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 abj

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 abj

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 = abj

= Net pressure force on an infinitesimal volume per unit volume abj = 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 abj

Q3: Given a motion, find the pressure field/distribution abj 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 abj

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 abj

Fluid Motion without Flow - Fluid in Rigid Body Motion - Static Fluid abj 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

for a fluid motion without flow abj 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

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, abj

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 abj

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

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 abj