Formula Sheet MECH-3220-S22
Fluid Kinematics
Acceleration field
𝑎⃗𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 (𝑥, 𝑦, 𝑧, 𝑡) =
𝛻=
Equation of a streamline
𝑑𝑟⃗ 𝑑𝑥 𝑑𝑦 𝑑𝑧
=
=
=
⃗⃗
𝑢
𝑣
𝑤
𝑉
⃗⃗ 𝜕𝑉
⃗⃗
𝑑𝑉
⃗⃗. 𝛻⃗⃗ )𝑉
⃗⃗
=
+ (𝑉
𝑑𝑡
𝜕𝑡
𝜕( )
𝜕( )
𝜕( )
𝑖⃗ +
𝑗⃗ +
𝑘⃗⃗
𝜕𝑥
𝜕𝑦
𝜕𝑥
𝐷( ) 𝜕( )
⃗⃗. 𝛻⃗⃗)( )
=
+ (𝑉
𝐷𝑡
𝜕𝑡
1 𝑑𝑉
𝜕𝑢 𝜕𝑣 𝜕𝑤
⃗⃗
= 𝜀𝑥𝑥 + 𝜀𝑦𝑦 + 𝜀𝑧𝑧 =
+
+
= 𝛻. 𝑉
𝑉 𝑑𝑡
𝜕𝑥 𝜕𝑦 𝜕𝑧
1 𝜕𝑤 𝜕𝑣 → 1 𝜕𝑢 𝜕𝑤 → 1 𝜕𝑣 𝜕𝑢 →
→
𝜔= (
− )𝑖 + ( −
) 𝑗 + ( − )𝑘
2 𝜕𝑦 𝜕𝑧
2 𝜕𝑧 𝜕𝑥
2 𝜕𝑥 𝜕𝑦
→
→
→
𝜕𝑢
𝜕𝑢
𝜕𝑢
𝜕𝑢
+𝑢
+𝑣
+𝑤
𝜕𝑡
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑣
𝜕𝑣
𝜕𝑣
𝜕𝑣
𝑎𝑦 =
+𝑢
+𝑣
+𝑤
𝜕𝑡
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑤
𝜕𝑤
𝜕𝑤
𝜕𝑤
𝑎𝑧 =
+𝑢
+𝑣
+𝑤
𝜕𝑡
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝑎𝑥 =
𝜀𝑥𝑥 =
𝜕𝑢
,
𝜕𝑥
→
𝑉𝑜𝑟𝑡𝑖𝑐𝑖𝑡𝑦 𝑣𝑒𝑐𝑡𝑜𝑟 𝑖𝑛 𝑐𝑦𝑙𝑖𝑛𝑑𝑟𝑖𝑐𝑎𝑙 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠:
→
1 𝜕𝑢𝑧 𝜕𝑢𝜃 →
𝜕𝑢𝑟 𝜕𝑢𝑧 →
1 𝜕(𝑟𝑢𝜃 ) 𝜕𝑢𝑟 →
𝜁 =(
−
) 𝑒𝑟 + (
−
) 𝑒𝜃 + (
−
) 𝑒𝑧
𝑟 𝜕𝜃
𝜕𝑧
𝜕𝑧
𝜕𝑟
𝑟
𝜕𝑟
𝜕𝜃
𝐶𝑎𝑢𝑐ℎ𝑦’𝑠 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛:
𝜕
⃗⃗) + 𝛻⃗⃗ ⋅ (𝜌𝑉
⃗⃗𝑉
⃗⃗) = 𝜌𝑔⃗ + 𝛻⃗⃗ · 𝜎𝑖𝑗
(𝜌𝑉
𝜕𝑡
𝜕𝑣
,
𝜕𝑦
𝜀𝑧𝑧 =
𝜕𝑤
𝜕𝑧
1
1
→
⃗⃗ = curl⃗⃗⃗⃗
𝜔= 𝛻×𝑉
𝑉
2
2
𝜁 = 2𝜔
⃗⃗ = 𝛻 × 𝑉 = curl 𝑉
𝜕𝑢
1 𝜕𝑢 𝜕𝑣
1 𝜕𝑢 𝜕𝑤
( + )
( +
)
𝜕𝑥
2 𝜕𝑦 𝜕𝑥
2 𝜕𝑧 𝜕𝑥
1 𝜕𝑣 𝜕𝑢
𝜕𝑣
1 𝜕𝑣 𝜕𝑤
𝜀𝑖𝑗 =
( + )
( +
)
2 𝜕𝑥 𝜕𝑦
𝜕𝑦
2 𝜕𝑧 𝜕𝑦
1 𝜕𝑤 𝜕𝑢
1 𝜕𝑤 𝜕𝑣
𝜕𝑤
(
+ )
(
+ )
2 𝜕𝑦 𝜕𝑧
𝜕𝑧 )
( 2 𝜕𝑥 𝜕𝑧
𝜀𝑦𝑦 =
1 𝜕𝑢 𝜕𝑣
𝜀𝑥𝑦 = ( + )
2 𝜕𝑦 𝜕𝑥
1 𝜕𝑤 𝜕𝑢
𝜀𝑧𝑥 = (
+ )
2 𝜕𝑥 𝜕𝑧
1 𝜕𝑣 𝜕𝑤
𝜀𝑦𝑧 = ( +
)
2 𝜕𝑧 𝜕𝑦
Continuity:
𝜕𝜌 𝜕(𝜌𝑢) 𝜕(𝜌𝜐) 𝜕(𝜌𝑤)
+
+
+
=0
𝜕𝑡
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝜌 1 𝜕(𝑟𝜌𝑢𝑟 ) 1 𝜕(𝜌𝑢𝜃 ) 𝜕(𝜌𝑢𝑧 )
+
+
+
=0
𝜕𝑡 𝑟 𝜕𝑟
𝑟 𝜕𝜃
𝜕𝑧
