Pressure distribution in a fluid Pressure and pressure gradient Lecture 4 Mecânica de Fluidos Ambiental 2015/2016

Chapter 2 Concept of pressure and pressure gradient Hydrostatic pressure distribution Hydrostatic forces Buoyance and stability Mecânica de Fluidos Ambiental 2015/2016

Forces in fluids Surface or volumetric (or mass) Surface forces can be normal (pressure) or tangential (friction) Friction Forces are always parallel to velocity. This is why they are often called “tangential forces”. Their equation can be complex if velocity is not parallel to any reference axis. Mecânica de Fluidos Ambiental 2015/2016

Pressure Many fluid problems do not involve motion. They concern the pressure distribution in a static fluid and its effect on solid surfaces and on floating and submerged bodies. Fluids at rest cannot support shear stress. Pressure is used to indicate the normal force per unit of area at a given point acting on a given plane within the fluid mass of interest. How the pressure at a point varies with the orientation of the plane passing through the point? Mecânica de Fluidos Ambiental 2015/2016

Pressure at a point Small wedge (“pequena cunha”) of fluid at rest of size  x by  z by  s and depth b into the paper. There is no shear by definition, but we postulate that the pressures p x, p z, and p n may be different on each face The weight of the element also may be important. The element is assumed small, so the pressure is constant on each face. Summation of forces must equal zero (no acceleration) in both the x and z directions. Mecânica de Fluidos Ambiental 2015/2016

Pressure at a point Mecânica de Fluidos Ambiental 2015/2016

Pressure force in fluid element Mecânica de Fluidos Ambiental 2015/2016

Equilibrium of fluid element Mecânica de Fluidos Ambiental 2015/2016 Weight=  g(dxdydz)

Equilibrium of fluid element Mecânica de Fluidos Ambiental 2015/2016

General equation Mecânica de Fluidos Ambiental 2015/2016

Hydrostatic pressure Mecânica de Fluidos Ambiental 2015/2016

Hydrostatic pressure Mecânica de Fluidos Ambiental 2015/2016

Efect of variable gravity Mecânica de Fluidos Ambiental 2015/2016

Hydrostatic pressure in liquids Mecânica de Fluidos Ambiental 2015/2016

Hidrostatic pressure distribution Mecânica de Fluidos Ambiental 2015/2016

Hydrostatic pressure in gases Mecânica de Fluidos Ambiental 2015/2016

Hydrostatic forces on plane surfaces Objective: compute the hydrostatic force over flat and curve surfaces and the application point (center of pressure). When a surface is submerged in a fluid, forces develop on the surface due to the fluid. The determination of these forces is important in the design of storage tanks, ships, dams, and other hydraulic structures. Mecânica de Fluidos Ambiental 2015/2016

Hydrostatic forces on plane surfaces Plane panel of arbitrary shape completely submerged in a liquid Panel plane makes an arbitrary angle  with the horizontal free surface, so that the depth varies over the panel surface. If h is the depth to any element area dA of the plate, from hydrostatic pressure equation in liquids the pressure there is p=p a +  h. Mecânica de Fluidos Ambiental 2015/2016

Hydrostatic forces on plane surfaces Mecânica de Fluidos Ambiental 2015/2016

Therefore, since  is constant along the plate, becomes Finally, unravel this by noticing that, the depth straight down from the surface to the plate centroid. Thus 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  at which it is slanted. Mecânica de Fluidos Ambiental 2015/2016

Center of pressure However, to balance the bending-moment portion of the stress, the resultant force F acts not through the centroid but below it toward the high-pressure side. Its line of action passes through the center of pressure CP of the plate (see figure). 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. To compute y CP, we equate I xx is the area moment of inertia of the plate area about its centroidal x axis Mecânica de Fluidos Ambiental 2015/2016 Vanishes by def. of centroidal axes

Center of pressure The negative sign shows that y CP is below the centroid at a deeper level (below the gravity center) and, unlike F, depends on angle . The determination of x CP is exactly similar: I xy the product of inertia of the plate, again computed in the plane of the plate If I xy = 0, usually implying symmetry, x CP = 0 and the center of pressure lies directly below the centroid on the y axis. Mecânica de Fluidos Ambiental 2015/2016

In most cases the ambient pressure p a 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 =  h CG, and the center of pressure becomes independent of specific weight: Mecânica de Fluidos Ambiental 2015/2016

Resume Mecânica de Fluidos Ambiental 2015/2016