MECH 221 FLUID MECHANICS (Fall 06/07) Chapter 2: FLUID STATICS. Instructor: Professor C. T. HSU. 2.1. Hydrostatic Pressure. Fluid mechanics is the study of fluids in macroscopic motion. For a special static case: No Motion at All
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Instructor: Professor C. T. HSU
V= x y z/2
∑Fx = 0 :
F1cos - F2 = 0
p1 A1cos - p2 A2 = 0
SinceA1cos = A2 = y z
p1 = p2
∑Fz = 0 :
F1 sin + m.g = F3
p1 A1 sin + V g = p3 A3
p1(x/sin ) y sin + g x y z/2 = p3 x y
p1 + g z/2 = p3
• Normal stress at any point in a fluid in equilibrium is the same in all directions.
• This stress is called hydrostatic pressure.
• Pressure has units of force per unit area.
P = F/A [N/m2]
• The objective of hydrostatics is to find the pressure field (distribution) in a given body of fluid at rest.
(P+ P) A + g A y = P A
P/ y = - g
P2 - P1 = - g (y2 - y1)
If = (y), then: ∫dP = -g ∫ (y)dy
If = (p,y)
such as for ideal gas P = RT where T=T (y)
∫dP/P = -(g/R) ∫dy/T(y)
The integration at the right hand side
depends on the distribution of T(y).
P1A = P2A
P1 = P2
* The pressure varies with depth, P= gh.
* The pressure acts perpendicularly to an immersed surface
2.4.1. Plane Surface
• Let the surface be infinitely thin, i.e. NO volume
• Plate has arbitrary planform, and is set at an arbitrary angle, , with the horizontal.
P = Patm + gh
F = ∫PdA = PatmA + g ∫h dA
F = PatmA + g sin∫y dA
yc.g. = (1/A) ∫y dA
on a submerge plane surface is equal to the pressure at the c.g. of the plane multiplied by the area of the plane.
Similar to c.g., the point on the surface where the resultant force is applied is called the Center of Pressure, c.p.
Iox = ∫y2 dA = y2c.g.A + Ic.g.x
= g sin(y2c.g.A + Ic.g.x) +Patmyc.g.A
=(g sin yc.g.A + PatmA) yc.g.+ g sin Ic.g.x
xc.p. = xc.g. + (g sinIc.g.y) / (Pc.g.A)
Ic.g.y - moment of inertia about the y-axis at c.g.
* Tables of Ic.g. for common shapes are available
* For simple pressure distribution profiles, the c.p. is usually at "c.g." of the profile
2.4.2. Curved Surface
22.214.171.124. Horizontal Force
126.96.36.199. Vertical Force
FaV = Fa cos = Pa Aacos
WhereFVab =g (vol. 1-a-b-2-1), FVcd =g (vol. 1'-d-c-2'-1')
* The line of action is through the center of
the mass of the displaced fluid volume
* Direction of buoyant force is upward
W = FB
a = -(g + al+ ar)
al = axi + ayj
ar = -rω2er where er is the unit vector in r direction