Fluids Dynamics Formula Sheet

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 𝑡𝑕𝑒 𝑟𝑎𝑡𝑖𝑜 𝑜𝑓 the pressure at every point within the fluid increases 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑡𝑕𝑒 𝑠𝑢𝑏𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑜 𝑡𝑕𝑒 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 𝑊𝑒𝑖𝑔𝑕𝑡 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) 𝑤𝑒𝑖𝑔𝑕𝑡𝑜𝑏𝑗𝑒𝑐𝑡 𝑖𝑛 𝑎𝑖𝑟 > 𝑤𝑒𝑖𝑔𝑕𝑡𝑜𝑏𝑗𝑒𝑐𝑡 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 𝑊𝑕𝑒𝑟𝑒 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 𝑔𝑕2 Pressure on the bottom surface ----------------------------------------------------------𝑃2 = 𝜌𝐹 𝑔𝑕2 𝐴𝑡𝑚𝑜𝑠𝑝𝑕𝑒𝑟𝑖𝑐 𝑝𝑟𝑒𝑠𝑢𝑟𝑒 & 𝑔𝑎𝑢𝑔𝑒 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 Force on then bottom surface 𝐏𝐚𝐛𝐬𝐨𝐥𝐮𝐭𝐞= 𝐏𝐠𝐚𝐮𝐠𝐞 + 𝐏𝐚𝐭𝐦𝐬 [email protected] 𝐹2 = 𝑃2 𝐴 = 𝜌𝐹 𝑔𝑕2 𝐴 ----------------------------------------------------------FB is the net force exerted by the fluid on the 𝑩𝒖𝒍𝒌 𝑴𝒐𝒅𝒖𝒍𝒖𝒔 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑠 𝑡𝑕𝑒 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑𝑠 𝑜𝑟 𝑠𝑜𝑙𝑖𝑑𝑠 𝑡𝑜 𝑐𝑕𝑎𝑛𝑔𝑒 𝑡𝑕𝑒𝑟𝑒 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 𝑠𝑕𝑒𝑎𝑟 𝑠𝑡𝑟𝑒𝑠𝑠 𝜼= 𝝆𝟏 𝑨𝟏 𝑽𝟏 = 𝝆𝟐 𝑨𝟐 𝑽𝟐 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑐𝑕𝑎𝑛𝑔𝑒 𝑜𝑓 𝑠𝑕𝑒𝑎𝑟 𝑠𝑡𝑟𝑒𝑠𝑠 (𝜌𝐴𝑉)𝑖𝑛 − (𝜌𝐴𝑉)𝑜𝑢𝑡 = 0 𝑎𝑠 Δ𝑡 𝑡𝑕𝑒 𝑢𝑝𝑝𝑒𝑟 𝑝𝑙𝑎𝑡𝑒𝑠 𝑚𝑜𝑣𝑒 𝑥 𝑑𝑖𝑠𝑡 Δ𝑥 = 𝑣Δ𝑡 For multiple inputs & outputs 𝐹 𝑠𝑕𝑒𝑎𝑟 𝑠𝑡𝑟𝑒𝑠𝑠 = 𝐴 𝝆𝒊 𝑨𝒊 𝑽𝒊 = 𝝆𝒐 𝑨𝒐 𝑽𝒐 𝒊𝒏𝒑𝒖𝒕𝒔 𝒐𝒖𝒕𝒑𝒖𝒕𝒔 𝑆𝑕𝑒𝑎𝑟 𝑠𝑡𝑟𝑎𝑖𝑛 = Δ𝑥 𝑙 Δ𝑥 𝑣Δ𝑡 -------------------------------------------------------------𝑙= 𝑙 Δ𝑡 Δ𝑡