Mathematical Problems in Engineering

Research Article

Group-Invariant Solutions for Two-Dimensional Free, Wall, and Liquid Jets Having Finite Fluid Velocity at Orifice

Table 1

Comparison between two-dimensional jets with finite velocity at orifice and infinite velocity at orifice.
Finite velocity at orificeInfinite velocity at orifice
2-D free jet 𝜓 = 9 𝜈 𝐽 2 𝜌 𝑥 + 2 𝑐 2 3 𝑐 1 1 / 3 𝑓 ( 𝜂 ) 𝜓 = 9 𝜈 𝐽 𝑥 2 𝜌 1 / 3 𝑓 ( 𝜂 )
𝑢 ( 𝑥 , 𝑦 ) = 3 𝐽 2 4 𝜌 2 𝜈 ( 𝑥 + 2 𝑐 2 / 3 𝑐 1 ) 1 / 3 𝑓 ′ ( 𝜂 ) 𝑢 ( 𝑥 , 𝑦 ) = 3 𝐽 2 4 𝜌 2 𝜈 ( 𝑥 + 2 𝑐 2 / 3 𝑐 1 ) 1 / 3 𝑓 ( 𝜂 )
𝐽 𝜂 = 6 𝜌 𝜈 2 ( 𝑥 + 2 𝑐 2 / 3 𝑐 1 ) 2 1 / 3 𝑦 𝐽 𝜂 = 6 𝜌 𝜈 2 1 / 3 𝑦 𝑥 2 / 3
𝑓 + 𝑓 𝑓 + 𝑓 ′ 2 = 0 𝑓 + 𝑓 𝑓 + 𝑓 ′ 2 = 0
3 𝑋 = 2 𝑐 𝑥 + 2 𝑐 1 𝜕 𝜕 𝜕 𝑥 + 𝑦 + 1 𝜕 𝑦 2 𝜓 𝜕 𝜕 𝜓 3 𝑋 = 2 𝑥 𝜕 𝜕 𝜕 𝑥 + 𝑦 + 1 𝜕 𝑦 2 𝜓 𝜕 𝜕 𝜓
2-D wall jet 𝜓 = 4 0 𝐹 𝜈 𝑥 + 3 𝑐 2 4 𝑐 1 1 / 4 𝑓 ( 𝜂 ) 𝜓 = [ 4 0 𝐹 𝜈 𝑥 ] 1 / 4 𝑓 ( 𝜂 )
𝜂 = 5 𝐹 3 2 𝜈 3 ( 𝑥 + 3 𝑐 2 / 4 𝑐 1 ) 3 1 / 4 𝑦 𝜂 = 5 𝐹 3 2 𝜈 3 1 / 4 𝑦 𝑥 3 / 4
𝑢 ( 𝑥 , 𝑦 ) = 5 𝐹 2 𝜈 ( 𝑥 + 3 𝑐 2 / 4 𝑐 1 ) 1 / 2 𝑓 ( 𝜂 ) 𝑢 ( 𝑥 , 𝑦 ) = 5 𝐹 2 𝜈 𝑥 1 / 2 𝑓 ( 𝜂 )
𝑓 + 𝑓 𝑓 + 2 𝑓 ′ 2 = 0 𝑓 + 𝑓 𝑓 + 2 𝑓 ′ 2 = 0
4 𝑋 = 3 𝑐 𝑥 + 2 𝑐 1 𝜕 𝜕 𝜕 𝑥 + 𝑦 + 1 𝜕 𝑦 3 𝜓 𝜕 𝜕 𝜓 4 𝑋 = 3 𝑥 𝜕 𝜕 𝜕 𝑥 + 𝑦 + 1 𝜕 𝑦 3 𝜓 𝜕 𝜕 𝜓
2-D liquid jet 3 √ 𝜓 ( 𝑥 , 𝑦 ) = 3 𝑀 𝜋 𝑓 ( 𝜂 ) 3 √ 𝜓 ( 𝑥 , 𝑦 ) = 3 𝑀 𝜋 𝑓 ( 𝜂 )
√ 𝜂 = 3 𝑀 𝜈 𝜋 ( 𝑥 + 𝑐 2 / 𝑐 1 ) 𝑦 √ 𝜂 = 3 𝑀 𝑦 𝜈 𝜋 𝑥
𝑢 ( 𝑥 , 𝑦 ) = 9 𝑀 2 𝜈 𝜋 2 ( 𝑥 + 𝑐 2 / 𝑐 1 ) 𝑓 ( 𝜂 ) 𝑢 ( 𝑥 , 𝑦 ) = 9 𝑀 2 𝜈 𝜋 2 𝑥 𝑓 ( 𝜂 )
𝑓 + 3 𝑓 ′ 2 = 0 𝑓 + 3 𝑓 ′ 2 = 0
𝑐 𝑋 = 𝑥 + 2 𝑐 1 𝜕 𝜕 𝜕 𝑥 + 𝑦 𝜕 𝑦 𝜕 𝑋 = 𝑥 𝜕 𝜕 𝑥 + 𝑦 𝜕 𝑦