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* from \"-!../../../../node_modules/mini-css-extract-plugin/dist/loader.js??ref--8-oneOf-1-0!../../../../node_modules/css-loader/index.js??ref--8-oneOf-1-1!../../../../node_modules/vue-loader/lib/loaders/stylePostLoader.js!../../../../node_modules/postcss-loader/src/index.js??ref--8-oneOf-1-2!../../../../node_modules/sass-loader/dist/cjs.js??ref--8-oneOf-1-3!../../../../node_modules/cache-loader/dist/cjs.js??ref--0-0!../../../../node_modules/vue-loader/lib/index.js??vue-loader-options!./Lesson.vue?vue&type=style&index=0&id=50fc2e10&prod&lang=scss&\"","var render = function render(){var _vm=this,_c=_vm._self._c;return _c('div',[_c('h3',{ref:\"intro\"},[_vm._v(\"\\n    Statement of Bernoulli's Principle\\n  \")]),_c('p',[_vm._v(\"\\n    Bernoulli's principle states that for an inviscid, incompressible flow, the sum of static head,\\n    kinetic head and hydrostatic head remains constant along a streamline.\\n    Let \\\\(P\\\\) be the static presure, \\\\(v\\\\) be the velocity and \\\\(h\\\\) be the elevation of a point along the streamline.\\n    Then, according to Bernoulli's principle:\\n    $$ P + \\\\frac{1}{2} \\\\rho v^2 + \\\\rho g h = constant $$\\n    Thus, Bernoulli's principle can be used to relate pressure, velocity and elevation\\n    at any two points along a streamline. For example:\\n    $$ P_1 + \\\\frac{1}{2} \\\\rho v_{1}^{2} + \\\\rho g z_1 = P_2 + \\\\frac{1}{2} \\\\rho v_{2}^{2} + \\\\rho g z_{2} $$\\n  \")]),_c('p',[_vm._v(\"\\n    Further, for a constant elevation flow i.e. \\\\(z_1 = z_2\\\\), Bernoulli's equation becomes:\\n    $$ P_1 + \\\\frac{1}{2} \\\\rho v_{1}^{2} = P_2 + \\\\frac{1}{2} \\\\rho v_{2}^{2} $$\\n    along a streamline in an inviscid, incompressible flow.\\n    Thus, an increase in fluid velocity is accompanied by a decrease in fluid pressure, and vice versa.\\n\\n    Bernoulli's principle has a number of applications ranging from velocity measurement in a pitot tube,\\n    design of lift in airfoils, to design of hydraulic ducts in chemical plants.\\n  \")]),_c('h3',{ref:\"appl\"},[_vm._v(\"\\n    Conditions for Applicabiilty of Bernoulli's Principle\\n  \")]),_vm._m(0),_c('h3',{ref:\"derivations\"},[_vm._v(\"\\n    Derivation of Bernoulli's principle\\n  \")]),_c('p',[_vm._v(\"\\n    Bernoulli's principle is essentially a statement of conservation of total energy. The real fluids always have viscous losses due to viscosity (resulting in inter-layer frictions).\\n    Bernoulli's principle is an idealization based on the notion of an inviscid flow. During such a flow, the total energy of the fluid remains constant along a streamline. The total energy of a fluid is comprised of\\n    the potential energy (due to the action of gravity on the fluid) and the kinetic energy (due to velocity of the fluid elements). Now, consider a fluid element of volume \\\\(\\\\Delta V\\\\) traveling from point '1' to point '2' along a duct\\n    as shown in the figure below.\\n    The change in gravitational potential energy of this fluid element is given as:\\n    $$ \\\\Delta \\\\mathcal U = \\\\rho g \\\\Delta V (h_1 - h_2) $$\\n    where \\\\(\\\\rho\\\\) is the density of the fluid element, \\\\(h_1\\\\) is the elevation of section '1' and \\\\(h_2\\\\) is the elevation of section '2'.\\n    The change in kinetic energy of this fluid element is given as:\\n    $$ \\\\Delta \\\\mathcal K = \\\\frac{1}{2} \\\\rho \\\\Delta V (v_2^{2} - v_1^{2}) $$\\n    where \\\\(v_1\\\\) and \\\\(v_2\\\\) are fluid velocities at sections '1' and '2', respectively.\\n    The external work done on the fluid element is given as:\\n    $$ \\\\mathcal W_{ext} = \\\\delta(P\\\\ \\\\Delta V) = \\\\delta P \\\\Delta V + P \\\\delta \\\\Delta V$$\\n    Since the flow is incompressible, we have \\\\(\\\\delta V = 0\\\\). Thus, we have:\\n    $$ \\\\mathcal W_{ext} = (P_2 -  P_1) V $$\\n    where  \\\\(P_1 \\\\) is the fluid pressure at point '1' of the streamline, and \\\\(P_2\\\\) is the fluid pressure at point '2' of the streamline.