I'm basically reading on how the velocity profile is found for a laminar flow of a Newtonian fluid down an inclined plane surface. (x is along the incline, y is perpendicular to the incline)(adsbygoogle = window.adsbygoogle || []).push({});

The assumptions being made are

- The fluid is Newtonian

- It's laminar

- It's fully developed

- It's incompressible

What the book did, was to take an infinitesimal control volume, find the forces acting on it, and equate it to the sum of the linear momentum flux and rate of accumulation of momentum in the c.v (along the x-direction)

I understand how the sum of the flux and the accumulation is zero. Next, the book evaluates the forces.

It says,

[tex] \sum F_x = P \Delta y|_x - P \Delta y|_{x+\Delta x} + \tau_{yx} \Delta x|_{y+\Delta y} - \tau_{yx} \Delta x|_y + \rho g \Delta x \Delta y \sin \theta [/tex]

which I understand.

Then it says

Note that the pressure-force terms also cancel because of the presence of a free liquid surfaces.

This is what I don't understand. Why should the pressure be constant for a free liquid surface? For example, if we take a fluid between two cylinders, and rotate the inner cylinder (and make the same assumptions), then the centrifugal force (you know what I mean) would cause a pressure gradient along the radial direction. So, even at the free surface at the top, the pressure won't be constant.

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# Flow of a Newtonian fluid down an inclined plane.

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