Fluid Mechanics hydrostatic pressure.pptx

hydrostatic on inclined surface

1. Fluid Mechanics By Samar Singhal

2. Hydrostatic Force on Inclined Surface

3. First Moment of Area The first moment of area of a shape, about a certain axis, equals the sum over all the infinitesimal parts of the shape of the area of that part times its distance from the axis . First moment of area is used to determine the  centroid  of an area.

4. Second and Product of moment of Area The second moment of area  I xx  (often denoted as  I x ) can be computed as The  product moment of area is defined as

5. Product of moment of Area Green area need to balance the red area. Automatically satisfied when x and y are symmetry axes.

6. Hydrostatic Force on Inclined Surface The pressure on any submerged surface varies linearly with depth. If h is the depth to any element area dA of the plate, the pressure there is From the figure First Moment of Area Therefore θ is constant along the plate Force on the Plate

7. Hydrostatic Pressure on Inclined Surface But, Therefore Force on the plate can be written as whereas h cg the depth straight down from the surface to the plate centroid p cg is the pressure at centroid of the plate. The force on one side of any plane submerged surface in a uniform fluid equals the pressure at the plate centroid times the plate area, independent of the shape of the plate or the angle θ at which it is slanted.

8. Pressure Distribution The hydrostatic pressure force on a plane surface is equal, regardless of its shape, to the resultant of the three-dimensional linear pressure distribution on that surface.

9. Special Cases

10. Center of Pressure To find the coordinates ( x CP , y CP ), we sum moments of the elemental force p dA about the centroid and equate to the moment of the resultant F . vanishes due to on centroid of centroidal axes Let Then I xx is the area moment.

11. Center of Pressure The negative sign in Eq. shows that y CP is below the centroid at a deeper level and, unlike F , depends on angle θ . If we move the plate deeper, y CP approaches the centroid because every term remains constant except p CG , which increases. determination of x CP For positive I xy , x CP is negative because the dominant pressure force acts in the third, or lower left, quadrant of the panel. If I xy = , usually implying symmetry, x CP = 0 and the center of pressure lies directly below the centroid on the y axis.

12. Hydrostatic Force on Inclined Surface In most cases the ambient pressure pa is neglected because it acts on both sides of the plate; for example, the other side of the plate is inside a ship or on the dry side of a gate or dam. In this case p CG = γh CG , and the center of pressure becomes independent of specific weight:

13. Pressure Prism Length of prism is the linearly varying pressure. I ts volume is equal to the magnitude of the resultant hydrostatic force acting on the plate since FR = ʃ P dA , and the line of action of this force passes through the centroid of this homogeneous prism. The projection of the centroid on the plate is the pressure center .

14. Centroidal moments of inertia for various cross sections:

15. Hydrostatic thrust on a submerged curved surface

16. Hydrostatic thrust on a submerged curved surface The hydrostatic force on the elemental area dA is T he force acts in a direction normal to the area dA . The components of the force dF in x, y and z directions are The components of the surface element dA projected on yz , xz and xy planes are, respectively

17. Hydrostatic thrust on a submerged curved surface Forces can be written as: Integrating the forces: z c is the z cordinate of centroid of area A x and A y the projected areas on yz and and xz planes respectively.

18. Center of Pressure Point of action of F x Point of action of F y F or a curved surface, the component of hydrostatic force in a horizontal direction is equal to the hydrostatic force on the projected plane surface perpendicular to that direction and acts through the centre of pressure of the projected area.

19. Vertical component of force V ertical component of the hydrostatic force: where V is the volume of the body of liquid within the region extending vertically above the submerged surface to the free surface of the liquid. Therefore, the vertical component of hydrostatic force on a submerged curved surface is equal to the weight of the liquid volume vertically above the solid surface to the free surface of the liquid and acts through the centre of gravity of the liquid in that volume.

20. Surface submerged in a multilayered fluid The hydrostatic force on a surface submerged in a multilayered fluid can be determined by considering parts of the surface in different fluids as different surfaces.

21. Hydrostatic thrust on a submerged curved surface We could sum the separate three components of these elemental pressure forces, but it turns out that we need not perform a laborious three-way integration. The horizontal component of force on a curved surface equals the force on the plane area formed by the projection of the curved surface onto a vertical plane normal to the component. The vertical component of pressure force on a curved surface equals in magnitude and direction the weight of the entire column of fluid, both liquid and atmosphere, above the curved surface.

22. Hydrostatic thrust on a submerged curved surface

23. Hydrostatic thrust on a submerged curved surface

24. Hydrostatic thrust on a submerged curved surface The horizontal component of the hydrostatic force acting on a curved surface is equal (in both magnitude and the line of action) to thehydrostatic force acting on the vertical projection of the curved surface. The vertical component of the hydrostatic force acting on a curved surface is equal to the hydrostatic force acting on the horizontal projection of the curved surface , plus (minus, if acting in the opposite direction ) the weight of the fluid block.

25. Buoyancy When a body is either wholly or partially immersed in a fluid, the hydrostatic lift due to the net vertical component of hydrostatic pressure forces experienced by the body is called the buoyant force and the phenomenon is called buoyancy.

26. Buoyancy T he resultant horizontal force in any direction for such a closed surface is always zero. The vertical forces acting on the two ends of such a prism Therefore, the buoyant force (the net vertically upward force) acting on the elemental prism is Integrating will give net buoyant force:

27. Line of action of the force The line of action of the force can be found by taking moment of the force with respect to z-axis. Substituting the values: W hich is the centroid of the displaced volume. B uoyant force F B equals to the weight of liquid displaced by the submerged body of volume V and known as the Archimedes principle. The principle states that the buoyant force on a submerged body is equal to the weight of liquid displaced by the body, and acts vertically upward through the centroid of the displaced volume.

28. Float, Suspended and Sinking Body

29. Numericals A tank of oil has a right-triangular panel near the bottom, as in Fig. Omitting p a , find the (a) hydrostatic force and (b) C P on the panel.

30. Numericals A long solid cylinder of radius 0.8 m hinged at point A is used as an automatic gate, as shown in Fig. When the water level reaches 5 m, the gate opens by turning about the hinge at point A. Determine (a) the hydrostatic force acting on the cylinder and its line of action when the gate opens and ( b) the weight of the cylinder per m length of the cylinder.