It is the distance perpendicular to the boundary, by which the boundary should be displaced outward to compensate for reduction in the discharge in the boundary layer formation.

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

Consider the fluid flow having free-stream velocity ‘U’ over a thin smooth plate as shown in.


Consider elementary strip at a distance ‘y’ from the plate and ‘dy’ is thickness.

Let u be the velocity at elemental strip and b is width of plate.
Mass per second through strip

= ρ . V. A
= ρ . u. (dy x b)
= ρu . b dy

Mass per second through strip if no plate is placed i.e. u = U.

= ρ U b. dy

Total reduction of mass / second of strip,

= ρ U b. dy - ρ  ub dy

= ρ b dy (U - U)
                                                                                                           δ
Total reduction of mass / second for whole boundary laye = ∫  ρ b dy (U -η)
                                                                                                           0

Let the plate displaced by a distance ( ) and velocity of flow for the distance ( ) is free stream velocity ‘U’.

Loss of mass / second flowing through distance δ*

= ρ x Velocity x Area
= ρ x U x b δ

Equating Equations

                   δ
ρ U b δ*  = ∫  ρ b dy (U - u)
                   0
        δ
δ* = ∫ U - u / U dy
       0

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

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