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Published byΞ Ξ±Ξ½ΞΊΟΞ±ΟΞΉΞΏΟ Ξ€ΞΏΞΊΞ±ΟΞ»Ξ―Ξ΄Ξ·Ο Modified over 5 years ago
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12. Navier-Stokes Applications
CH EN 374: Fluid Mechanics
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Problem Consider a steady, two-dimensional, incompressible velocity field: π£ = ππ₯+π π + βππ¦+ππ₯ π Calculate the pressure as a function of x and y.
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Problem Consider a steady, two-dimensional, incompressible velocity field: π£ = ππ₯+π π + βππ¦+ππ₯ π Calculate the pressure as a function of x and y. π π π£ π₯ ππ‘ + π£ π₯ π π£ π₯ ππ₯ + π£ π¦ π π£ π₯ ππ¦ + π£ π§ π π£ π₯ ππ§ =β ππ ππ₯ +π π π₯ +π π 2 π£ π₯ π π₯ π 2 π£ π₯ π π¦ π 2 π£ π₯ π π§ 2 π π π£ π¦ ππ‘ + π£ π₯ π π£ π¦ ππ₯ + π£ π¦ π π£ π¦ ππ¦ + π£ π§ π π£ π¦ ππ§ =β ππ ππ¦ +π π π¦ +π π 2 π£ π¦ π π₯ π 2 π£ π¦ π π¦ π 2 π£ π¦ π π§ 2
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Laminar Pipe Flow Iβve shown you pictures of the velocity profile of laminar pipe flow. Now letβs find the profile ourselves.
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1 π π(π π£ π ) ππ + 1 π π π£ π ππ + π π£ π§ ππ§ =0
Continuity Incompressible continuity equation (incompressible): 1 π π(π π£ π ) ππ + 1 π π π£ π ππ + π π£ π§ ππ§ =0 For laminar flow, we know all flow is in the z direction π£ π =0 π£ π =0 So what does the continuity equation tell us about π π£ π§ ππ§ ?
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Simplifying NS We did this Friday:
π π π£ π§ ππ‘ + π£ π π π£ π§ ππ + π£ π π π π£ π§ ππ + π£ π§ π π£ π§ ππ§ =β ππ ππ§ +π π π§ +π 1 π π ππ (π π π£ π§ ππ )+ 1 π 2 π 2 π£ π§ ππ 2 + π π£ π§ 2 π π§ 2 PS: What about pressure and gravity in the other directions?
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Boundary Conditions At what π values do we know what π£ π§ is?
What else do we know?
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Solve
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Whatβs the velocity profile good for?
Shear stress
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Whatβs the velocity profile good for?
Average velocity π£ ππ£π = π π΄ = 1 π΄ π΄ π£ππ΄
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