\\n    Now, according to the work-energy theorem, the change in total energy should equal the external work done on the fluid element. Mathematically, it can be expressed as:\\n    $$\\\\mathcal W_{ext} = \\\\Delta \\\\mathcal K + \\\\Delta \\\\mathcal U$$\\n    Substituting the various values, we obtain:\\n    $$ (P_1 - P_2) \\\\Delta V = \\\\frac{1}{2} \\\\rho \\\\Delta V (v_2^{2} - v_1^{2}) + \\\\rho g \\\\Delta V (h_2 - h_1) $$\\n    Upon rearranging, we obtain:\\n    $$ P_1 + \\\\frac{1}{2} \\\\rho v_1^{2} + \\\\rho g h_1 = P_2 + \\\\frac{1}{2} \\\\rho v_2^{2} + \\\\rho g h_2 $$\\n    Then, the general expression for Bernoulli's equation is:\\n    $$ P +\\\\frac{1}{2} \\\\rho v_1^{2} + \\\\rho gh = constant $$\\n  \")]),_c('h3',{ref:\"continuity\"},[_vm._v(\"\\n    Continuity Equation\\n  \")]),_c('br'),_c('p',[_vm._v(\"\\n    For incompressible flows, the velocity of a fluid element can be related to the geometry of the duct via continuity equation. The continuity equation states:\\n    $$ A_1 v_1 = A_2 v_2 $$\\n    where \\\\(A_1\\\\) and \\\\(A_2\\\\) are the cross-sectional areas at the two sections, and \\\\(v_1\\\\) and \\\\(v_2\\\\) are the respective velocities at those sections. While Bernoulli's equation is a statement of conservation of energy, the continuity equation is a statement of the conservation of mass.\\n    Bernoulli's equation in conjunction with continuity equation allows us\\n    to fully determined fluid pressure and velocity at any section along\\n    the streamline.\\n  \")]),_c('h3',{ref:\"PlayGraph\"},[_vm._v(\"\\n    MagicGraph | Fluid Flow in a Vertical Duct\\n  \")]),_c('h5',[_vm._v(\" To Get Started\")]),_c('p',[_vm._v(\"\\n    You are given a duct carrying an incompressible fluid.\\n    The height and the cross-section area of the inlet and outlet sections of the duct can be changed by using the sliders.\\n    For the velocity and the pressure prescribed at the inlet (section '1'),\\n    the MagicGraph determines the pressure and the velocity at the outlet (section '2').\\n  \")]),_c('v-responsive',[_c('v-layout',{attrs:{\"align-center\":\"\",\"justify-center\":\"\"}},[_c('div',{staticClass:\"edliy-box-about\",attrs:{\"id\":\"jxgbox1\"}})]),_c('v-layout',{attrs:{\"justify-center\":\"\"}},[_c('h5',[_vm._v(\" Follow this link \"),_c('a',{staticClass:\"icn\",attrs:{\"href\":\"https://edliy.com/magic\"}},[_c('i',{staticClass:\"fa fa-external-link-square\"})]),_vm._v(\" to learn how MagicGraphs help students accelerate their learning curve. \")])])],1)],1)\n}\nvar staticRenderFns = [function (){var _vm=this,_c=_vm._self._c;return _c('p',[_vm._v(\"\\n    Bernoulli's equation, in the strictest sense, is applicable only when the following flow conditions\\n    are satisfied:\\n    \"),_c('ul',{staticStyle:{\"list-style-type\":\"square\"}},[_c('li',[_vm._v(\"\\n        Fluid is inviscid ― which means it has zero viscosity and hence the viscous losses are absent.\\n      \")]),_c('li',[_vm._v(\"\\n        Flow is incompressible ― which means the density of a fluid element along a streamline\\n        remains constant.\\n      \")])])])\n}]\n\nexport { render, staticRenderFns }","// Import the edliy_utils\r\nimport {\r\n    makeResponsive,\r\n    placeTitle,\r\n    placeImage,\r\n    placeInput,\r\n    placeSlider,\r\n    hoverMe,\r\n    placeRec,\r\n    hiddenPt,\r\n    fixedPt,\r\n    clearInputFields,\r\n    dragMe,\r\n    placeArrow,\r\n    placeGravity,\r\n    placeText,\r\n    placeLine,\r\n    placePoint,\r\n    placeGlider,\r\n    placeRuler,\r\n    placeLeftText,\r\n    placeRightText,\r\n    placeSliderSwitch,\r\n    placeDash\r\n} from '../../../common/edliy_utils-bernoulli';\r\nconst Boxes = {\r\n  box1: function () {\r\n  var brd1 = JXG.JSXGraph.initBoard('jxgbox1',{boundingbox: [-1, 15, 15, -1],\r\n  keepaspectratio: true, axis:true, ticks:{visible:false},\r\n  grid:true, showCopyright:false, showNavigation:false,\r\n  pan:{enabled:false}, zoom:{enabled:false}});\r\n  brd1.suspendUpdate();\r\n  brd1.options.board.minimizeReflow = 'none';\r\n  brd1.options.text.highlight =false;\r\n  brd1.options.text.fixed =true;\r\n  