Flow of a Newtonian fluid down an inclined plane.

In summary, the conversation discusses the process of finding the velocity profile for a laminar flow of a Newtonian fluid down an inclined plane surface. The book takes an infinitesimal control volume and equates the forces acting on it to the sum of the linear momentum flux and rate of accumulation of momentum. The pressure force terms cancel due to the presence of a free liquid surface, where the pressure is constant and at atmospheric pressure. However, this is not the case if the top is left open and the fluid is rotating, as the pressure will vary along the streamlines and maintain hydrostatic equilibrium normal to the plane.
  • #1
siddharth
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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)

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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  • #2
Free liquid surfaces means it is at atm pressure, ie. zero pressure gauge.
 
  • #3
Ouch. Yeah, it's kinda obvious now :blushing:
 
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  • #4
siddharth said:
w 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.

That's not true. If you leave the top open, as Cyrus said, the pressure is the atmospheric one at the top of the upmost film. The centrifugal force will curve the shape of the surface as it were a paraboloid, such that the hydrostatic pressure balances the centrifugal overpressure generated.

In your problem, the book should say that pressure is consant along the streamlines, but it is holding hydrostatic equilibrium normal to the plane, but instead with the gravitational acceleration g, with [tex]gcos\theta[/tex]. So it is not uniform in the normal direction to the plane.
 
  • #5
Clausius2 said:
That's not true. If you leave the top open, as Cyrus said, the pressure is the atmospheric one at the top of the upmost film. The centrifugal force will curve the shape of the surface as it were a paraboloid, such that the hydrostatic pressure balances the centrifugal overpressure generated.

In your problem, the book should say that pressure is consant along the streamlines, but it is holding hydrostatic equilibrium normal to the plane, but instead with the gravitational acceleration g, with [tex]gcos\theta[/tex]. So it is not uniform in the normal direction to the plane.

Yeah, I get it. Thanks
 

FAQ: Flow of a Newtonian fluid down an inclined plane.

What is a Newtonian fluid?

A Newtonian fluid is a type of liquid that exhibits a constant viscosity, meaning that its resistance to flow remains the same regardless of the amount of force applied.

How does the inclination of a plane affect the flow of a Newtonian fluid?

The inclination of a plane, or the angle at which it is positioned, can significantly affect the flow of a Newtonian fluid. As the angle increases, the fluid experiences a greater force of gravity pulling it down the plane, causing it to flow faster.

What factors can affect the flow of a Newtonian fluid down an inclined plane?

Aside from the angle of inclination, other factors that can affect the flow of a Newtonian fluid down an inclined plane include the viscosity of the fluid, the surface roughness of the plane, and the presence of any obstacles or barriers that may impede the flow.

How is the flow rate of a Newtonian fluid down an inclined plane calculated?

The flow rate of a Newtonian fluid down an inclined plane can be calculated using the equation Q = Av, where Q is the flow rate, A is the cross-sectional area of the fluid, and v is the velocity of the fluid.

Can the flow of a Newtonian fluid down an inclined plane be affected by external forces?

Yes, the flow of a Newtonian fluid down an inclined plane can be affected by external forces such as wind or vibrations. These forces can cause disturbances in the fluid's flow, resulting in changes in velocity and flow rate.

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