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Volume and Surface forces. Surface forces Short range forces that act only on a thin layer adjacent to the boundary of a fluid element. Surface forces are negligible unless there is direct mechanical contact between the interacting elements.-The total force exerted across the surface element
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1. Volume and Surface forces Volume forces – Long range forces capable of penetrating into the interior of the fluid and act on all elements.
- Contact of source of force with material is not necessary.
- Also called body forces
Examples of body forces in fluid systems:
a. Gravity
b. Electromagnetic Radiation
c. Apparent forces due to coordinate system motion
2. Volume and Surface forces Surface forces – Short range forces that act only on a thin layer adjacent to the boundary of a fluid element.
Surface forces are negligible unless there is direct mechanical contact between the interacting elements.
-The total force exerted across the surface element will be proportional to its area, .
Examples of surface forces in fluids:
a. Pressure Fields
b. Friction
3. Newton’s second law Newton’s law, in its general form, equates the net force on an object to the time rate of change of momentum. For a fixed volume in space, momentum is defined as
And Newton’s second law states
4. Newton’s second law With some subtle work (to be worked out SO 414) it can be shown the previous expression can be expressed as
The net force can be due to any of the external volume or surface forces discussed in the first two slides.
5. Pressure Gradient Force The only surface force we will concern ourselves with this semester is the pressure gradient force.
Recall that pressure is a force per unit area, therefore to obtain a force we must integrate the pressure field over an arbitrary surface area.
Why the minus sign?
6. Pressure Gradient Force The minus sign is a matter of convention to account for the fact that pressure forces are directed in the opposite direction to the normal of the surface.
7. Pressure Gradient Force Now we wish to convert the area integral into a volume integral using a vector form of Gauss theorem.
Equate the above force in the original equation for Newton’s second law
8. Pressure Gradient Force
We are interested in maintaining an arbitrary fixed volume
and not consider specific boundaries for integration . This,
the only way the integral can be 0 is for the integrand to be 0
over the domain. This leads to Euler’s equation of motion
In index notation, this is expressed as
9. Body forces There are two types of body forces that we may
Consider:
1. Apparent forces: Apparent forces due to the rotation of the earth are important and will be considered in the next chapter.
2. Gravitation forces: These forces are obtained by use of the gravitational force equation
10. Body forces
z is much smaller than Rearth and it is safe to
assume that
The gravitational equation can then be
approximated as
Where g=9.81 m/s2
11. Equation of motion Combining this Force contribution with the
pressure gradient force we obtain
Since we wish for the above equality to be true throughout
the entire integral domain, we require the integrand to be
zero and obtain the equations of motion for an
incompressible fluid in a non-rotating inviscid frame
12. Equation of motion Sometimes all conservative forces, including gravity, are
lumped into a general vector,
In this case the equation of motion is expressed as
the gravitational contribution to the total body force is still
13. Exercise A hydrostatic fluid is defined as a medium
What does the equation of motion
Reduce to in a hydrostatic medium?
