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                                         Reynold's Theory Fig: Transfer of momentum in turbulent flow Relative velocity of layer 'A' w.r.t. layer 'B' is So, fluctuating components in X and Y directions due to turbulence are Vx and Vy . Now if over a surface of area 'A' perpendicular to the Y-direction and separating two adjacent fluid layers the component 'Vy ' is uniformly distributed.  So, Mass of fluid transferred across that surface from one layer to another per second =                                                        pAVy This mass of fluid has started moving with a relative velocity 'Vx'. So, Transfer of momentum = pAVxVy takes place, resulting in developing tangential forces on each of the layers. Now, the corresponding turbulent shear stress exerted on fluid layers (  𝜏 )  = pVx Vy  Since, Vx and Vy are varying, the magnitude of ' 𝜏'  will also vary. Hence, usually the time average value of  shear stress is considered

  Reynold's Experiment • Osborne Reynolds (Mathematician & Physicist, UK) • In 1883, he developed a laboratory set up in which he injected Dye (i.e. a fine, threadlike stream  of colored liquid having the same density as water) at the entrance to a large glass tube through  which water was flowing from a tank .    Fig: Reynold's experiment Ad: https://happyshirtsnp.com/ • Procedure: ✓ The water from the tank was allowed to flow through glass tube into atmosphere. ✓ The velocity of flow was varied by the valve. ✓ Dye was injected into the flow through a small tube. • Observation: ✓ At the initial stage of flow, dye filament in the glass tube was in the form of straight line. This was  a laminar flow. ✓ When increasing the velocity of flow, the dye filament was no longer a straight line. The dye  filament starts to become wavy. This was a transitional flow. ✓ While further increase in velocity of flow, the wavy dye filament broke up and finally diffused in  water. This was a