$$<br \/>\nin general, we obtain the <strong>Navier-Stokes equation<\/strong>:<br \/>\n$$<br \/>\n\\rho\\frac{D u}{D t}=f+\\div(-pI_n+2\\mu\\dd(u)+\\lambda(\\div u)I_n)\\tag{NSE}.<br \/>\n$$<\/p>\n<h4 id=\"conservation-of-mass\">Conservation of Mass<\/h4>\n<p>By the conservation of mass, we have<br \/>\n$$<br \/>\n\\int_O\\rho_t=\\frac{\\rd}{\\rd t}\\int_O\\rho\\rd x<br \/>\n=\\int_{\\partial O}\\rho u\\cdot \\nu\\rd\\sigma<br \/>\n=-\\int_O\\div(\\rho u),<br \/>\n$$<br \/>\nthat is<br \/>\n$$<br \/>\n\\rho_t+\\div(\\rho u)=0.<br \/>\n$$<br \/>\nThus, the fluid is incompressible iff $\\rho_t+u\\div \\rho=0$.<\/p>\n<h4 id=\"homogenous-nse\">Homogenous NSE<\/h4>\n<p>If $\\rho\\equiv1$, then the NSE becomes<br \/>\n$$<br \/>\n\\begin{cases}<br \/>\nu_t+u\\cdot\\div u+\\div p=f+\\mu\\Delta u,\\\\<br \/>\n\\div u=0\\\\<br \/>\nu|_{t=0}=u_0,\\:\\div u_0=0.<br \/>\n\\end{cases}<br \/>\n$$<\/p>\n<p>\\begin{rem}<\/p>\n<ol>\n<li>For $n=2$, we know almost all the things, but for $n=3$, we almost know nothing!<\/li>\n<li>If $\\mu=0$, then the fluid is called <strong>ideal fluid<\/strong>, and the NSE reduce to <em>Euler equation<\/em>:<br \/>\n$$<br \/>\n\\begin{cases}<br \/>\nu_t+u\\cdot \\div u+\\div p=f,\\\\<br \/>\n\\div u=0.<br \/>\n\\end{cases}<br \/>\n$$<\/li>\n<li>If, furthermore, we have $f=0$, then, by a simple calculation (recall that we have $\\div u=0$),<br \/>\n$$<br \/>\n-\\Delta p=-\\div((u\\cdot\\nabla)u)=\\begin{cases}\\tr[(\\nabla u)^2]\\\\div^2(u\\otimes u)\\end{cases},<br \/>\n$$<br \/>\nwhere $(u\\otimes v)_{ij}=u^iv^i$, they are both useful in the study of fluid.<\/li>\n<\/ol>\n<p>\\end{rem}<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Lecture notes taken from C.Y. Wang&#038;#8217 &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"more-link\" href=\"https:\/\/lttt.vanabel.cn\/?p=209\"> <span class=\"screen-reader-text\">Navier\u2013Stokes equations<\/span> \u9605\u8bfb\u66f4\u591a &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[12],"tags":[575],"class_list":["post-209","post","type-post","status-publish","format-standard","hentry","category-mathnotes","tag-navier-stokes-equations"],"_links":{"self":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/209","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=209"}],"version-history":[{"count":0,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/209\/revisions"}],"wp:attachment":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=209"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=209"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=209"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}