Mecânica de Fluídos Ambiental 2015/2016 Lecture 5 Pressure distribution in a fluid Hydrostaic forces on curved surfaces Bouyancy and stability Mecânica de Fluídos Ambiental 2015/2016

From the previous lecture Forças hidrostáticas sobre superfícies planas horizontais Superfície plana, idealmente sem espessura (placa metálica fina), imersa num fluido em repouso Cada elemento de superfície está sujeita a uma força elementar exercida, perpendicularmente, pelo fluido. Esta força é originada pelo campo de pressão existente no próprio fluido. A pressão p é uniforme ao longo de toda a superfície. Se A for a área da superfície, a força a que a placa está sujeita será: F=pA Como p é uniforme em toda a superfície placa, o ponto de aplicação de 𝐹 , centro de pressões (CP), coincide com o centro geométrico (CG) da superfície 𝐹 Mecânica de Fluidos Ambiental 2015/2016

From the previous lecture Forças hidrostáticas sobre superfícies planas inclinadas Superfície plana imersa num fluido de peso específico , com uma inclinação  , em relação ao plano da superfície livre. Sistema de coordenadas cartesianas planas (x,y), localizada no plano da superfície inclinada, com origem no CG da superfície. É introduzida uma coordenada auxiliar, . Cada elemento de área dA da superficie mergulhada está sujeita a uma força elementar de módulo 𝑑𝐹=𝑝𝑑𝐴= 𝑝 𝑎 +𝛾ℎ 𝑑𝐴=( 𝑝 𝑎 +𝛾𝑠𝑒𝑛𝜃)𝑑𝐴 p=pa hCG 𝒅𝑭 𝑭 CG CP dA  x y h  𝒈 O Mecânica de Fluidos Ambiental 2015/2016

From the previous lecture Forças hidrostáticas sobre superfícies planas inclinadas (cont.) 𝑑𝐹=𝑝𝑑𝐴= 𝑝 𝑎 +𝛾ℎ 𝑑𝐴=( 𝑝 𝑎 +𝛾𝑠𝑒𝑛𝜃)𝑑𝐴 A força resultante exercida sobre a totalidade da fase superior da superfície inclinada é obtida pela integração da eq. anterior em toda a superfície de área A: 𝐹= 𝐴 𝑑𝐹= 𝑝 𝑎 𝐴+𝛾𝑠𝑒𝑛𝜃𝐴 𝐹= 𝑝 𝑎 +𝛾 ℎ 𝐶𝐺 𝐴= 𝑝 𝐶𝐺 𝐴 Para a completa definição da força resultante é necessário localizar o ponto de aplicação, ou seja, o centro de pressões (CP) p=pa hCG 𝒅𝑭 𝑭 CG CP dA  x y h  𝒈 O Mecânica de Fluidos Ambiental 2015/2016

From the previous lecture 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 pCG=hCG, and the center of pressure becomes independent of specific weight: Mecânica de Fluidos Ambiental 2015/2016

Mecânica de Fluidos Ambiental 2015/2016

Mecânica de Fluidos Ambiental 2015/2016

Mecânica de Fluidos Ambiental 2015/2016

Hydrostatic forces on curved surfaces Many surface of interest are nonplanar (such as those associated with dams, pipes and tanks). In this cases we cannot apply the equations obtained for plane surface. The resultant pressure force on a curved surface is most easily computed by separating it into horizontal and vertical components. Consider the arbitrary curved surface in figure: The incremental pressure forces, being normal to the local area element, vary in direction along the surface and thus cannot be added numerically. Mecânica de Fluidos Ambiental 2015/2016

Hydrostatic forces on curved surfaces Figure shows a free-body diagram of the column of fluid contained in the vertical projection above the curved surface. FH and FV are exerted by the surface on the fluid column. Other forces are fluid weight and horizontal pressure on the vertical sides of this column. On the lower part, the summation of horizontal forces shows that FH due to the curved surface is exactly equal to the force FH on the vertical left side of the fluid column. Summation of vertical forces on the fluid free body shows that FV = w1+w2+wair Two rules can be state: 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. Mecânica de Fluidos Ambiental 2015/2016

