30750844 Fluids Dynamics Formula Sheet PDF

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𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 =

𝐹𝑜𝑟𝑐𝑒 𝑁 𝐴𝑟𝑒𝑎 𝑚3

𝐔𝐍𝐈𝐅𝐎𝐑𝐌 𝐅𝐎𝐑𝐂𝐄 𝐎𝐕𝐄𝐑 𝐀𝐑𝐄𝐀 𝐏 =

𝐅 𝐀

∆𝐅 ∆𝐀 ----------------------------------------------------------𝐍𝐎𝐍 − 𝐔𝐍𝐈𝐅𝐎𝐑𝐌 𝐅𝐎𝐑𝐂𝐄 𝐏 =

𝐦 𝑀𝑎𝑠𝑠 𝐾𝑔 𝐷𝑒𝑛𝑠𝑖𝑡𝑦 𝛒 = = ∀ 𝑉𝑜𝑙𝑢𝑚𝑒 𝑚3

𝑳 → 𝒎𝟑 =× 𝟏𝟎−𝟑

𝒎𝟑 → 𝑳 = × 𝟏𝟎𝟑

----------------------------------------------------------𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝐺𝑟𝑎𝑣𝑖𝑡𝑦 𝑺𝑮 𝑖𝑠 𝑡𝑕𝑒 𝑟𝑎𝑡𝑖𝑜 𝑜𝑓 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑡𝑕𝑒 𝑠𝑢𝑏𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑜 𝑡𝑕𝑒 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟 𝑎𝑡 4° 𝐶

𝛒𝐬𝐮𝐛𝐬𝐭𝐚𝐧𝐜𝐞 𝛒𝐬𝐮𝐛𝐬𝐭𝐚𝐧𝐜𝐞 𝐒𝐆 = = 𝛒𝐰𝐚𝐭𝐞𝐫 𝐚𝐭 𝟒° 𝐂 𝟏. 𝟎𝟎𝟎 × 𝟏𝟎𝟑 𝐤𝐠 𝐦𝟑

----------------------------------------------------------Pressure vs depth (incompressible fluids) 𝑊𝑒𝑖𝑔𝑕𝑡 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 𝑭 𝝆. 𝑨. 𝒉. 𝒈 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑷 = = 𝑨 𝒄𝒂𝒏𝒄𝒆𝒍𝒔 𝑨 𝑨 ∴ 𝐏 = 𝛒𝐠𝐡

𝑭 𝑭𝒍 𝜼= 𝒗𝑨= 𝚫𝒗 𝒍 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’

𝑩𝒆𝒓𝒏𝒐𝒖𝒍𝒍𝒊𝒔 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 (conservation of energy)

1 2 2 𝑃1 + 2 𝜌1 𝑉1 + 𝜌𝑔𝑦1 = 𝑃2 + 2 𝜌𝑉2 + 𝜌𝑔𝑦2 1

Further common assumptions ONLY FOR SV 𝑃1 + 𝑃2 = 𝐴𝑇𝑀𝑂𝑆𝑃𝐻𝐸𝑅𝐼𝐶 𝑃𝑅𝐸𝑆𝑆𝑈𝑅𝐸 𝑉1 = 0 -------------------------------------------------------------Ideal Gas equation 𝑷𝒗 = 𝑵𝑨 𝒌𝑩 𝑻 = 𝒏𝑹𝑻

𝑃1 = 𝑃2 𝑅 = 𝑔𝑎𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 8.3145 𝐽𝐾−1 𝑚𝑜𝑙 −1 𝐹2 -------------------------------------------------------------𝐴2 𝐴1 = Real Gas equation Can be used to obtain mechanical advantage 𝐴2 𝒑𝑽 𝐹2 = 𝐹1 =𝒁 𝐴1 𝒏𝑹𝑻 Work done is the same by which the surface A2 rises is smaller than the change in the height of surface Z= compressibility & is dimensionless with area A 𝑭𝟏 𝚫𝒙𝟏 = 𝑭𝟐 𝚫𝒙𝟐

eg Hydraulic Lift 𝐹1

-------------------------------------------------------------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. 𝑤𝑒𝑖𝑔𝑕𝑡𝑜𝑏𝑗𝑒𝑐𝑡 𝑖𝑛 𝑎𝑖𝑟 > 𝑤𝑒𝑖𝑔𝑕𝑡𝑜𝑏𝑗𝑒𝑐𝑡 𝑖𝑛 𝑤𝑎𝑡𝑒𝑟

