If the velocity profile in laminar flow is parabolic, then the shear s...
In laminar flow, the velocity profile is often observed to be parabolic. This means that the velocity of the fluid at different distances from the boundary or centerline varies in a particular manner. However, the shear stress profile is not the same as the velocity profile. The shear stress profile describes the distribution of shear stress within the fluid.
Explanation:
1. Velocity profile in laminar flow:
- In laminar flow, the fluid moves in smooth layers or laminae with no mixing between them.
- The velocity of the fluid at different distances from the boundary or centerline can be represented by a parabolic curve.
- The maximum velocity occurs at the centerline, and the velocity decreases symmetrically towards the boundaries.
- This parabolic velocity profile is a consequence of the no-slip condition at the boundaries, where the fluid velocity is zero.
2. Shear stress profile in laminar flow:
- Shear stress is the force per unit area acting tangentially to a surface.
- In laminar flow, the shear stress is directly proportional to the velocity gradient, which is the rate of change of velocity with respect to distance.
- The relationship between shear stress and velocity gradient is described by Newton's law of viscosity: τ = μ(dv/dy), where τ is the shear stress, μ is the dynamic viscosity, v is the velocity, and y is the distance perpendicular to the flow direction.
- For a parabolic velocity profile, the velocity gradient is not constant throughout the fluid.
- As the velocity decreases towards the boundaries, the velocity gradient increases, resulting in a higher shear stress near the boundaries.
- Conversely, the maximum velocity at the centerline corresponds to a lower velocity gradient and a lower shear stress.
- Therefore, the shear stress profile in laminar flow is not parabolic but rather a straight line, with higher values near the boundaries and lower values towards the centerline.
Conclusion:
The shear stress profile in laminar flow is a straight line, not a parabolic curve. The parabolic shape is characteristic of the velocity profile, but the shear stress profile is influenced by the velocity gradient, resulting in higher shear stress near the boundaries and lower shear stress towards the centerline.
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