Mecânica de Fluidos Ambiental 2015/2016

Mecânica de Fluidos Ambiental 2015/2016

Mecânica de Fluidos Ambiental 2015/2016

Hydrostatic forces in layered fluids The formulas for planed and curved surfaces are only valid for fluids of uniform density If the fluid is layered with different densities a single formula cannot solve the problem because the slope of the linear pressure distribution changes between layers We can apply the formulas separately to each layer, and then sum the separate layer forces and moments Mecânica de Fluidos Ambiental 2015/2016

Mecânica de Fluidos Ambiental 2015/2016 Exemplo Mecânica de Fluidos Ambiental 2015/2016

Mecânica de Fluidos Ambiental 2015/2016

Mecânica de Fluidos Ambiental 2015/2016 Buoyancy (impulsão) The same principles used to compute hydrostatic forces on surfaces can be applied to the net pressure force on a completely submerged or floating body The resultant fluid force acting on the body is called buoyant force The results are the two laws of buoyancy discovered by Archimedes in the third century B.C.: 1. A body immersed in a fluid experiences a vertical buoyant force equal to the weight of the fluid it displaces. 2. A floating body displaces its own weight in the fluid in which it floats. 𝑭 𝑩 =𝜸 𝒃𝒐𝒅𝒚 𝒗𝒐𝒍𝒖𝒎𝒆 =𝝆𝒈(𝒃𝒐𝒅𝒚 𝒗𝒐𝒍𝒖𝒎𝒆) Mecânica de Fluidos Ambiental 2015/2016

Mecânica de Fluidos Ambiental 2015/2016 Buoyancy 1 Downward force: weight of the fluid above the upper surface (1). Upward force: weight of the fluid above the lower surface (2). 2 Resulting force= (Upward force) – (Downward force)= = weight of fluid equivalent to body volume Assumes that the fluid has uniform specific weight. Resulting force is applied in the gravity center of the displaced volume: buoyancy center. For a layered fluid (LF): 𝐹 𝐵 𝐿𝐹 = 𝜌 𝑖 𝑔 (𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑑 𝑣𝑜𝑙𝑢𝑚𝑒) 𝑖 Mecânica de Fluidos Ambiental 2015/2016

Mecânica de Fluidos Ambiental 2015/2016

Mecânica de Fluidos Ambiental 2015/2016

Stability of floating body A floating body may not approve of the position in which it is floating. If so, it will overturn at the first opportunity and is said to be statically unstable. If the body gravity center is located below the buoyancy center, the body would be stable. Mecânica de Fluidos Ambiental 2015/2016

Stability of floating body Outline of the basic principle of the static stability calculation Fig. 2.18 - Calculation of the metacenter M of the floating body shown in (a). Tilt the body a small Angle . Either (b) B’ moves far out (point M above G denotes stability); or (c) B’ moves slightly (point M below G denotes instability). Mecânica de Fluidos Ambiental 2015/2016

Stability of floating body 1. The basic floating position is calculated from Eq. (2.36). The body’s center of mass G and center of buoyancy B are computed. 2. The body is tilted a small angle , and a new waterline is established for the body to float at this angle. The new position B’ of the center of buoyancy is calculated. A vertical line drawn upward from B’ intersects the line of symmetry at a point M, called the metacenter, which is independent of  for small angles. 3. If point M is above G (that is, if the metacentric height is positive), a restoring moment is present and the original position is stable. If M is below G (negative ), the body is unstable and will overturn if disturbed. Stability increases with increasing. Mecânica de Fluidos Ambiental 2015/2016

Mecânica de Fluidos Ambiental 2015/2016 Problems The Mediterranean Sea salinity is 39 (g/l) and the Atlantic’s Ocean at Gibraltar salinity is 36. If the straight’s depth is 400 m and temperature was the same at any depth on both sides (Atlantic and Mediterranean). What would be the level difference between both sides of the straight? Atlantic Mediterranean Mecânica de Fluidos Ambiental 2015/2016

Mecânica de Fluidos Ambiental 2015/2016