It is the distance perpendicular to solid boundary, by which the boundary should be displaced to compensate for the reduction in kinetic energy of the flowing fluid in the formation of boundary layers.

Kinetic energy of fluid, if plate is placed  = ½ m V² = (ρ u b dy) u²

Kinetic energy of fluid if no plate is placed = ½ m V² = (ρ u b dy) U²

Loss of Kinetic energy = ½  ρ u b (U² - u²) dy

Total loss of Kinetic energy in BC

= 1/2 ρ u b ∫ ( U² - u²) dy

Loss of Kinetic energy through ( ) of fluid flowing with velocity U.

= ½ mass x velocity2 = ½  (ρ b δ_e U) U³

= ½ (ρ b δe) u²

Equating Equations

                                              δ
1/2 ρ b δ_e U³  = 1/2 ρ b ∫ U (U² - u²) dy
                                              0

         δ
δ_e  = ∫  u/U (U² - U² / U²) dy
         0

         δ
δ_e  = ∫ u/U (1 - u/U²) dy
         0

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