ENB221 Lecture Fluid Mechanics
Fluid Statics
Fluid statics deals with problems associated with fluids at rest, fluids can be either liquid or gaseous:
Hydrostatics when it is a liquid
Aerostatics when it is a gas
No relative motion between adjacent fluids layers, no shear stresses in the fluid deforming it. The only stress of
concern is the normal stress, which is the pressure – the variation of which only due to the weight of the fluid.
Hence, gravity is the major factor in fluid statics.
What do we use fluid statics for?
Used to determine the forces on floating or submerged bodies and the forces developed by devices like hydraulic
presses and car jacks. Also the design of water dams and liquid storage tanks.
The complete description of the resultant hydrostatic force acting on a submerged surface requires the
determination of:
1. The magnitude
2. The direction
3. The line of action of the force
Hydrostatic Forces on Submerged Plane Surfaces
A plate is subjected to fluid pressure distributed over its surface when exposed to a liquid
A gate valve in a dam
The wall of a liquid storage tank
The hull of a ship at rest
Centre of Pressure
On a Plane surface, the hydrostatic forces form a system of parallel forces, and we often need to determine the
magnitude of the force and its point of application, which is called the centre of pressure. In most cases, the other
side of the plate is open to the atmosphere, thus the atmospheric pressure acts on both sides of the plate, yielding a
zero resultant. Therefore in most cases it is convenient to simply work with gauge pressure only.
The plane of this surface (normal to the page) intersects the horizontal free surface at an angle θ. We take the line of
intersection to be the x-axis (out of the page).
Resultant Force
The absolute pressure at any point is given by:
The resultant hydrostatic force F acting on the surface is determined by integrating the force P dA acting on a
R
differential area dA over the entire surface:
The first moment of area is related to the y-coordinate of the centroid of the surface:
1 Week 4 Tuesday, 27 August 2013 ENB221 Lecture Fluid Mechanics
Substituting:
The magnitude of the resultant force acting on a plane
surface of a completely submerged plate in a
homogenous (constant density) fluid is equal to the
product of the pressure P aC the centroid of the surface
and the area A of the surface.
Line of action of the Resultant Force
Two parallel force systems are equivalent if they have the same magnitude and the same moment about any point.
The line of action of the resultant hydrostatic force, in general, does not pass through the centroid of the surface.
It lies underneath where the pressure is higher
The intersection of the line of action and the surface is the centre of pressure
The vertical location of the line of action is determined by equating the moment of the resultant force to the
moment of the distributed force about the x-axis.
yp= is the distance of the centre of pressure from the x-axis (point 0) and
Ixx,0 is the area moment of inertia about the x-axis
Second moment of Area – Area moment of Inertia
Usually given about the axes passing through the centroid of the area. Second moments of area about two parallel
axes are related to each other by the parallel axis theorem.
I = Second moment of area about the x-axis passing through the centroid of the area
xx,c
Yc= the distance between two parallel axes
Calculating the distance of the Centre of Pressure from the x-axis
Combining:
Ignoring Atmospheric Pressure
The case where P =00 (which is the usual case) this expression simplifes greatly.
The vertical distance of the centre of pressure from the free surface is given by
Rearranging so all distances are from the free surface
2 Week 4 Tuesday, 27 August 2013 ENB221 Lecture Fluid Mechanics
Common values for second moment of area about the x-axis
Centre of Pressure
If there is a symmetry about the y-axis the centre of pressure lies on the y-axis directly below the centroid. The point
H prom the free surface lies on the vertical plane of symmetry.
Pressure Prism
Pressure acts normal to the surface, and the hydrostatic
forces acting on a flat plate of any shape form a volume
whose base is the plate area and whose length is the
linearly varying pressure.
The volume of the pressure prism is equal to the magnitude of the resultant hydrostatic force,RF :
The line of action passes through the centroid of the pressure prism, hence, the projection of the centroid of the
prism onto the plate occurs at the centre of pressure. Problem of determining the hydrostatic force reduces to
finding the two points and the volume of the pressure prism.
3 Week 4 Tuesday, 27 August 2013 ENB221 Lecture Fluid Mechanics
Tilted Rectangular Plate Vertical Plate
Distance to the centre of pressure For a completely submerged vertical plate whose top is
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