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For my application, I have water entering a small diameter (D1) tubing (a hypodermic needle tubing) with pressure (P1) and exiting into the atmosphere. The tubing is also very long with respect to it's diameter (Length/Diameter > 1000). How do I calculate the total force acting on the tubing due to the shear forces at the fluid-wall interface along the entire length of the tubing? I would also like to be able to calculate the flow rate. For now, I would like to just focus on the portion where the flow is fully developed. If anyone can point me in the direction of the correct equations or some literature that you would recommend, I would very much appreciate the assistance.

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This physical scenario can be described well by the Hagen-Poiseuille equation for incompressible viscous flow. See the formulae for $v(r)$ (i.e., the axial velocity of the fluid as a function of the distance from the center of the tube) and $F_{viscosity, top}$ (i.e., the force of one layer of fluid acting on an adjacent layer). These expressions can be combined and evaluated at the wall of the tube to determine the net shear force acting on the wall.

The flow rate can also be found by integrating the axial velocity over the cross-sectional area. An expression for this is found in the same article.

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