𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 = 𝐹𝑜𝑟𝑐𝑒 𝑁 𝐴𝑟𝑒𝑎 𝑚3 𝐔𝐍𝐈𝐅𝐎𝐑𝐌 𝐅𝐎𝐑𝐂𝐄 𝐎𝐕𝐄𝐑 𝐀𝐑𝐄𝐀 𝐏 = 𝐅 𝐀 ∆𝐅 ∆𝐀 ----------------------------------------------------------𝐍𝐎𝐍 − 𝐔𝐍𝐈𝐅𝐎𝐑𝐌 𝐅𝐎𝐑𝐂𝐄 𝐏 = 𝑭 𝑭𝒍 𝜼= 𝒗𝑨= 𝒍 𝚫𝒗 𝑽𝒊𝒔𝒄𝒐𝒔𝒊𝒕𝒚 Temperature has a srong effect on viscosity May depend on the rate of shear strain Assumptions often used in fluid mechanics*viscosity is constant (Newtonian fluid) *viscosity is 0 (ideal fluid, inviscid fluid, flow is frictionless) 𝑩𝒆𝒓𝒏𝒐𝒖𝒍𝒍𝒊𝒔 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 (conservation of energy) 1 1 𝑃1 + 2𝜌1 𝑉1 2 + 𝜌𝑔𝑦1 = 𝑃2 + 2𝜌𝑉2 2 + 𝜌𝑔𝑦2 -------------------------------------------------------------- Further common assumptions ONLY FOR SV 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝑇𝑒𝑛𝑠𝑖𝑜𝑛 𝜸 𝑃1 + 𝑃2 = 𝐴𝑇𝑀𝑂𝑆𝑃𝐻𝐸𝑅𝐼𝐶 𝑃𝑅𝐸𝑆𝑆𝑈𝑅𝐸 𝑉1 = 0 𝜸=𝑭 𝑳 𝟑 −𝟑 𝟑 𝟑 -------------------------------------------------------------𝑳 → 𝒎 =× 𝟏𝟎 𝒎 → 𝑳 = × 𝟏𝟎 -------------------------------------------------------------Pascals principle ----------------------------------------------------------- ‘if an external pressure is applied to a confined fluid, Ideal Gas equation 𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝐺𝑟𝑎𝑣𝑖𝑡𝑦 𝑺𝑮 𝑖𝑠 𝑡𝑒 𝑟𝑎𝑡𝑖𝑜 𝑜𝑓 the pressure at every point within the fluid increases 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑡𝑒 𝑠𝑢𝑏𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑜 𝑡𝑒 by that amount’ 𝑷𝒗 = 𝑵𝑨 𝒌𝑩 𝑻 = 𝒏𝑹𝑻 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟 𝑎𝑡 4° 𝐶 eg Hydraulic Lift 𝑃1 = 𝑃2 𝑅 = 𝑔𝑎𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 8.3145 𝐽𝐾 −1 𝑚𝑜𝑙−1 𝛒𝐬𝐮𝐛𝐬𝐭𝐚𝐧𝐜𝐞 𝛒𝐬𝐮𝐛𝐬𝐭𝐚𝐧𝐜𝐞 𝐹1 𝐹2 -------------------------------------------------------------𝐒𝐆 = = 𝐴1 = 𝐴2 𝛒𝐰𝐚𝐭𝐞𝐫 𝐚𝐭 𝟒° 𝐂 𝟏. 