Shear stress in a turbulent flow is due to:

a) Viscous property of the fluid:
- Viscosity is the property of a fluid that determines its resistance to flow.
- In a turbulent flow, the fluid particles move in a chaotic and irregular manner, causing the fluid to mix and the velocity to fluctuate.
- The viscosity of the fluid plays a significant role in determining the shear stress in a turbulent flow.
- As the fluid particles move at different velocities and directions, they exert frictional forces on each other, resulting in shear stress.
- The magnitude of shear stress is directly proportional to the viscosity of the fluid.

b) Fluid density:
- Fluid density refers to the mass per unit volume of the fluid.
- In a turbulent flow, the density of the fluid does not directly contribute to the shear stress.
- However, the density of the fluid affects the momentum transfer and the overall behavior of the turbulent flow.
- The turbulent flow is characterized by the presence of eddies and vortices, which have a significant impact on the shear stress.
- The density of the fluid influences the formation and behavior of these eddies, indirectly affecting the shear stress.

c) Fluctuation of velocity in the direction of flow:
- In a turbulent flow, the velocity of the fluid particles fluctuates in the direction of flow.
- These fluctuations are caused by the chaotic and irregular motion of the fluid particles.
- The velocity fluctuations result in variations in the momentum of the fluid, leading to the generation of shear stress.
- The magnitude of shear stress is directly related to the intensity of velocity fluctuations in the direction of flow.

d) Fluctuation of velocity in the direction of flow as well as transverse to it:
- In addition to the fluctuations in the direction of flow, turbulent flow also exhibits fluctuations in the transverse direction.
- The fluid particles in a turbulent flow move in a swirling and chaotic manner, resulting in velocity fluctuations both in the direction of flow and transverse to it.
- These fluctuations in both directions contribute to the generation of shear stress.
- The presence of velocity fluctuations transverse to the flow direction enhances the mixing and turbulence in the flow, leading to an increase in the shear stress magnitude.

In summary, shear stress in a turbulent flow is primarily due to the viscous property of the fluid and the fluctuations of velocity in both the direction of flow and transverse to it. The density of the fluid indirectly affects the shear stress by influencing the behavior of the turbulent flow.