brd1.options.text.highlight =false;\r\n  brd1.options.text.cssStyle='fontFamily:Oswald',\r\n  brd1.options.image.highlight =false;\r\n  brd1.options.polygon.highlight =false;\r\n  brd1.options.line.highlight =false;\r\n  brd1.options.point.highlight =false;\r\n  brd1.options.curve.highlight =false;\r\n  brd1.options.slider.highlight =false;\r\n  brd1.options.point.showInfoBox =false;\r\n//brd1.options.layer['image'] =14;\r\n//Title and tetxs\r\n  makeResponsive(brd1);\r\n  placeTitle(brd1, 'Bernoulli\\'s Principle', '');\r\n  //var title1 = brd1.create('text', [7, 14, \"<b> Bernoulli's Principle <b>\"],{fixed: true, anchorX: 'middle', anchorY: 'middle', CssStyle:'fontFamily:Oswald',fontSize:function(){return Math.round(28*brd1.canvasWidth/500.)}},);\r\n\r\n//Sliders\r\n//var p1 = brd1.create('slider',[[0,12],[2,12],[100000,600000,1000000]], {snapToGrid: false ,face:'square', size:4, strokeWidth: 2, name:'P_1', strokeColor:'black', fillColor: 'grey',  label:{fontSize:function(){return 18*brd1.canvasHeight/800}, cssStyle:'fontFamily:Oswald'}});\r\n//var v1 = brd1.create('slider',[[0,11],[2,11],[1,3,5]], {snapToGrid: false ,face:'square', size:4, strokeWidth: 2, name:'v_1', strokeColor:'black', fillColor: 'grey',  label:{fontSize:function(){return 18*brd1.canvasHeight/800}, cssStyle:'fontFamily:Oswald'}});\r\nvar p1 = placeSlider(brd1, 1, 12, 100000, 200000, 600000, 2, '&nbsp; P_1', ' Pa');\r\nvar v1 = placeSlider(brd1, 1, 11, 0.1, 1, 2, 2, '&nbsp; v_1', ' m/s');\r\n//var r1 = brd1.create('slider',[[0,10],[2,10],[0.1,1,5]], {snapToGrid: false ,face:'square', size:4, strokeWidth: 2, name:'r_1', strokeColor:'black', fillColor: 'grey',  label:{fontSize:function(){return 18*brd1.canvasHeight/800}, cssStyle:'fontFamily:Oswald'}});\r\n//var r2 = brd1.create('slider',[[0,9],[2,9],[0.1,0.5,5]], {snapToGrid: false ,face:'square', size:4, strokeWidth: 2, name:'r_2', strokeColor:'black', fillColor: 'grey',  label:{fontSize:function(){return 18*brd1.canvasHeight/800}, cssStyle:'fontFamily:Oswald'}});\r\nvar r1=placeSlider(brd1, 1, 9, 0.1, 1, 2, 2, '&nbsp; r_1', ' m');\r\nvar r2=placeSlider(brd1, 11, 9, 0.1, 0.5, 2, 2, '&nbsp; r_2', ' m');\r\n//Height\r\nvar segh1 = brd1.create('segment', [[0,0], [0, 8]], {visible: false, fixed: true, strokeColor: 'black', strokeWidth: 3});\r\nvar segh2 = brd1.create('segment', [[14,0], [14, 13]], {visible: false, fixed: true, strokeColor: 'black', strokeWidth: 3});\r\n\r\nbrd1.create('axis', [[14, 0],[14, 10]]);\r\nvar h1 = brd1.create('glider', [0,5, segh1],{face:'>' , name:'', size:6, strokeColor:'black', withLabel:true, fixed:false, visible:true, label:{fontSize:function(){return 18*brd1.canvasHeight/800}, cssStyle:'fontFamily:Oswald'}});\r\nvar h2 = brd1.create('glider', [0,3, segh2],{face:'<' , name:'', size:6, withLabel:true, strokeColor:'black', fixed:false, visible:true, label:{fontSize:function(){return 18*brd1.canvasHeight/800}, cssStyle:'fontFamily:Oswald'}});\r\n//brd1.create('arrow', [[13, 0],[13, ()=>h2.Y()]]);\r\nplaceArrow(brd1,[13, 0],[13, ()=>h2.Y()], 2, 'grey');\r\nplaceArrow(brd1,[1, 0],[1, ()=>h1.Y()], 2, 'grey');\r\nplaceDash(brd1, [0, ()=>h1.Y()], [2, ()=>h1.Y()], 2, 'black');\r\nplaceDash(brd1, [12, ()=>h2.Y()], [14, ()=>h2.Y()], 2, 'black');\r\nplaceRightText(brd1, 13.5, ()=>0.5*h2.Y(), 'h_2');\r\nplaceRightText(brd1, 1.5, ()=>0.5*h1.Y(), 'h_1');\r\n//\r\n//Circle 1\r\nvar pC1 = brd1.create('point', [4, function(){return h1.Y()}],{face:'o' , name:'A', size:2, withLabel:true, fixed:false, visible:false, label:{fontSize:function(){return 18*brd1.canvasHeight/800}, cssStyle:'fontFamily:Oswald'}});\r\nvar pCup = brd1.create('point', [2, function(){return h1.Y() + r1.Value()}],{face:'o' , name:'A', size:2, withLabel:true, fixed:false, visible:false, label:{fontSize:function(){return 18*brd1.canvasHeight/800}, cssStyle:'fontFamily:Oswald'}});\r\nvar pCdown = brd1.create('point', [2, function(){return h1.Y() - r1.Value()}],{face:'o' , name:'A', size:2, withLabel:true, fixed:false, visible:false, label:{fontSize:function(){return 18*brd1.canvasHeight/800}, cssStyle:'fontFamily:Oswald'}});\r\n\r\nvar