Dynamics of Fluid Flow
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The document discusses fluid flow dynamics, particularly the behavior of shear stress and velocity profiles within a pipe. It highlights how shear stress is maximal at the pipe boundary and minimal at the center, leading to a parabolic velocity distribution characterized by varying layers of flow. Additionally, it covers the formation of a boundary layer and the impact of turbulence on momentum transfer, resulting in a more uniform velocity profile in turbulent flow conditions.
Dynamics of Fluid Flow
- 1. Dynamics of Fluid Flow Presented By….. Abhishek Anand Thakur Manash Pratim
- 2. * u u+ du y dy 𝝉 = 𝝁 𝒅𝒖 𝒅𝒚 𝜏 ∝ 𝑑𝑢 𝑑𝑦 Stress is proportional to the rate of change of velocity w.r.t y μ is the constant of proportionality & known as coefficient of viscosity u Velocity variation boundary near a solid boundary When two layer of fluid a distance “dy” apart. Move one over the other at different velocity u & u+du Viscosity together with relative velocity cause a shear stress acting b/w the fluid layer
- 3. Direction of Flow a τ ∗ 2𝜋𝑟∆𝑥 A B C D 𝑝 + 𝜕𝑝 𝜕𝑥 𝑑𝑥 𝜋𝑟2p𝜋𝑟2 ∆x r R Neglecting the acceleration of fluid element ,net force =o p 𝜋𝑟2 − 𝑝 + 𝜕𝑝 𝜕𝑥 𝑑𝑥 𝜋𝑟2 − τ ∗ 2𝜋𝑟∆𝑥 = 0
- 4. From it we can find out shear stress distribution Shear stress is maximum at the boundary of pipe and minimum at the center of pipe 𝝉 ∝ 𝒓 r=0 r=R r=R 𝝉 𝒎𝒂𝒙 𝝉 𝒎𝒂𝒙 𝝉 𝒎𝒊𝒏 Shear stress increase linearly with distance from the center
- 5. After integration and using boundary condition at r=R, u=0 R r y a b c d y=R−𝐫 , dy =−𝐝𝐫 𝜏 = −𝜇 𝑑𝑢 𝑑𝑟 ……. 2 By eq 1 and 2 𝒅𝒖 = 𝒓 𝟐𝒖 𝝏𝒑 𝝏𝒙 . 𝒅𝒓 U=− 𝟏 𝟒𝒖 . 𝝏𝒑 𝝏𝒙 𝑹 𝟐 − 𝒓 𝟐 … … … . 𝟑 Here it is clear when r=0 𝑎𝑡 𝑡ℎ𝑒 𝑐𝑒𝑛𝑡𝑟𝑒 𝑜𝑓 𝑝𝑖𝑝𝑒 then velocity is maximum and minimum at boundary where 𝑟 = 𝑅
- 6. * At the pipe wall Velocity of the fluid will be Zero. The velocity will increase As we move towards the Centre of the pipe. The change in velocity across the direction of flow is known as Velocity profile. U∝ 𝒓 𝟐 Velocity distribution across the section of pipe is parabolic U=− 𝟏 𝟒𝒖 . 𝝏𝒑 𝝏𝒙 𝑹 𝟐 − 𝒓 𝟐 𝜇, 𝜕𝑝 𝜕𝑥 , 𝑎𝑛𝑑 R is constant ` 𝒗 𝒎𝒂𝒙 𝒗 𝒎𝒊𝒏 𝒗 𝒎𝒊𝒏
- 7. *If a fluid is along way from the boundary and all the particle moving with the same velocity then the velocity profile look some thing like this • In this flow there are 4 layer • Viscous sub layer-viscous effect are dominant so velocity profile in this layer is linear & flow is stream line • Buffer layer –turbulent effect are significant but flow is still dominated by viscous effect • Overlap layer- turbulent effect are more significant but still not dominated • Turbulent layer – turbulent effect are dominated on viscous effect V Velocity profile for turbulent flow
- 8. Pipe Entrance v vv Because of the shear force near the pipe wall, a boundary layer forms on the inside surface and occupies a large portion of the flow area as the distance downstream from the pipe entrance increase. At some value of this distance the boundary layer fills the flow area. The velocity profile becomes independent of the axis in the direction of flow, and the flow is said to be fully developed. *
- 9. * *Turbulence causes transfer of momentum from center of pipe to fluid closer to the pipe wall. *Mixing of fluid (transfer of momentum) causes the central region of the pipe to have relatively uniform velocity (compared to laminar flow) *Close to the pipe wall, eddies are smaller (size proportional to distance to the boundary)