FLUID STATICS HYDROSTATIC FORCES AND BUOYANCY Explain the hydrostatic forces applied on submerged plane surface. Hydrostatic forced on submerged curved surface. Define and explain Buoyancy and Stability.

FLUID STATICS Deals with problems associated with fluid at rest. No relative motion between adjacent layers. Thus, no shear stress (tangential stress) to deform the fluid. Liquid – hydrostatic; Gas – aerostatic.

FLUID STATICS The only stress in fluid statics is normal stress (perpendicular to surface) Normal stress is due to pressure (Pressure: gravity field-weight of fluid) Variation of pressure is due only to the weight of the fluid → fluid statics is only relevant in presence of gravity fields.

HYDROSTATIC FORCES F=PA Only when the pressure distribution is uniform over the entire area of interest.

HYDROSTATIC FORCES What about the areas where the pressure is non-uniformly distributed ?? FIND THE AVERAGE PRESSURE AND WHERE IT ACTS

HYDROSTATIC FORCES ON PLANE SURFACES Problem To find the resultant force FR (magnitude) and its point of application (center of pressure) for non-uniformly distributed pressure. **Atmospheric pressure Patm can be neglected when it acts on both sides of the surface.

WHAT IS A RESULTANT FORCE AND CENTRE OF PRESSURE ? Engineering Mechanics Resultant force: A force that results from the combination of two or more forces. COP: A point where the entire forces (Resultant force) were concentrated at a single point.

HOW TO DETERMINE THE RESULTANT FORCE Consider a flat plate completely submerged in a liquid. Plane of the top surface intersects with horizontal surface with an angle Ө, the line of intersection will be the x axis.

HOW TO DETERMINE THE RESULTANT FORCE, FR Absolute pressure at any point of the fluid, P = Po + ρgh, h = y sin Ө = Po + ρgy sin Ө, FR = ∫PdA =∫ (Po + ρgy sin Ө)dA = PoA + ρg sin Ө ∫ydA **∫ydA is the first moment of area is related to the y coordinate of the centroid (or centre) of the surface by , yc= 1/A ∫ydA. FR = (Po + ρgyc sin Ө) A = (Po + ρghc) A = PcA The magnitude of the resultant force, FR acting on a plane surface of a completely submerged plate in a homogeneous (constant density) fluid is = the product of the pressure Pc at the centroid of the surface and the area A of the surface.

CENTROID AND CENTROIDAL MOMENTS OF INERTIA FOR SOME COMMON GEOMETRIES

SUBMERGED RECTANGULAR PLATE : HOW TO DETERMINE THE RESULTANCE FORCE, FR.

How to determine the location of the COP? Line of action of resultant force FR=PCA does not pass through the centroid of the surface. In general, it lies below the centroid because pressure increase with depth.

How to determine the location of the resultant force (Center of Pressure) ? The vertical location of the line of action is determined by equating the moment of the resultant force, FR to the moment of the distributed pressure force about the x-axis. Then, ypFR = ∫yPdA =∫ y(Po + ρgy sin Ө)dA = Po ∫ ydA + ρgsin Ө ∫y2dA = PoycA + ρgsin Ө Ixx,o

Moment of the resultance force = Moment of the distributed pressure force about the x axis. Ixx,o = ∫y2dA is actually the second moment of area about the x axis passing through point O.

Normally, the second moment of area is given about the axes passing through the centroid of the area. Therefore, we need parallel axis theorem to relate the Ixx,o and Ixx,c Therefore, If Po = 0

SUBMERGED RECTANGULAR PLATE : HOW TO DETERMINE THE C.O.P.

If Ө=90

Example 11-1; Calculate FR, C. O Example 11-1; Calculate FR, C.O.P and discuss whether the driver can open the door or not. The moment acting on the mid point of the submerged door is 101.3kN x 0.5m = 50.6kNm, which is 50 times of the moment the driver can possibly generate. CAN’T OPEN.

Hydrostatic Forces on Curved Surfaces Complicated: FR on a curved surface requires integration of the pressure forces that change direction along the surface. Easiest approach: Determine horizontal and vertical components FH and FV separately.

Hydrostatic Forces on Curved Surfaces 1. Vertical surface of the liquid block, BC = projection of the curved surface on a vertical plane (vertical projection). 2. Horizontal surface of the liquid block, AB = projection of the curved surface on a horizontal plane (horizontal projection). 3. Newton’s 3rd Law – Action and reaction. The resultant force acting on the curve liquid surface = the force on curved solid surface.

Assume that the direction to the right and up is positive, Horizontal force component on curved surface: FH – FX = 0. The horizontal component of the hydrostatic force (FH) acting on a curved surface is (both magnitude and line of action) equal to the hydrostatic force (FX) acting on the vertical projection of the curved surface. Vertical force component on curved surface: FV - FY – W = 0, where W is the weight of the liquid in the enclosed block W=rgV. The vertical component of the hydrostatic force (FV) acting on a curved surface is (both magnitude and line of action) equal to the hydrostatic force (FY) acting on the horizontal projection of the curved surface, plus or minus (depend on direction) the weight of the fluid block.

Hydrostatic Forces on Curved Surfaces Magnitude of force FR=(FH2+FV2)1/2 Angle of force is a = tan-1(FV/FH)

A GRAVITY CONTROLLED CYLINDRICAL GATE Example 11-2 A long solid cylinder, R=0.8, hinged at point A. When water level at 5m, gate open. Determine: 1. The hydrostatic force acting on the cylinder and its line of action when the gates open 2. The weight of cylinder per m length of the cylinder

Friction at hinge is negligible The other side of the gate is exposed to the atmosphere, therefore, the Patm is cancelled out. A) Determine the net horizontal and vertical force, FH and FV respectively. FH = FX = PCA = ρghcA; hc = 4.2 + 0.8/2 FY = ρghcA; hc = 5m ; W=ρg(R2 – (πR2/4)) FY - FV - W = 0; Solve for resultant force and angle. B) Think about the moment acting at point A due to the cylinder weight and also resultant force

BUOYANCY An objects lighter in fluid compare to in an air. Fluid exerts an upward force on a body immersed in it.

BUOYANCY Buoyant force is caused by the increase of pressure with depth in fluid. Figure 10-13 Difference between pressure at the top and bottom surface is ρg(s+h)A - ρghA = ρgsA = ρgV, V = hA, volume of the plate.

Buoyancy Archimedes principal : The buoyant force acting on a body immersed in a fluid is equal to the weight of the fluid displaced by the body, and it acts upward through the centroid of the displaced volume.

Buoyancy The tendency of fluid to exert a supporting force on a body placed in the fluid. The force = weight of the fluid displaced by the body. Its act upward through centroid of the displaced volume. FB=rfgV Density of fluid

Bouyancy of Floating Bodies FB = W (The weight of the entire body must be equal to the buoyant force) ρf gVsub = ρave,bodygVtotal Vsub / Vtotal = ρave,body / ρf (The submerged volume fraction is equal to the ratio of the average density of the body to the density of the fluid) Completely submerged when, ρave,body >= ρf Submerged portion (yellow) LIQUID BOUYANT FORCE = WEIGHT OF DISPLACED FLUID Displaced fluid whose volume = yellow volume

BUOYANCY Buoyance force by air is so small, 0.1m3, ρair = 1.2 kg/m3 , 1.2N. If mass = 80 kg, weigth = 788 N. Ignore the buoyancy, error is 0.15%, so small. Rise of warm air – natural convection currents. The rise of hot air or Helium balloons, and air movements in the atmosphere.

Buoyancy Calculation Example 11-3 and 4 Q 11-33 and 11-34