1. abj
4.2.2: Pressure, Pressure Force, and Fluid Motion Without Flow [Q2 and Q3]
Area as A Vector
Component of Area Vector – Projected Area
Net Area Vector for A Two-Dimensional Surface
Resultant Due to Pressure
Resultant Force and Moment (on A General Curved Surface)
Questions of Interest
Q1: Given the pressure field/distribution , find the net/resultant pressure force and moment on a finite surface
4.2.2
Q2: Given the pressure field/distribution , find the net pressure force (per unit volume) on an infinitesimal volume
Q3: Given a motion (fluid motion without flow), find the pressure field/distribution
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2. Very Brief Summary of Important Points and Equations
Q2: Given the pressure field/distribution , find the net pressure force (per unit volume) on an infinitesimal volume
Q3: Given a motion (fluid motion without flow), find the pressure field/distribution
Governing Equation of Motion
Differential Change in p
Calculation: Substitute and integrate (line integral)
= Net pressure force on an infinitesimal volume per unit volume
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3. abj
Q2: Given the pressure field/distribution , find the net pressure force (per unit volume) on an infinitesimal volume
In this FBD, only the x component is shown. x
x x+dx
Magnitude of force at x on -x-plane =
Use Taylor series expansion, we have
Magnitude of force at x+dx on +x-plane =
Net x-force =
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4. = Net pressure force on an infinitesimal volume per unit volume
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= Net pressure force on an infinitesimal
volume per unit volume
Similarly, we find the resultant forces on y-planes and z-planes and in the y- and z-directions, respectively,
= Net pressure force on an infinitesimal volume per unit volume
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5. Q3: Given a motion, find the pressure field/distribution
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Q3: Given a motion, find the pressure field/distribution
Here, we are interested in the motion of fluid where the only forces are
Surface force  Pressure force alone (no friction)
Body force  mg alone
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6. Examples of Fluid Motion where The Only Forces are Pressure Force and mg
Fluid Motion without Flow
Static Fluid
Fluid in Rigid-Body Motion
Inviscid Flow
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7. Fluid Motion without Flow - Fluid in Rigid Body Motion - Static Fluid
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Fluid Motion without Flow - Fluid in Rigid Body Motion - Static Fluid
Definition of Fluid: A fluid is a substance that deforms continuously under the application of a shear (tangential) stress no matter how small the shear stress may be. (Fox, et al., 2004)
a (t)
Shear  Deformation (Flow)
No Deformation (Flow)  No shear
From Surface forces = Pressure Viscous/Friction,
then the only surface force in fluid motion without flow is the pressure force.
abj

8. for a fluid motion without flow
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for a fluid motion without flow
Increasing p Decreasing p
Direction of maximum spatial rate of decrease in pressure
Surface Force:
Resultant force due to pressure per unit volume
Direction of constant pressure
Body Force:
Resultant gravitational force per unit volume
Direction of maximum spatial rate of increase in pressure
From: Newton’s Second law
if evaluated per unit volume, we have
or,
where = net surface force, = net body force
In the case of fluid motion without flow, since
the only surface force is pressure, 
the only body force present is gravitational force, 
We therefore have the equation of motion for fluid motion without flow
abj

9. abj
Increasing p Decreasing p
Surface Force:
Resultant force due to pressure per unit volume
Body Force:
Resultant gravitational force per unit volume
Direction of constant pressure
Direction of maximum spatial rate of decrease in pressure
Direction of maximum spatial rate of increase in pressure
Net force due to pressure force and gravitational force results in an acceleration of a fluid element.
Pressure, p, has the maximum spatial rate of change in the direction of, and has constant value in the direction perpendicular to,
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10. Q3: Given a motion (fluid motion without flow),
- find the pressure field/distribution , or
- find the pressure difference between any two points in the flow
Two Main Equations
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11. abj
Example: Finding The Pressure Difference between Two Points / Pressure Field of Fluid in Rigid Body Motion with Constant Linear Acceleration x z y
Problem: Find the pressure difference between any two points (a reference point and any point ) within a fluid in rigid body motion that moves with constant linear acceleration.
NOTE: Since is any point in the flow, we in effect solve for the pressure field
Assumption 1: Fluid in rigid body motion.
Assumption 2: The only body force is the gravitational force.
Assumption 3:
ANS
NOTE: 1) In general, the “flow” is not steady with respect to the stationary frame of reference.
2) Here, we analyze the effect of the change in space at any one particular fixed time t.
Surface of constant pressure can be found from letting
abj

12. abj
Example: Finding Pressure Difference between Two Points / Pressure Field of Fluid in Rigid Body Motion Rotating with Constant Angular Velocity
Problem: Find the pressure difference between any two points (a reference point and any point ) within a fluid in rigid body motion rotating in a cylinder with constant angular velocity. x z q
Assumptions: 1) Fluid in rigid body motion. 2) The only body force is the gravitational force. 3: r, g, w = constant
Surface of constant pressure can be found from letting
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