Shear stress in laminar flow

In laminar flow, a fluid moves in smooth, parallel layers with little to no mixing between them. The velocity of the fluid varies across the flow profile, with the highest velocity at the center of the flow and decreasing velocities towards the walls. This variation in velocity leads to a corresponding variation in shear stress.

Velocity profile in laminar flow

The velocity profile in laminar flow is typically parabolic. This means that the velocity of the fluid is highest at the center of the flow and decreases symmetrically towards the walls. The parabolic shape of the velocity profile is a result of the no-slip condition at the walls, where the fluid velocity is zero, and the fluid velocity at the center being the highest due to the absence of any wall effects.

Relationship between velocity and shear stress

The shear stress in a fluid is a measure of the force per unit area that is exerted on a fluid layer parallel to the direction of flow. It is directly proportional to the velocity gradient of the fluid. In other words, the shear stress increases as the velocity gradient increases.

Shear stress profile in laminar flow

Since the velocity profile in laminar flow is parabolic, the velocity gradient is highest at the walls, where the velocity is lowest. This means that the shear stress is highest at the walls and decreases towards the center of the flow. The shear stress profile in laminar flow is therefore a straight line, with the highest shear stress at the walls and zero shear stress at the center.

Conclusion

In summary, in laminar flow, the velocity profile is parabolic and the shear stress profile is a straight line. The parabolic velocity profile leads to a variation in velocity across the flow, which in turn leads to a variation in shear stress. The highest shear stress occurs at the walls and decreases towards the center of the flow.