pC11 = brd1.create('point', [function(){return pC1.X() - 4}, function(){return pC1.Y()}],{face:'o' , name:'A', size:2, withLabel:true, fixed:false, visible:false, label:{fontSize:function(){return 18*brd1.canvasHeight/800}, cssStyle:'fontFamily:Oswald'}});\r\n\r\nvar c1 = brd1.create('arc', [pC1, pCup, pCdown], {strokeColor: 'blue', strokeWidth: 3});\r\nvar c11 = brd1.create('arc', [pC11, pCdown , pCup], {strokeColor: 'blue', strokeWidth: 3});\r\n\r\n//Circle 2\r\nvar pC1_2 = brd1.create('point', [14, function(){return h2.Y()}],{face:'o' , name:'A', size:2, withLabel:true, fixed:false, visible:false, label:{fontSize:function(){return 18*brd1.canvasHeight/800}, cssStyle:'fontFamily:Oswald'}});\r\nvar pCup_2 = brd1.create('point', [12, function(){return h2.Y() + r2.Value()}],{face:'o' , name:'A', size:2, withLabel:true, fixed:false, visible:false, label:{fontSize:function(){return 18*brd1.canvasHeight/800}, cssStyle:'fontFamily:Oswald'}});\r\nvar pCdown_2 = brd1.create('point', [12, function(){return h2.Y() - r2.Value()}],{face:'o' , name:'A', size:2, withLabel:true, fixed:false, visible:false, label:{fontSize:function(){return 18*brd1.canvasHeight/800}, cssStyle:'fontFamily:Oswald'}});\r\n\r\nvar pC11_2 = brd1.create('point', [function(){return pC1_2.X() - 4}, function(){return pC1_2.Y()}],{face:'o' , name:'A', size:2, withLabel:true, fixed:false, visible:false, label:{fontSize:function(){return 18*brd1.canvasHeight/800}, cssStyle:'fontFamily:Oswald'}});\r\n\r\nvar c2 = brd1.create('arc', [pC1_2, pCup_2, pCdown_2], {strokeColor: 'blue', strokeWidth: 3});\r\nvar c22 = brd1.create('arc', [pC11_2, pCdown_2 , pCup_2], {strokeColor: 'blue', strokeWidth: 3});\r\n\r\n//Intemediate points\r\n\r\nvar intUp_1 = brd1.create('point', [6, function(){return h1.Y() + r1.Value()}],{face:'o' , name:'A', size:2, withLabel:true, fixed:false, visible:false, label:{fontSize:function(){return 18*brd1.canvasHeight/800}, cssStyle:'fontFamily:Oswald'}});\r\nvar intDown_1 = brd1.create('point', [6, function(){return h1.Y() - r1.Value()}],{face:'o' , name:'A', size:2, withLabel:true, fixed:false, visible:false, label:{fontSize:function(){return 18*brd1.canvasHeight/800}, cssStyle:'fontFamily:Oswald'}});\r\n\r\nvar intUp_2 = brd1.create('point', [8, function(){return h2.Y() + r2.Value()}],{face:'o' , name:'A', size:2, withLabel:true, fixed:false, visible:false, label:{fontSize:function(){return 18*brd1.canvasHeight/800}, cssStyle:'fontFamily:Oswald'}});\r\nvar intDown_2 = brd1.create('point', [8, function(){return h2.Y() - r2.Value()}],{face:'o' , name:'A', size:2, withLabel:true, fixed:false, visible:false, label:{fontSize:function(){return 18*brd1.canvasHeight/800}, cssStyle:'fontFamily:Oswald'}});\r\n\r\n//Tube\r\n\r\nvar seg1 = brd1.create('segment', [pCup,intUp_1 ], {visible: true, fixed: true, strokeColor: 'blue', strokeWidth: 3});\r\nvar seg2 = brd1.create('segment', [pCdown,intDown_1 ], {visible: true, fixed: true, strokeColor: 'blue', strokeWidth: 3});\r\n\r\nvar seg3 = brd1.create('segment', [pCup_2,intUp_2 ], {visible: true, fixed: true, strokeColor: 'blue', strokeWidth: 3});\r\nvar seg4 = brd1.create('segment', [pCdown_2,intDown_2 ], {visible: true, fixed: true, strokeColor: 'blue', strokeWidth: 3});\r\n\r\nvar seg4 = brd1.create('segment', [intUp_1,intUp_2 ], {visible: true, fixed: true, strokeColor: 'blue', strokeWidth: 3});\r\nvar seg4 = brd1.create('segment', [intDown_1,intDown_2 ], {visible: true, fixed: true, strokeColor: 'blue', strokeWidth: 3});\r\n\r\n//Velocity 2\r\nvar density = 1000;\r\n\r\nvar v2 = function(){return Math.pow(r1.Value(), 2)*v1.Value()/Math.pow(r2.Value(), 2)};\r\n\r\nvar p2 = function(){return p1.Value()  + 0.5*density* ( Math.pow(v1.Value(), 2) - Math.pow(v2(), 2) ) + 9.81*density*(h1.Y() -h2.Y())  };\r\n\r\nbrd1.create('text', [9 , 12,  function(){return 'v_2 = ' + v2().toFixed(2)+' m/s';}],{visible: true, fixed: true, anchorX: 'left', anchorY: 'middle', fontSize:function(){return Math.round(12*brd1.canvasWidth/500.)