ENB221 Fluid Mechanics Week 4.docx

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Negareh Ghasemi

ENB221 Lecture Fluid Mechanics Fluid Statics Fluid statics deals with problems associated with fluids at rest, fluids can be either liquid or gaseous:  Hydrostatics when it is a liquid  Aerostatics when it is a gas No relative motion between adjacent fluids layers, no shear stresses in the fluid deforming it. The only stress of concern is the normal stress, which is the pressure – the variation of which only due to the weight of the fluid. Hence, gravity is the major factor in fluid statics. What do we use fluid statics for? Used to determine the forces on floating or submerged bodies and the forces developed by devices like hydraulic presses and car jacks. Also the design of water dams and liquid storage tanks. The complete description of the resultant hydrostatic force acting on a submerged surface requires the determination of: 1. The magnitude 2. The direction 3. The line of action of the force Hydrostatic Forces on Submerged Plane Surfaces A plate is subjected to fluid pressure distributed over its surface when exposed to a liquid  A gate valve in a dam  The wall of a liquid storage tank  The hull of a ship at rest Centre of Pressure On a Plane surface, the hydrostatic forces form a system of parallel forces, and we often need to determine the magnitude of the force and its point of application, which is called the centre of pressure. In most cases, the other side of the plate is open to the atmosphere, thus the atmospheric pressure acts on both sides of the plate, yielding a zero resultant. Therefore in most cases it is convenient to simply work with gauge pressure only. The plane of this surface (normal to the page) intersects the horizontal free surface at an angle θ. We take the line of intersection to be the x-axis (out of the page). Resultant Force The absolute pressure at any point is given by: The resultant hydrostatic force F acting on the surface is determined by integrating the force P dA acting on a R differential area dA over the entire surface: The first moment of area is related to the y-coordinate of the centroid of the surface: 1 Week 4 Tuesday, 27 August 2013 ENB221 Lecture Fluid Mechanics Substituting: The magnitude of the resultant force acting on a plane surface of a completely submerged plate in a homogenous (constant density) fluid is equal to the product of the pressure P aC the centroid of the surface and the area A of the surface. Line of action of the Resultant Force Two parallel force systems are equivalent if they have the same magnitude and the same moment about any point. The line of action of the resultant hydrostatic force, in general, does not pass through the centroid of the surface.  It lies underneath where the pressure is higher  The intersection of the line of action and the surface is the centre of pressure The vertical location of the line of action is determined by equating the moment of the resultant force to the moment of the distributed force about the x-axis. yp= is the distance of the centre of pressure from the x-axis (point 0) and Ixx,0 is the area moment of inertia about the x-axis Second moment of Area – Area moment of Inertia Usually given about the axes passing through the centroid of the area. Second moments of area about two parallel axes are related to each other by the parallel axis theorem. I = Second moment of area about the x-axis passing through the centroid of the area xx,c Yc= the distance between two parallel axes Calculating the distance of the Centre of Pressure from the x-axis Combining: Ignoring Atmospheric Pressure The case where P =00 (which is the usual case) this expression simplifes greatly. The vertical distance of the centre of pressure from the free surface is given by Rearranging so all distances are from the free surface 2 Week 4 Tuesday, 27 August 2013 ENB221 Lecture Fluid Mechanics Common values for second moment of area about the x-axis Centre of Pressure If there is a symmetry about the y-axis the centre of pressure lies on the y-axis directly below the centroid. The point H prom the free surface lies on the vertical plane of symmetry. Pressure Prism Pressure acts normal to the surface, and the hydrostatic forces acting on a flat plate of any shape form a volume whose base is the plate area and whose length is the linearly varying pressure. The volume of the pressure prism is equal to the magnitude of the resultant hydrostatic force,RF : The line of action passes through the centroid of the pressure prism, hence, the projection of the centroid of the prism onto the plate occurs at the centre of pressure. Problem of determining the hydrostatic force reduces to finding the two points and the volume of the pressure prism. 3 Week 4 Tuesday, 27 August 2013 ENB221 Lecture Fluid Mechanics Tilted Rectangular Plate Vertical Plate Distance to the centre of pressure For a completely submerged vertical plate whose top is
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