abj : Pressure, Pressure Force, and Fluid Motion Without Flow [Q1] 1.Area as A Vector  Component of Area Vector – Projected Area  Net Area Vector for A Two-Dimensional Surface 2.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  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

abj2 1.Area as a vector 1.Projected area and Component of area vector 2.Net area vector 2.Surface Force: Resultant (Force and Moment) Due to Pressure  Resultant Force and Moment (on A General Curved Surface) Question of Interest Q1: Given the pressure field/distribution, find the net/resultant pressure force and moment on a finite surface. Very Brief Summary of Important Points and Equations [1]

abj3  Special Case:Resultant force due to uniform pressure  Component of the pressure force  Net pressure force Very Brief Summary of Important Points and Equations [2]

abj4 Area as A Vector

abj5 Area as a vector Magnitude of =Magnitude of the area Direction of =Outward normal (from the system to the surrounding) = Outward unit normal vector pointing from the system to the surroundings system surrounding system surrounding

abj6 Component of Area Vector – Projected Area [1] x y xx yy l Magnitude of the projected area xx yy w l x y z xx yy

abj7 Component of Area Vector – Projected Area [2] x y z In 3D

abj8 Component of Area Vector – Projected Area [3] A z x y

abj9 How to Find The Net Area Vector for A Two-Dimensional Surface w l x y z  x y  1 2 The net area vector can be found simply by summing its projected area components.

abj10 How to Find The Net Area Vector A z x y A z x y Similar approach can be used in 3-D: The net area vector can be found simply by summing its projected area components.

abj11 Example:Find The Net Area Vector Questions: 1.Find the net area vector for the 2-D surfaces (a) and (b). Both have depth w. 2.Are they equal? If so, why? Surface (b) is plotted here as dotted line in order to compare its size to (a). (a)(b) h l  y x z

abj12 Resultant Due to Pressure  Resultant Force and Moment (on A General Curved Surface) Question of Interest  Q1: Given the pressure field/distribution, find the net/resultant pressure force and moment on a finite surface

abj13 system surrounding Resultant Due to Pressure (on A General Curved Surface): Finding 1) Resultant Force and 2) Resultant Moment 1 Infinitesimal area element 2 Infinitesimal pressure force on the element 3 Resultant pressure force on the finite area A 4Infinitesimal moment of pressure force about C 5 Resultant moment of pressure force about C Minus sign: is always opposite to is compressively normal to the surface

abj14 Some Properties of Pressure Force [1] 1. is always opposite to is compressively normal to the surface 2. If the pressure field is uniform over A

abj15 3.Component of the pressure force as a force due to pressure (distribution on the surface S being projected) onto the projected area. Some Properties of Pressure Force [2] x y z

abj16 Some Properties of Pressure Force [3] How to find the net pressure force via its components A z x y A z x y

abj17 Example:Find The Net Pressure Force Due to Uniform Pressure Questions: 1.Find the net pressure force due to uniform pressure in two cases: a.uniform p 1 (on the left side only), b.both uniform pressures p 1 and p 2, on plates (a) and (b). 2.In the corresponding cases, Are they equal? If so, why? (a)(b) h l  y x z p1p1 p2p2 p1p1 p2p2

abj18 Problem 1.Find the resultant pressure force (magnitude, direction, and line of action) due to all fluid pressures on the curved plate of width w. 2.Find the center of pressure (CP) for the case of a parabolic gate (n = 2), and D = 1 m and a = 1 m -1. Resultant Due to Pressure (Hydrostatic Force/Moment) on A Curved Submerged Surface Water,  Air, p o x D y o

abj19 Resultant Force Force on upper surface Force on lower surface Net force on plate due to all fluid pressures pressures Thus, the net force x D y o h 1 2’ 2 How to write the area vector vectorially for the integration from 1  2 (‘) To integrate from 1  2, points from 1  2 No need for (+/-) signs in dx or dy since dx and dy are components of vectors, they have sign embedded in them. is the result of 90 o -rotation of about the axis; hence,

abj20 You know how to integrate these already. x D y o h 1 2’ 2

abj21 NOTE on Direction of Integration,, and x D y o h 1 2’ 2 1  2 : Rotate about x D y o h 2 1’ 1 1  2 : Rotate about

abj22 CP Resultant Moment and Line of Action x D y o h 1 2’ 2 p Infinitesimal force Infinitesimal moment due to is the position vector to any point p on the line of action of Net/Resultant moment due to Principle of Moment Thus, This equation is in fact a linear equation Y(X), representing the locus of points on the line of action of

abj23 Thus, the line of action of the resultant force is given by (1) (2) CP is the intersection of this line with the surface: Thus, solve 2 equations in 2 unknowns X and Y, we have for the parabolic gate

abj24 Special Case: Resultant Pressure Force/Moment on A Flat Surface

abj25 x y O Problem:Find the resultant force (magnitude, direction, and line of action) on a plate submerged in fluid with pressure at the free surface of p o. Resultant Force Due to Pressure (Hydrostatic Force) on A Plane Submerged Surface  y O popo z h Interest in force on this side of the plate. C = Centroid of the plate area CP = Center of pressure CP Consider an area element located at dA This area, located at depth h, has infinitesimal force acting on it. C hChC xCxC yCyC If it is not too confusing, recall that we also have another characteristic point that has a role to play, the centroid of the volume of pressure distribution. Draw first to identify the system of interest. Let the resultant force, which is in the +z direction, be denoted by. It acts through the point on the surface called the center of pressure (CP), which is located at. As we shall later see, another characteristic point is the centroid C of the plate area. Let this centroid be located at depth h C and point (x C, y C ) As we shall also see, the parallel coordinate system whose origin is at the centroid C will also have a role to play.

abj26 x y O  y O popo z dA CP C Interest in force on this side of the plate. C CP

abj27 Assumptions: Static fluid. Gravity is the only body force.  =  g is constant wrt depth. Resultant Force (Magnitude and Direction): where p c is the hydrostatic pressure at the depth, h c, of the centroid of the plate area; A is the total area of the plate.  x y y z CP dA O O C popo C CP

abj28 Resultant Moment (Line of action of the resultant force): Moment about point O : Application 4.2: Hydrostatic Force on A Plane Submerged Surface  x y y z CP dA O O C popo C CP

abj29 Application 4.2: Hydrostatic Force on A Plane Submerged Surface Resultant Moment (Line of action of the resultant force): Moment about the x -axis (that passes through point O ): Consequence: Since the second term on the RHS is always >= 0, the equation indicates that the y -location of the center of pressure ( y’, the point on the plate at which the resultant force acts) always lies at equal or lower depth than that of the plate centroid, y C. NOTE: If the other side of the plate is exposed to free surface pressure p o and the net force due to pressure distributions on both sides of the plate is desired, we have

abj30 Resultant Moment (Line of action of the resultant force): Moment about the y -axis (that passes through point O ): NOTE: If the other side of the plate is exposed to free surface pressure p o and the net force due to pressure distributions on both sides of the plate is desired, we have