𝟎𝟎𝟎 × 𝟏𝟎𝟑 𝐤𝐠 𝐦𝟑 Real Gas equation 𝐦 𝑀𝑎𝑠𝑠 𝐾𝑔 𝐷𝑒𝑛𝑠𝑖𝑡𝑦 𝛒 = = ∀ 𝑉𝑜𝑙𝑢𝑚𝑒 𝑚3 Can be used to obtain mechanical advantage 𝐴2 𝐹2 = 𝐹1 𝐴1 Work done is the same by which the surface A2 rises is smaller than the change in the height of surface with area A 𝑭𝟏 𝚫𝒙𝟏 = 𝑭𝟐 𝚫𝒙𝟐 ----------------------------------------------------------𝒑𝑽 =𝒁 Pressure vs depth (incompressible fluids) 𝒏𝑹𝑻 𝑊𝑒𝑖𝑔𝑡 𝑜𝑓 𝑙𝑖𝑞𝑢𝑖𝑑 𝑾 = 𝒎. 𝒈 𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑙𝑖𝑞𝑢𝑖𝑑 𝑽 = 𝑨. 𝒉 Z= compressibility & is dimensionless 𝑀𝑎𝑠𝑠 𝑜𝑓 𝑙𝑖𝑞𝑢𝑖𝑑 𝒎 = 𝝆𝑽 = 𝝆. 𝑨. 𝒉 𝐹𝑜𝑟𝑐𝑒 𝑭 = 𝑾 = 𝝆. 𝑨. 𝒉. 𝒈 -------------------------------------------------------------- -------------------------------------------------------------𝑭 𝝆. 𝑨. 𝒉. 𝒈 Root-mean-square atomic velocity Buoyancy 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑷 = = 𝑨 𝒄𝒂𝒏𝒄𝒆𝒍𝒔 𝑨 𝑨 Pressure increases with depth. So the pressure at 𝟑𝑲𝑩 𝑻 𝟑𝑹𝑻 the bottom of a floating object is greater than on 𝑽𝑹𝑴𝑺 = ∴ 𝐏 = 𝛒𝐠𝐡 𝒎 𝑴 top. Thus the water exerts a net upward force on the object. This is the boyant force. Pressure vs depth (compressible fluids) 𝑤𝑒𝑖𝑔𝑡𝑜𝑏𝑗𝑒𝑐𝑡 𝑖𝑛 𝑎𝑖𝑟 > 𝑤𝑒𝑖𝑔𝑡𝑜𝑏𝑗𝑒𝑐𝑡 𝑖𝑛 𝑤𝑎𝑡𝑒𝑟 T= Temperature Kelvins 𝑃 + ∆𝑃 𝐴 − 𝑃𝐴 − 𝜌𝐴∆𝑔 = 0 m= mass (𝑃 + ∆𝑃) − 𝑃 − 𝜌∆𝑔 = 0 Archimedes’ Principal M= Molar mass of gas ∴ ∆𝐏 = 𝛒𝐠∆𝐡 The boyant force on an object immersed in fluid is ----------------------------------------------------------------------------------------------------------------------- equal to the weight of fluid displaced by that object. STP 𝑭𝑩 = 𝑾′ = 𝒎′𝒈 For pressure of fluid in container with lid open. P=101.325 kPa T=273.15K 22.414L Assume fluid is incompressible. -------------------------------------------------------------Pressure