𝐵𝑒𝑟𝑛𝑜𝑢𝑙𝑙𝑖 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑓𝑜𝑟 𝑖𝑟𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑓𝑙𝑜𝑤:
𝑃 𝑉2
+
+ 𝑔𝑦 = 𝐶 = constant everywhere
𝜌
2
Potential flow
1
Flow over Stationary
Circular Cylinder
𝑎2
𝜓 = 𝑉∞ 𝑟 (1 − 𝑟 2 ) sin 𝜃;
𝐾
𝑎=√
𝑉∞
𝑝𝑠 = 𝑝 +
𝜌𝑉∞ 2
(1 − 4 𝑠𝑖𝑛2 𝜃 )
2
Stress Tensor in Cylindrical Coordinates
Newtonian Incompressible Flow
Navier-Stokes Equations in Cartesian Coordinates
𝜕𝑢
𝜕𝑢
𝜕𝑢
𝜕𝑢
𝑥 − 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡: 𝜌 ( + 𝑢
+𝜐
+𝑤 )
𝜕𝑡
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑃
𝜕2𝑢 𝜕2𝑢 𝜕2𝑢
=−
+ 𝜌𝑔𝑥 + 𝜇 ( 2 + 2 + 2 )
𝜕𝑥
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝜐
𝜕𝜐
𝜕𝜐
𝜕𝜐
𝑦 − 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡: 𝜌 ( + 𝑢
+𝜐
+𝑤 )
𝜕𝑡
𝜕𝑥
𝜕𝑦
𝜕𝑧
2
𝜕𝑃
𝜕 𝜐 𝜕2𝜐 𝜕2𝜐
=−
+ 𝜌𝑔𝑦 + 𝜇 ( 2 + 2 + 2 )
𝜕𝑦
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑤
𝜕𝑤
𝜕𝑤
𝜕𝑤
𝑧 − 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡 : 𝜌 (
+𝑢
+𝜐
+𝑤
)
𝜕𝑡
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑃
𝜕2𝑤 𝜕2𝑤 𝜕2𝑤
=−
+ 𝜌𝑔𝑧 + 𝜇 ( 2 + 2 + 2 )
𝜕𝑧
𝜕𝑥
𝜕𝑦
𝜕𝑧
Navier Stokes Equations in Cylindrical Coordinates
𝜕𝑢
𝜕𝑢 𝜕𝜐
𝜕𝑢 𝜕𝑤
𝜇( + ) 𝜇( +
)
𝜕𝑥
𝜕𝑦 𝜕𝑥
𝜕𝑧 𝜕𝑥
𝜕𝜐 𝜕𝑢
𝜕𝜐
𝜕𝜐 𝜕𝑤
𝜏𝑖𝑗 = 𝜇 ( + )
2𝜇
𝜇( +
)
𝜕𝑥 𝜕𝑦
𝜕𝑦
𝜕𝑧 𝜕𝑦
𝜕𝑤 𝜕𝑢
𝜕𝑤 𝜕𝜐
𝜕𝑤
𝜇(
+ ) 𝜇(
+ )
2𝜇
(
𝜕𝑥 𝜕𝑧
𝜕𝑦 𝜕𝑧
𝜕𝑧 )
2𝜇
2
Fully developed laminar flow in a pipe
Viscous flow in pipes
Non-circular Ducts
Hydraulic diameter: Dh = 4Ac/P Ac = cross-section area, P = wetted perimeter
𝑈𝐿 𝜌𝑈𝐿
=
𝑣
𝜇
Reynolds number
𝑅𝑒 =
Energy equation:
Head Basis:
𝑝in
𝛾
+
𝛼in 𝑉in2
2𝑔
+ 𝑧in + ℎ𝑝 =
𝑝out
𝛾
+
2
𝛼out 𝑉out
2𝑔
+ 𝑧out + ℎ𝑡 + ℎ𝐿
𝛼𝑖𝑛 𝑎𝑛𝑑 𝛼𝑜𝑢𝑡 = kinetic energy correction coefficients
3
Moody diagram
4