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30750844 fluids-dynamics-formula-sheet
- 1. 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 = 𝐹𝑜𝑟𝑐𝑒 𝑁 𝐴𝑟𝑒𝑎𝑚3 𝐔𝐍𝐈𝐅𝐎𝐑𝐌 𝐅𝐎𝐑𝐂𝐄 𝐎𝐕𝐄𝐑 𝐀𝐑𝐄𝐀 𝐏 = 𝐅 𝐀 𝐍𝐎𝐍 − 𝐔𝐍𝐈𝐅𝐎𝐑𝐌 𝐅𝐎𝐑𝐂𝐄 𝐏 = ∆𝐅 ∆𝐀 ----------------------------------------------------------- 𝐷𝑒𝑛𝑠𝑖𝑡𝑦 𝛒 = 𝐦 ∀ = 𝑀𝑎𝑠𝑠 𝐾𝑔 𝑉𝑜𝑙𝑢𝑚𝑒 𝑚3 𝑳 → 𝒎 𝟑 =× 𝟏𝟎−𝟑 𝒎 𝟑 → 𝑳 = × 𝟏𝟎 𝟑 ----------------------------------------------------------- 𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝐺𝑟𝑎𝑣𝑖𝑡𝑦 𝑺𝑮 𝑖𝑠 𝑡𝑒 𝑟𝑎𝑡𝑖𝑜 𝑜𝑓 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑡𝑒 𝑠𝑢𝑏𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑜 𝑡𝑒 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟 𝑎𝑡 4° 𝐶 𝐒𝐆 = 𝛒 𝐬𝐮𝐛𝐬𝐭𝐚𝐧𝐜𝐞 𝛒 𝐰𝐚𝐭𝐞𝐫 𝐚𝐭 𝟒° 𝐂 = 𝛒 𝐬𝐮𝐛𝐬𝐭𝐚𝐧𝐜𝐞 𝟏. 𝟎𝟎𝟎 × 𝟏𝟎 𝟑 𝐤𝐠 𝐦 𝟑 ----------------------------------------------------------- Pressure vs depth (incompressible fluids) 𝑊𝑒𝑖𝑔𝑡 𝑜𝑓 𝑙𝑖𝑞𝑢𝑖𝑑 𝑾 = 𝒎. 𝒈 𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑙𝑖𝑞𝑢𝑖𝑑 𝑽 = 𝑨. 𝒉 𝑀𝑎𝑠𝑠 𝑜𝑓 𝑙𝑖𝑞𝑢𝑖𝑑 𝒎 = 𝝆𝑽 = 𝝆. 𝑨. 𝒉 𝐹𝑜𝑟𝑐𝑒 𝑭 = 𝑾 = 𝝆. 𝑨. 𝒉. 𝒈 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑷 = 𝑭 𝑨 = 𝝆. 𝑨. 𝒉. 𝒈 𝑨 𝑨 𝒄𝒂𝒏𝒄𝒆𝒍𝒔 ∴ 𝐏 = 𝛒𝐠𝐡 Pressure vs depth (compressible fluids) 𝑃 + ∆𝑃 𝐴 − 𝑃𝐴 − 𝜌𝐴∆𝑔 = 0 (𝑃 + ∆𝑃) − 𝑃 − 𝜌∆𝑔 = 0 ∴ ∆𝐏 = 𝛒𝐠∆𝐡 ----------------------------------------------------------- For pressure of fluid in container with lid open. Assume fluid is incompressible. 𝑊𝑒𝑟𝑒 𝑃2 = 𝑃𝐴 = 𝑃𝐴𝑡𝑚𝑜𝑠𝑝 𝑒𝑟𝑒 = 1.01325 × 105 ∆𝑃 = 𝜌𝑔∆ 𝑃1 − 𝑃2 = 𝜌𝑔 ∴ 𝐏 = 𝐏 𝐀 + 𝛒𝐠𝐡 ----------------------------------------------------------- 𝐴𝑡𝑚𝑜𝑠𝑝𝑒𝑟𝑖𝑐 𝑝𝑟𝑒𝑠𝑢𝑟𝑒 & 𝑔𝑎𝑢𝑔𝑒 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝐏𝐚𝐛𝐬𝐨𝐥𝐮𝐭𝐞= 𝐏𝐠𝐚𝐮𝐠𝐞 + 𝐏𝐚𝐭𝐦𝐬 ----------------------------------------------------------- 𝑩𝒖𝒍𝒌 𝑴𝒐𝒅𝒖𝒍𝒖𝒔 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑠 𝑡𝑒 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑𝑠 𝑜𝑟 𝑠𝑜𝑙𝑖𝑑𝑠 𝑡𝑜 𝑐𝑎𝑛𝑔𝑒 𝑡𝑒𝑟𝑒 𝑣𝑜𝑙𝑢𝑚𝑒. 