Total force on surface of a plate?

[eterna]

Homework Statement

http://scienceofdoom.files.wordpress.com/2010/12/convection-basics-1b.png

How do I find the total force on that surface of the plate?

Homework Equations

T = μ * u'(y)

where T is the shear stress, μ is a scalar constant - the viscosity and u'(y) is the velocity gradient in the perpendicular direction

Force = T * A

The Attempt at a Solution

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Does every point along the plate have a different u(y) distribution so that the shear stress at the surface, μ * u'(y=0) , is different for all of these points?
And so the total force is the sum of μ * u'(y=0) * A for all of those points?

 

[Chestermiller]

Yes.

Chet

 

[eterna]

So is the correct integral

from 0 to L(length of the plate) : μ u'(0) width dl ?And so for the total force( top + bottom of plate), is it just two times that?

 

[Chestermiller]

Yes, if the flow is on both sides of the plate.

Chet

 

[HalfLostAndSearching]

Yes, every point along the plate will have a different velocity gradient, u'(y), and therefore a different shear stress, T. This is due to the varying velocity profile in the fluid layer above the plate. The total force on the surface of the plate can be found by integrating the shear stress over the entire surface, using the equation T = μ * u'(y) and the given velocity profile. This integration will give you the total force on the surface of the plate, which is the sum of all the individual forces acting on each point along the surface.