}},);\r\nbrd1.create('text', [9 , 11,  function(){return 'P_2 = ' + p2().toFixed(2)+' Pa';}],{visible: true, fixed: true, anchorX: 'left', anchorY: 'middle', fontSize:function(){return Math.round(12*brd1.canvasWidth/500.)}},);\r\n//placeLeftText(brd1, 10, 12, function(){return 'P_2 = ' + p2().toFixed(2);});\r\n//placeLeftText(brd1, 10, 11, function(){return 'v_2 = ' + v2().toFixed(2);});\r\n//Water segments\r\nvar pWat1_1 = brd1.create('point', [2, function(){return h1.Y() + r1.Value() - 1*2*r1.Value()/4}],{face:'o' , name:'', size:1, withLabel:true, fixed:false, visible:false, label:{fontSize:function(){return 18*brd1.canvasHeight/800}, cssStyle:'fontFamily:Oswald'}});\r\nvar pWat2_1 = brd1.create('point', [2, function(){return h1.Y() + r1.Value() - 2*2*r1.Value()/4}],{face:'o' , name:'', size:1, withLabel:true, fixed:false, visible:false, label:{fontSize:function(){return 18*brd1.canvasHeight/800}, cssStyle:'fontFamily:Oswald'}});\r\nvar pWat3_1 = brd1.create('point', [2, function(){return h1.Y() + r1.Value() - 3*2*r1.Value()/4}],{face:'o' , name:'', size:1, withLabel:true, fixed:false, visible:false, label:{fontSize:function(){return 18*brd1.canvasHeight/800}, cssStyle:'fontFamily:Oswald'}});\r\n\r\nvar pWat1_1int = brd1.create('point', [6, function(){return h1.Y() + r1.Value() - 1*2*r1.Value()/4}],{face:'o' , name:'', size:1, withLabel:true, fixed:false, visible:false, label:{fontSize:function(){return 18*brd1.canvasHeight/800}, cssStyle:'fontFamily:Oswald'}});\r\nvar pWat2_1int = brd1.create('point', [6, function(){return h1.Y() + r1.Value() - 2*2*r1.Value()/4}],{face:'o' , name:'', size:1, withLabel:true, fixed:false, visible:false, label:{fontSize:function(){return 18*brd1.canvasHeight/800}, cssStyle:'fontFamily:Oswald'}});\r\nvar pWat3_1int = brd1.create('point', [6, function(){return h1.Y() + r1.Value() - 3*2*r1.Value()/4}],{face:'o' , name:'', size:1, withLabel:true, fixed:false, visible:false, label:{fontSize:function(){return 18*brd1.canvasHeight/800}, cssStyle:'fontFamily:Oswald'}});\r\n\r\nvar pWat1_2 = brd1.create('point', [12, function(){return h2.Y() + r2.Value() - 1*2*r2.Value()/4}],{face:'o' , name:'', size:1, withLabel:true, fixed:false, visible:false, label:{fontSize:function(){return 18*brd1.canvasHeight/800}, cssStyle:'fontFamily:Oswald'}});\r\nvar pWat2_2 = brd1.create('point', [12, function(){return h2.Y() + r2.Value() - 2*2*r2.Value()/4}],{face:'o' , name:'', size:1, withLabel:true, fixed:false, visible:false, label:{fontSize:function(){return 18*brd1.canvasHeight/800}, cssStyle:'fontFamily:Oswald'}});\r\nvar pWat3_2 = brd1.create('point', [12, function(){return h2.Y() + r2.Value() - 3*2*r2.Value()/4}],{face:'o' , name:'', size:1, withLabel:true, fixed:false, visible:false, label:{fontSize:function(){return 18*brd1.canvasHeight/800}, cssStyle:'fontFamily:Oswald'}});\r\n\r\nvar pWat1_2int = brd1.create('point', [8, function(){return h2.Y() + r2.Value() - 1*2*r2.Value()/4}],{face:'o' , name:'', size:1, withLabel:true, fixed:false, visible:false, label:{fontSize:function(){return 18*brd1.canvasHeight/800}, cssStyle:'fontFamily:Oswald'}});\r\nvar pWat2_2int = brd1.create('point', [8, function(){return h2.Y() + r2.Value() - 2*2*r2.Value()/4}],{face:'o' , name:'', size:1, withLabel:true, fixed:false, visible:false, label:{fontSize:function(){return 18*brd1.canvasHeight/800}, cssStyle:'fontFamily:Oswald'}});\r\nvar pWat3_2int = brd1.create('point', [8, function(){return h2.Y() + r2.Value() - 3*2*r2.Value()/4}],{face:'o' , name:'', size:1, withLabel:true, fixed:false, visible:false, label:{fontSize:function(){return 18*brd1.canvasHeight/800}, cssStyle:'fontFamily:Oswald'}});\r\n\r\nvar segWat1_1 = brd1.create('segment', [pWat1_1, pWat1_1int], {visible: true, fixed: true, strokeColor: 'blue', strokeWidth: 1});\r\nvar segWat2_1 = brd1.create('segment', [pWat2_1, pWat2_1int], {visible: true, fixed: true, strokeColor: 'blue', strokeWidth: 1});\r\nvar segWat3_1 = brd1.create('segment', [pWat3_1, pWat3_1int], {visible: true, fixed: true, strokeColor: 'blue', strokeWidth: 1});\r\n\r\nvar segWat1_int = brd1.create('segment', [pWat1_1int, pWat1_2int], {visible: true, fixed: true, strokeColor: 'blue', strokeWidth: 