on the top surface 𝑊𝑒𝑟𝑒 𝑃2 = 𝑃𝐴 = 𝑃𝐴𝑡𝑚𝑜𝑠𝑝 𝑒𝑟𝑒 = 1.01325 × 105 𝑃1 = 𝜌𝐹 𝑔 ∆𝑃 = 𝜌𝑔∆ 𝑃1 − 𝑃2 = 𝜌𝑔 Force on the top surface ∴ 𝐏 = 𝐏𝐀 + 𝛒𝐠𝐡 𝐹1 = 𝑃1 𝐴 = 𝜌𝐹 𝑔2 Pressure on the bottom surface ----------------------------------------------------------𝑃2 = 𝜌𝐹 𝑔2 𝐴𝑡𝑚𝑜𝑠𝑝𝑒𝑟𝑖𝑐 𝑝𝑟𝑒𝑠𝑢𝑟𝑒 & 𝑔𝑎𝑢𝑔𝑒 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 Force on then bottom surface 𝐏𝐚𝐛𝐬𝐨𝐥𝐮𝐭𝐞= 𝐏𝐠𝐚𝐮𝐠𝐞 + 𝐏𝐚𝐭𝐦𝐬 markriley1985@hotmail.com 𝐹2 = 𝑃2 𝐴 = 𝜌𝐹 𝑔2 𝐴 ----------------------------------------------------------FB is the net force exerted by the fluid on the 𝑩𝒖𝒍𝒌 𝑴𝒐𝒅𝒖𝒍𝒖𝒔 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑠 𝑡𝑒 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑𝑠 𝑜𝑟 𝑠𝑜𝑙𝑖𝑑𝑠 𝑡𝑜 𝑐𝑎𝑛𝑔𝑒 𝑡𝑒𝑟𝑒 𝑣𝑜𝑙𝑢𝑚𝑒. submerged object 𝐹 = 𝐹 − 𝐹 𝐅 𝐵 2 1 = 𝜌𝐹 𝑔𝐴 2 − 1 = 𝜌𝐹 𝑔𝐴Δ Mark Riley 𝐁≡ 𝐯𝐨𝐥𝐮𝐦𝐞 𝐬𝐭𝐫𝐞𝐬𝐬 𝐀 = − ∆𝐏 =− ∆∀ ∆∀ 𝐯𝐨𝐥𝐮𝐦𝐞 𝐬𝐭𝐫𝐚𝐢𝐧 ∀𝟎 ∀𝟎 𝑭𝑩 = 𝝆𝑭𝒍𝒖𝒊𝒅 𝑽𝒅𝒊𝒔𝒑𝒈 𝑭𝑩 = 𝒎𝑭𝒍𝒖𝒊𝒅 𝒈 ----------------------------------------------------------- -------------------------------------------------------------𝑪𝒐𝒏𝒕𝒊𝒏𝒖𝒊𝒕𝒚 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝑉𝑖𝑠𝑐𝑜𝑠𝑖𝑡𝑦 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑𝑠 − 𝐼𝑛𝑡𝑒𝑟𝑛𝑎𝑙 𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛 (conservation of mass) 𝒄𝒐𝒆𝒇𝒇𝒊𝒄𝒊𝒆𝒏𝒕 𝒐𝒇 𝒗𝒊𝒔𝒄𝒐𝒔𝒊𝒕𝒚 𝜼 𝐼𝑛𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑏𝑙𝑒 𝐹𝑙𝑢𝑖𝑑𝑠 𝜌1 = 𝜌2 𝑜𝑟 𝜌𝑖 = 𝜌𝑜 𝑠𝑒𝑎𝑟 𝑠𝑡𝑟𝑒𝑠𝑠 𝜼= 𝝆𝟏 𝑨𝟏 𝑽𝟏 = 𝝆𝟐 𝑨𝟐 𝑽𝟐 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑐𝑎𝑛𝑔𝑒 𝑜𝑓 𝑠𝑒𝑎𝑟 𝑠𝑡𝑟𝑒𝑠𝑠 (𝜌𝐴𝑉)𝑖𝑛 − (𝜌𝐴𝑉)𝑜𝑢𝑡 = 0 𝑎𝑠 Δ𝑡 𝑡𝑒 𝑢𝑝𝑝𝑒𝑟 𝑝𝑙𝑎𝑡𝑒𝑠 𝑚𝑜𝑣𝑒 𝑥 𝑑𝑖𝑠𝑡 Δ𝑥 = 𝑣Δ𝑡 For multiple inputs & outputs 𝐹 𝑠𝑒𝑎𝑟 𝑠𝑡𝑟𝑒𝑠𝑠 = 𝐴 𝝆𝒊 𝑨𝒊 𝑽𝒊 = 𝝆𝒐 𝑨𝒐 𝑽𝒐 𝒊𝒏𝒑𝒖𝒕𝒔 𝒐𝒖𝒕𝒑𝒖𝒕𝒔 𝑆𝑒𝑎𝑟 𝑠𝑡𝑟𝑎𝑖𝑛 = Δ𝑥 𝑙 Δ𝑥 𝑣Δ𝑡 -------------------------------------------------------------𝑙= 𝑙 Δ𝑡 Δ𝑡