-------------------------------------------------------------Root-mean-square atomic velocity 𝑽𝑹𝑴𝑺 =

𝟑𝑲𝑩 𝑻 𝟑𝑹𝑻 𝑴 𝒎

Pressure vs depth (compressible fluids) 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 ----------------------------------------------------------𝐴𝑡𝑚𝑜𝑠𝑝𝑕𝑒𝑟𝑖𝑐 𝑝𝑟𝑒𝑠𝑢𝑟𝑒 & 𝑔𝑎𝑢𝑔𝑒 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑃2 = 𝜌𝐹 𝑔𝑕2 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 𝐹𝐵 = 𝐹2 − 𝐹1 = 𝜌𝐹 𝑔𝐴 𝑕2 − 𝑕1 = 𝜌𝐹 𝑔𝐴Δ𝑕 𝐅 𝐯𝐨𝐥𝐮𝐦𝐞 𝐬𝐭𝐫𝐞𝐬𝐬 ∆𝐏

Mark Riley

𝐁≡

𝐯𝐨𝐥𝐮𝐦𝐞 𝐬𝐭𝐫𝐚𝐢𝐧

=−

∆∀

𝐀 =− ∆∀ ∀𝟎 ∀𝟎

----------------------------------------------------------𝑉𝑖𝑠𝑐𝑜𝑠𝑖𝑡𝑦 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑𝑠 − 𝐼𝑛𝑡𝑒𝑟𝑛𝑎𝑙 𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛 𝒄𝒐𝒆𝒇𝒇𝒊𝒄𝒊𝒆𝒏𝒕 𝒐𝒇 𝒗𝒊𝒔𝒄𝒐𝒔𝒊𝒕𝒚 𝜼 𝑠𝑕𝑒𝑎𝑟 𝑠𝑡𝑟𝑒𝑠𝑠 𝜼= 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑐𝑕𝑎𝑛𝑔𝑒 𝑜𝑓 𝑠𝑕𝑒𝑎𝑟 𝑠𝑡𝑟𝑒𝑠𝑠 𝑎𝑠 Δ𝑡 𝑡𝑕𝑒 𝑢𝑝𝑝𝑒𝑟 𝑝𝑙𝑎𝑡𝑒𝑠 𝑚𝑜𝑣𝑒 𝑥 𝑑𝑖𝑠𝑡 Δ𝑥 = 𝑣Δ𝑡 𝑠𝑕𝑒𝑎𝑟 𝑠𝑡𝑟𝑒𝑠𝑠 = 𝐹 𝐴 𝑆𝑕𝑒𝑎𝑟 𝑠𝑡𝑟𝑎𝑖𝑛 = Δ𝑥 𝑙 𝑣Δ𝑡 Δ𝑥 𝑙 𝑙= Δ𝑡 Δ𝑡

𝑭𝑩 = 𝝆𝑭𝒍𝒖𝒊𝒅 𝑽𝒅𝒊𝒔𝒑𝒈

𝑭𝑩 = 𝒎𝑭𝒍𝒖𝒊𝒅 𝒈

-------------------------------------------------------------𝑪𝒐𝒏𝒕𝒊𝒏𝒖𝒊𝒕𝒚 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏 (conservation of mass) 𝐼𝑛𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑏𝑙𝑒 𝐹𝑙𝑢𝑖𝑑𝑠 𝜌1 = 𝜌2 𝑜𝑟 𝜌𝑖 = 𝜌𝑜 𝝆𝟏 𝑨𝟏 𝑽𝟏 = 𝝆𝟐 𝑨𝟐 𝑽𝟐 (𝜌𝐴𝑉)𝑖𝑛 − (𝜌𝐴𝑉)𝑜𝑢𝑡 = 0

For multiple inputs & outputs 𝝆𝒊 𝑨𝒊 𝑽𝒊 𝒊𝒏𝒑𝒖𝒕𝒔

=

𝝆𝒐 𝑨𝒐𝑽𝒐 𝒐𝒖𝒕𝒑𝒖𝒕𝒔

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