𝐁 ≡ 𝐯𝐨𝐥𝐮𝐦𝐞 𝐬𝐭𝐫𝐞𝐬𝐬 𝐯𝐨𝐥𝐮𝐦𝐞 𝐬𝐭𝐫𝐚𝐢𝐧 = − 𝐅 𝐀 ∆∀ ∀ 𝟎 = − ∆𝐏 ∆∀ ∀ 𝟎 ----------------------------------------------------------- 𝑉𝑖𝑠𝑐𝑜𝑠𝑖𝑡𝑦 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑𝑠 − 𝐼𝑛𝑡𝑒𝑟𝑛𝑎𝑙 𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛 𝒄𝒐𝒆𝒇𝒇𝒊𝒄𝒊𝒆𝒏𝒕 𝒐𝒇 𝒗𝒊𝒔𝒄𝒐𝒔𝒊𝒕𝒚 𝜼 𝜼 = 𝑠𝑒𝑎𝑟 𝑠𝑡𝑟𝑒𝑠𝑠 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑐𝑎𝑛𝑔𝑒 𝑜𝑓 𝑠𝑒𝑎𝑟 𝑠𝑡𝑟𝑒𝑠𝑠 𝑎𝑠 Δ𝑡 𝑡𝑒 𝑢𝑝𝑝𝑒𝑟 𝑝𝑙𝑎𝑡𝑒𝑠 𝑚𝑜𝑣𝑒 𝑥 𝑑𝑖𝑠𝑡 Δ𝑥 = 𝑣Δ𝑡 𝑠𝑒𝑎𝑟 𝑠𝑡𝑟𝑒𝑠𝑠 = 𝐹 𝐴 𝑆𝑒𝑎𝑟 𝑠𝑡𝑟𝑎𝑖𝑛 = Δ𝑥 𝑙 Δ𝑥 𝑙 Δ𝑡 = 𝑣Δ𝑡 𝑙 Δ𝑡 𝜼 = 𝑭 𝑨 𝒗 𝒍 = 𝑭𝒍 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) -------------------------------------------------------------- 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝑇𝑒𝑛𝑠𝑖𝑜𝑛 𝜸 𝜸 = 𝑭 𝑳 -------------------------------------------------------------- Pascals principle ‘if an external pressure is applied to a confined fluid, the pressure at every point within the fluid increases by that amount’ eg Hydraulic Lift 𝑃1 = 𝑃2 𝐹1 𝐴1 = 𝐹2 𝐴2 Can be used to obtain mechanical advantage 𝐹2 = 𝐹1 𝐴2 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 𝑭 𝟏 𝚫𝒙 𝟏 = 𝑭 𝟐 𝚫𝒙 𝟐 -------------------------------------------------------------- Buoyancy Pressure increases with depth. So the pressure at the bottom of a floating object is greater than on top. Thus the water exerts a net upward force on the object. This is the boyant force. 𝑤𝑒𝑖𝑔𝑡 𝑜𝑏𝑗𝑒𝑐𝑡 𝑖𝑛 𝑎𝑖𝑟 > 𝑤𝑒𝑖𝑔𝑡 𝑜𝑏𝑗𝑒𝑐𝑡 𝑖𝑛 𝑤𝑎𝑡𝑒𝑟 Archimedes’ Principal The boyant force on an object immersed in fluid is equal to the weight of fluid displaced by that object. 