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Pipe Flow Example Water flows steadily into the circular pipe with a uniform inlet velocity profile as shown. Due to the presence of viscosity, the velocity immediately adjacent to the inner pipe wall becomes zero and this phenomenon is called the no-slip boundary condition. It is found out that the velocity distribution reaches a parabolic profile at a distance downstream of the entrance and can be represented as V1(r)=Vmax[1-(r/R)2], where Vmax is the velocity at the center of the pipe and r is the radial distance measured away from the center axis. Use the mass conservation equation, determine Vmax. V1(r )=Vmax[1-(r/R)2] P1 VO PO R: radius of pipe

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Mass Conseration 0, steady state

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**Momentum Conservation**

Assume there is no significant forces acting between the pipe and the fluid except the pressure forces normal to the pipe inlet (PO) and the section 1 (P1). Use linear momentum conservation equation, estimate the pressure difference between these two sections in order to accelerate the velocity profile inside the pipe from the inlet to a parabolic profile at section 1. Assume the pressure is uniform both at the inlet and section 1.

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**Momentum Conservation (cont.)**

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