1});\r\nvar segWat2_int = brd1.create('segment', [pWat2_1int, pWat2_2int], {visible: true, fixed: true, strokeColor: 'blue', strokeWidth: 1});\r\nvar segWat3_int = brd1.create('segment', [pWat3_1int, pWat3_2int], {visible: true, fixed: true, strokeColor: 'blue', strokeWidth: 1});\r\n\r\nvar segWat1_2 = brd1.create('segment', [pWat1_2int,pWat1_2], {visible: true, fixed: true, strokeColor: 'blue', strokeWidth: 1});\r\nvar segWat2_2 = brd1.create('segment', [pWat2_2int,pWat2_2], {visible: true, fixed: true, strokeColor: 'blue', strokeWidth: 1});\r\nvar segWat3_2 = brd1.create('segment', [pWat3_2, pWat3_2int], {visible: true, fixed: true, strokeColor: 'blue', strokeWidth: 1});\r\n\r\n\r\nvar current1 =brd1.create('glider', [2, 0, segWat2_int], {strokeColor: 'black' , fillColor: 'black' , face: '>', size:0, name:'', visible: false, label:{fontSize:function(){return 18*brd1.canvasHeight/800}, cssStyle:'fontFamily:Oswald'}});\r\nvar current2 =brd1.create('glider', [2, 0, segWat2_1], {strokeColor: 'black' , fillColor: 'black' , face: '>', size:0, name:'', visible: false, label:{fontSize:function(){return 18*brd1.canvasHeight/800}, cssStyle:'fontFamily:Oswald'}});\r\nvar current3 =brd1.create('glider', [2, 0, segWat2_2], {strokeColor: 'black' , fillColor: 'black' , face: '>', size:0, name:'', visible: false, label:{fontSize:function(){return 18*brd1.canvasHeight/800}, cssStyle:'fontFamily:Oswald'}});\r\n\r\nbrd1.create('image', ['/assets/rightArrow.svg', [()=>current1.X()-0.15, ()=>current1.Y()-0.15], [0.3, 0.3]]);\r\nbrd1.create('image', ['/assets/rightArrow.svg', [()=>current2.X()-0.15, ()=>current2.Y()-0.15], [0.3, 0.3]]);\r\nbrd1.create('image', ['/assets/rightArrow.svg', [()=>current3.X()-0.15, ()=>current3.Y()-0.15], [0.3, 0.3]]);\r\ncurrent1.startAnimation(1, 30, 30);\r\ncurrent2.startAnimation(1, 30, 30);\r\ncurrent3.startAnimation(1, 30, 30);\r\n\r\n//Water drops\r\n\r\n//Function graph\r\nvar graph1 = brd1.create('functiongraph',\r\n                     [function(x){\r\n\r\n                      if(x>=2){ return h1.Y() + r1.Value() - 1*2*r1.Value()/4}\r\n                      else if(x> 2 && x< 8){ return 1}\r\n                      else if(x>= 8 && x<= 12){return h2.Y() + r2.Value() - 1*2*r2.Value()/4}\r\n\r\n\r\n\r\n                     }, -10, -5] , {visible: true, strokeColor: 'red'}\r\n);\r\n\r\n\r\nvar water1 = brd1.create('glider', [2, 6, graph1],{color: 'blue', face:'o' , name:'', size:2, withLabel:true, fixed:false, visible:true, label:{fontSize:function(){return 18*brd1.canvasHeight/800}, cssStyle:'fontFamily:Oswald'}});\r\n  brd1.unsuspendUpdate();\r\n    },\r\n}\r\nexport default Boxes;\r\n","<template>\r\n  <div>\r\n    <h3 ref=\"intro\">\r\n      Statement of Bernoulli's Principle\r\n    </h3>\r\n    <p>\r\n      Bernoulli's principle states that for an inviscid, incompressible flow, the sum of static head,\r\n      kinetic head and hydrostatic head remains constant along a streamline.\r\n      Let \\(P\\) be the static presure, \\(v\\) be the velocity and \\(h\\) be the elevation of a point along the streamline.\r\n      Then, according to Bernoulli's principle:\r\n      $$ P + \\frac{1}{2} \\rho v^2 + \\rho g h = constant $$\r\n      Thus, Bernoulli's principle can be used to relate pressure, velocity and elevation\r\n      at any two points along a streamline. For example:\r\n      $$ P_1 + \\frac{1}{2} \\rho v_{1}^{2} + \\rho g z_1 = P_2 + \\frac{1}{2} \\rho v_{2}^{2} + \\rho g z_{2} $$\r\n    </p>\r\n    <p>\r\n      Further, for a constant elevation flow i.e. \\(z_1 = z_2\\), Bernoulli's equation becomes:\r\n      $$ P_1 + \\frac{1}{2} \\rho v_{1}^{2} = P_2 + \\frac{1}{2} \\rho v_{2}^{2} $$\r\n      along a streamline in an inviscid, incompressible flow.\r\n      Thus, an increase in fluid velocity is accompanied by a decrease in fluid pressure, and vice versa.\r\n\r\n      Bernoulli's principle has a number of applications ranging from velocity measurement in a pitot tube,\r\n      design of lift in airfoils, to design of hydraulic ducts in chemical plants.