𝑭 𝑩 = 𝑾′ = 𝒎′𝒈 Pressure on the top surface 𝑃1 = 𝜌 𝐹 𝑔 Force on the top surface 𝐹1 = 𝑃1 𝐴 = 𝜌 𝐹 𝑔2 Pressure on the bottom surface 𝑃2 = 𝜌 𝐹 𝑔2 Force on then bottom surface 𝐹2 = 𝑃2 𝐴 = 𝜌 𝐹 𝑔2 𝐴 FB is the net force exerted by the fluid on the submerged object 𝐹𝐵 = 𝐹2 − 𝐹1 = 𝜌 𝐹 𝑔𝐴 2 − 1 = 𝜌 𝐹 𝑔𝐴Δ 𝑭 𝑩 = 𝝆 𝑭𝒍𝒖𝒊𝒅 𝑽 𝒅𝒊𝒔𝒑 𝒈 𝑭 𝑩 = 𝒎 𝑭𝒍𝒖𝒊𝒅 𝒈 -------------------------------------------------------------- 𝑪𝒐𝒏𝒕𝒊𝒏𝒖𝒊𝒕𝒚 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏 (conservation of mass) 𝐼𝑛𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑏𝑙𝑒 𝐹𝑙𝑢𝑖𝑑𝑠 𝜌1 = 𝜌2 𝑜𝑟 𝜌𝑖 = 𝜌𝑜 𝝆 𝟏 𝑨 𝟏 𝑽 𝟏 = 𝝆 𝟐 𝑨 𝟐 𝑽 𝟐 (𝜌𝐴𝑉)𝑖𝑛 − (𝜌𝐴𝑉) 𝑜𝑢𝑡 = 0 For multiple inputs & outputs 𝝆𝒊 𝑨𝒊 𝑽𝒊 𝒊𝒏𝒑𝒖𝒕𝒔 = 𝝆 𝒐 𝑨 𝒐 𝑽 𝒐 𝒐𝒖𝒕𝒑𝒖𝒕𝒔 -------------------------------------------------------------- 𝑩𝒆𝒓𝒏𝒐𝒖𝒍𝒍𝒊𝒔 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 (conservation of energy) 𝑃1 + 1 2 𝜌1 𝑉1 2 + 𝜌𝑔𝑦1 = 𝑃2 + 1 2 𝜌𝑉2 2 + 𝜌𝑔𝑦2 Further common assumptions ONLY FOR SV 𝑃1 + 𝑃2 = 𝐴𝑇𝑀𝑂𝑆𝑃𝐻𝐸𝑅𝐼𝐶 𝑃𝑅𝐸𝑆𝑆𝑈𝑅𝐸 𝑉1 = 0 -------------------------------------------------------------- Ideal Gas equation 𝑷𝒗 = 𝑵 𝑨 𝒌 𝑩 𝑻 = 𝒏𝑹𝑻 𝑅 = 𝑔𝑎𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 8.3145 𝐽𝐾−1 𝑚𝑜𝑙−1 -------------------------------------------------------------- Real Gas equation 𝒑𝑽 𝒏𝑹𝑻 = 𝒁 Z= compressibility & is dimensionless -------------------------------------------------------------- Root-mean-square atomic velocity 𝑽 𝑹𝑴𝑺 = 𝟑𝑲 𝑩 𝑻 𝒎 𝟑𝑹𝑻 𝑴 T= Temperature Kelvins m= mass M= Molar mass of gas ------------------------------------------------------------- STP P=101.325 kPa T=273.15K 22.414L -------------------------------------------------------------- Mark Riley markriley1985@hotmail.com