Fluids

1. 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 =
𝐹𝑜𝑟𝑐𝑒 𝑁
𝐴𝑟𝑒𝑎 𝑚3
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 𝑡𝑕𝑒 𝑟𝑎𝑡𝑖𝑜 𝑜𝑓
𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑡𝑕𝑒 𝑠𝑢𝑏𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑜 𝑡𝑕𝑒
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)
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
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.
𝑊𝑕𝑒𝑟𝑒 𝑃2 = 𝑃𝐴 = 𝑃𝐴𝑡𝑚𝑜𝑠𝑝 𝑕𝑒𝑟𝑒 = 1.01325 × 105
∆𝑃 = 𝜌𝑔∆𝑕 𝑃1 − 𝑃2 = 𝜌𝑔𝑕
∴ 𝐏 = 𝐏 𝐀 + 𝛒𝐠𝐡
-----------------------------------------------------------
𝐴𝑡𝑚𝑜𝑠𝑝𝑕𝑒𝑟𝑖𝑐 𝑝𝑟𝑒𝑠𝑢𝑟𝑒 & 𝑔𝑎𝑢𝑔𝑒 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒
𝐏𝐚𝐛𝐬𝐨𝐥𝐮𝐭𝐞= 𝐏𝐠𝐚𝐮𝐠𝐞 + 𝐏𝐚𝐭𝐦𝐬
-----------------------------------------------------------
𝑩𝒖𝒍𝒌 𝑴𝒐𝒅𝒖𝒍𝒖𝒔 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑠 𝑡𝑕𝑒 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓
𝑓𝑙𝑢𝑖𝑑𝑠 𝑜𝑟 𝑠𝑜𝑙𝑖𝑑𝑠 𝑡𝑜 𝑐𝑕𝑎𝑛𝑔𝑒 𝑡𝑕𝑒𝑟𝑒 𝑣𝑜𝑙𝑢𝑚𝑒.
𝐁 ≡
𝐯𝐨𝐥𝐮𝐦𝐞 𝐬𝐭𝐫𝐞𝐬𝐬
𝐯𝐨𝐥𝐮𝐦𝐞 𝐬𝐭𝐫𝐚𝐢𝐧
= −
𝐅
𝐀
∆∀
∀ 𝟎
= −
∆𝐏
∆∀
∀ 𝟎
-----------------------------------------------------------
𝑉𝑖𝑠𝑐𝑜𝑠𝑖𝑡𝑦 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑𝑠 − 𝐼𝑛𝑡𝑒𝑟𝑛𝑎𝑙 𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛
𝒄𝒐𝒆𝒇𝒇𝒊𝒄𝒊𝒆𝒏𝒕 𝒐𝒇 𝒗𝒊𝒔𝒄𝒐𝒔𝒊𝒕𝒚 𝜼
𝜼 =
𝑠𝑕𝑒𝑎𝑟 𝑠𝑡𝑟𝑒𝑠𝑠
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
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.
𝑤𝑒𝑖𝑔𝑕𝑡 𝑜𝑏𝑗𝑒𝑐𝑡 𝑖𝑛 𝑎𝑖𝑟 > 𝑤𝑒𝑖𝑔𝑕𝑡 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 𝑔𝑕2
Pressure on the bottom surface
𝑃2 = 𝜌 𝐹 𝑔𝑕2
Force on then bottom surface
𝐹2 = 𝑃2 𝐴 = 𝜌 𝐹 𝑔𝑕2 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