\n    </p>\r\n    <h3 ref=\"appl\">\r\n      Conditions for Applicabiilty of Bernoulli's Principle\r\n    </h3>\r\n    <p>\r\n      Bernoulli's equation, in the strictest sense, is applicable only when the following flow conditions\r\n      are satisfied:\r\n      <ul style=\"list-style-type:square;\">\r\n        <li>\r\n          Fluid is inviscid &#8213; which means it has zero viscosity and hence the viscous losses are absent.\r\n        </li>\r\n        <li>\r\n          Flow is incompressible &#8213; which means the density of a fluid element along a streamline\r\n          remains constant.\r\n        </li>\r\n      </ul>\r\n    </p>\r\n\r\n    <h3 ref=\"derivations\">\r\n      Derivation of Bernoulli's principle\r\n    </h3>\r\n    <p>\r\n      Bernoulli's principle is essentially a statement of conservation of total energy. The real fluids always have viscous losses due to viscosity (resulting in inter-layer frictions).\r\n      Bernoulli's principle is an idealization based on the notion of an inviscid flow. During such a flow, the total energy of the fluid remains constant along a streamline. The total energy of a fluid is comprised of\r\n      the potential energy (due to the action of gravity on the fluid) and the kinetic energy (due to velocity of the fluid elements). Now, consider a fluid element of volume \\(\\Delta V\\) traveling from point '1' to point '2' along a duct\r\n      as shown in the figure below.\r\n      The change in gravitational potential energy of this fluid element is given as:\r\n      $$ \\Delta \\mathcal U = \\rho g \\Delta V (h_1 - h_2) $$\r\n      where \\(\\rho\\) is the density of the fluid element, \\(h_1\\) is the elevation of section '1' and \\(h_2\\) is the elevation of section '2'.\r\n      The change in kinetic energy of this fluid element is given as:\r\n      $$ \\Delta \\mathcal K = \\frac{1}{2} \\rho \\Delta V (v_2^{2} - v_1^{2}) $$\r\n      where \\(v_1\\) and \\(v_2\\) are fluid velocities at sections '1' and '2', respectively.\r\n      The external work done on the fluid element is given as:\r\n      $$ \\mathcal W_{ext} = \\delta(P\\ \\Delta V) = \\delta P \\Delta V + P \\delta \\Delta V$$\r\n      Since the flow is incompressible, we have \\(\\delta V = 0\\). Thus, we have:\r\n      $$ \\mathcal W_{ext} = (P_2 -  P_1) V $$\r\n      where  \\(P_1 \\) is the fluid pressure at point '1' of the streamline, and \\(P_2\\) is the fluid pressure at point '2' of the streamline.\r\n      Now, according to the work-energy theorem, the change in total energy should equal the external work done on the fluid element. Mathematically, it can be expressed as:\r\n      $$\\mathcal W_{ext} = \\Delta \\mathcal K + \\Delta \\mathcal U$$\r\n      Substituting the various values, we obtain:\r\n      $$ (P_1 - P_2) \\Delta V = \\frac{1}{2} \\rho \\Delta V (v_2^{2} - v_1^{2}) + \\rho g \\Delta V (h_2 - h_1) $$\r\n      Upon rearranging, we obtain:\r\n      $$ P_1 + \\frac{1}{2} \\rho v_1^{2} + \\rho g h_1 = P_2 + \\frac{1}{2} \\rho v_2^{2} + \\rho g h_2 $$\r\n      Then, the general expression for Bernoulli's equation is:\r\n      $$ P +\\frac{1}{2} \\rho v_1^{2} + \\rho gh = constant $$\r\n    </p>\r\n    <h3 ref=\"continuity\">\n      Continuity Equation\n    </h3> <br>\r\n    <p>\r\n      For incompressible flows, the velocity of a fluid element can be related to the geometry of the duct via continuity equation. The continuity equation states:\r\n      $$ A_1 v_1 = A_2 v_2 $$\r\n      where \\(A_1\\) and \\(A_2\\) are the cross-sectional areas at the two sections, and \\(v_1\\) and \\(v_2\\) are the respective velocities at those sections. While Bernoulli's equation is a statement of conservation of energy, the continuity equation is a statement of the conservation of mass.\r\n      Bernoulli's equation in conjunction with continuity equation allows us\r\n      to fully determined fluid pressure and velocity at any section along\r\n      the streamline.\r\n    </p>\r\n    <h3 ref=\"PlayGraph\">\n      MagicGraph | Fluid Flow in a Vertical Duct\n    </h3>\r\n    <h5> To Get Started</h5>\r\n    <p>\r\n      You are given a duct carrying an incompressible fluid.\r\n      The height and the cross-section area of the inlet and outlet sections of the duct can be changed by using the sliders.\r\n      For the velocity and the pressure prescribed at the inlet (section '1'),\r\n      the MagicGraph determines the pressure and the velocity at the outlet (section '2').\r\n    </p>\r\n    <v-responsive>\r\n      <v-layout align-center justify-center>\r\n        <div id=\"jxgbox1\" class=\"edliy-box-about\" />\r\n      </v-layout>\r\n      <v-layout justify-center>\r\n        <h5> Follow this link <a class=\"icn\" href=\"https://edliy.com/magic\"> <i class=\"fa fa-external-link-square\" /></a> to learn how MagicGraphs help students accelerate their learning curve. </h5>\r\n      </v-layout>\r\n    </v-responsive>\r\n  </div>\r\n</template>\r\n\r\n<script>\r\nimport Boxes from './Boxes.js'\r\n\r\nexport default {\r\n  name: 'BernoulliPrinciple',\r\n  data () {\r\n    return {\r\n      v1: 1,\r\n      P1: 1,\r\n      rho: 1,\r\n      v2: undefined,\r\n      P2: undefined,\r\n    }\r\n  },\r\n  created: function () {\r\n    // Store mutations\r\n    let title = 'Bernoulli\\'s Principle';\r\n    this.$store.commit('navigation/changeTitle', title);\r\n    this.$store.commit('navigation/changeMenu', title);\r\n    let newTopics = [\r\n      {title: ' Statement of Bernoulli\\'s Principle', img:'/assets/number-1.svg',action: () => this.goto('intro')},\r\n      {title: 'Derivation of Bernoulli\\'s Principle', img:'/assets/number-2.svg',action: () => this.goto('derivation')},\r\n      {title: 'Continuity Equation', img:'/assets/number-3.svg',action: () => this.goto('continuity')},\r\n      {title: 'Conditions for Applicability', img:'/assets/number-4.svg',action: () => this.goto('appl')},\r\n      {title: 'MagicGraph: Fluid Flow in a Vertical Duct', img:'/assets/touch.svg',action: () => this.goto('PlayGraph')},\r\n    ];\r\n    this.$store.commit('navigation/replaceTopics', newTopics);\r\n    let newshowhome = false;\r\n    this.$store.commit('navigation/toggleshowhome', newshowhome);\r\n    let newPhys =true;\r\n    this.$store.commit('navigation/replacePhys', newPhys);\r\n    let newLeftArrow =true;\r\n    this.$store.commit('navigation/replaceLeftArrow', newLeftArrow);\r\n    let newModule =true;\r\n    this.$store.commit('navigation/replaceModule', newModule);\r\n  },\r\n  mounted () {\r\n    MathJax.Hub.Queue([\"Typeset\", MathJax.Hub]);\r\n    Boxes.box1();\r\n\r\n  },\r\n  updated () {\r\n    // MathJax.Hub.Queue([\"Typeset\", MathJax.Hub]);\r\n  }\r\n}\r\n</script>\r\n<style lang=\"scss\">\r\n@import 'src/styles/edliy-box.scss';\r\n@import 'src/styles/subtopic-menu.scss';\r\n@import 'src/styles/edliy-box-about.scss';\r\n@import 'src/styles/screen-sizes.scss';\r\n</style>\r\n","import mod from \"-!../../../../node_modules/cache-loader/dist/cjs.js??ref--12-0!../../../../node_modules/thread-loader/dist/cjs.js!../../../../node_modules/babel-loader/lib/index.js!../../../../node_modules/cache-loader/dist/cjs.js??ref--0-0!../../../../node_modules/vue-loader/lib/index.js??vue-loader-options!./Lesson.vue?vue&type=script&lang=js&\"; export default mod; export * from \"-!../../../../node_modules/cache-loader/dist/cjs.js??ref--12-0!../../../../node_modules/thread-loader/dist/cjs.js!../../../../node_modules/babel-loader/lib/index.js!../../../../node_modules/cache-loader/dist/cjs.js??ref--0-0!../../../../node_modules/vue-loader/lib/index.js??vue-loader-options!./Lesson.vue?vue&type=script&lang=js&\"","import { render, staticRenderFns } from \"./Lesson.vue?vue&type=template&id=50fc2e10&\"\nimport script from \"./Lesson.vue?vue&type=script&lang=js&\"\nexport * from \"./Lesson.vue?vue&type=script&lang=js&\"\nimport style0 from \"./Lesson.vue?vue&type=style&index=0&id=50fc2e10&prod&lang=scss&\"\n\n\n/* normalize component */\nimport normalizer from \"!../../../../node_modules/vue-loader/lib/runtime/componentNormalizer.js\"\nvar component = normalizer(\n  script,\n  render,\n  staticRenderFns,\n  false,\n  null,\n  null,\n  null\n  \n)\n\nexport default component.exports"],"sourceRoot":""}