CE 1501 CE 150 Fluid Mechanics G.A. Kallio Dept. of Mechanical Engineering, Mechatronic Engineering & Manufacturing Technology California State University, Chico

CE 1502 Fluid Statics Reading: Munson, et al., Chapter 2

CE 1503 Fluid Statics Problems Fluid statics refers to the study of fluids at rest or moving in such a manner that no shearing stresses exist in the fluid These are relatively simple problems since no velocity gradients exist - thus, viscosity does not play a role Applications include the hydraulic press, manometry, dams, and fluid containment (tanks)

CE 1504 Pressure at a Point Pressure is a scalar quantity that is defined at every point within a fluid Force analysis on a wedge-shaped fluid element is presented in the text (Figure 2.1)

CE 1505 Pressure at a Point The result shows that pressure at a point is independent of direction as long as there are no shearing stresses (or velocity gradients) present in the fluid This result is known as Pascal’s law For fluids in motion with shearing stresses, this result is not exactly true, but is still a very good approximation for most flows

CE 1506 The Pressure Field Now, we ask: how does pressure vary from point to point in a fluid w/o shearing stresses? Consider a small rectangular element of fluid (Figure 2.2):

CE 1507 The Pressure Field Two types of forces acting on element: –surface forces (due to pressure) –body forces (due to external fields such as gravity, electric, magnetic, etc.) Newton’s second law states that –where

CE 1508 The Pressure Field The differential volume terms cancel, leaving: This is the general equation of motion for a fluid w/o shearing stresses Recall: –known as the pressure gradient

CE 1509 Pressure Field for a Fluid at Rest For a fluid at rest, This vector equation can be broken down into component form: This shows that p only depends upon z, the direction in which gravity acts:

CE Pressure Field for a Fluid at Rest This equation can be used to determine how pressure varies with elevation within a fluid; integrating yields: If the fluid is incompressible (e.g., a liquid), then For a liquid with a free surface exposed to pressure p 0 :

CE Pressure Field for a Fluid at Rest If the fluid is compressible, then  (or  ) is not a constant; this is true for gases, however, the effect on pressure is not significant unless the elevation change is very large The pressure variation in our atmosphere is such an exception To integrate the pressure field equation, we need to know how atmospheric air density varies with elevation

CE Atmospheric Pressure Variation From the pressure field equation, Atmospheric air can be regarded an an ideal gas, P =  RT, so: Separating variables & integrating:

CE Atmospheric Pressure Variation In the troposphere (sea level to 11 km), temperature varies as: –where T a is the temperature at sea level and  is the lapse rate Completing the integration yields: –where the lapse rate for a standard atmosphere is  = K/m

CE Pressure Measurement Manometer – gravimetric device based upon liquid level deflection in a tube Bourdon tube – elliptical cross-section tube coil that straightens under under influence of gas pressure Mercury barometer – evacuated glass tube with open end submerged in mercury to measure atmospheric pressure Pressure transducer – converts pressure to electrical signal; i) flexible diaphragm w/strain gage ii) piezo- electric quartz crystal

CE The Manometer Simple, accurate device for measuring small to moderate pressure differences Rules of manometry: –pressure change across a fluid column of height h is  gh –pressure increases in the direction of gravity, decreases in the direction opposing gravity –two points at the same elevation in a continuous static fluid have the same pressure

CE Hydrostatic Force on a Plane Surface The forces on a plane surface submerged in a static fluid are due to pressure and are always perpendicular to that surface These forces can be resolved into a single resultant force F R, acting at a particular location (x R, y R ) along the surface For a horizontal surface: F R = pA x R, y R is at the centroid of the surface

CE Hydrostatic Force on a Plane Surface Resultant force on an inclined plane surface defined by angle  :

CE The resultant force is found by integrating the differential forces over the entire surface: –where h c is the vertical distance to the centroid of the area, which can be found from the y-direction distance to the centroid: Hydrostatic Force on a Plane Surface

CE Hydrostatic Force on a Plane Surface The location of the resultant force is found by integrating the differential moments over the area; this yields expressions which contain moments of inertia: –refer to Figure 2.18 for the centroids and moments of inertia of common shapes

CE Hydrostatic Force on a Plane Surface The resultant force on vertical, rectangular surfaces can be found using a graphical interpretation known as the pressure prism:

CE Hydrostatic Force on a Plane Surface The resultant force is Graphically, this can be interpreted as the volume of the pressure prism: This force passes through the centroid of the pressure prism, located a distance h/3 above the base

CE Hydrostatic Force on a Plane Surface The pressure prism can also be used for vertical surfaces that do not extend to the free surface of the fluid; here, the cross section is trapezoidal and the resultant force is: –located at:

CE Hydrostatic Force on a Curved Surface See section 2.10

CE Buoyancy The resultant buoyant force on a submerged or partially submerged object in a static fluid is given by Archimedes’ principle: –where is the submerged volume of the object “The buoyant force is equal to the weight of the fluid displaced by the object and is in a direction opposite the gravitational force” This force arises from the net pressure force acting on the object’s surface

CE Buoyancy The line of action of the buoyant force passes through the centroid of the displaced volume, often called the center of buoyancy (COB) The stability of submerged objects is determined by the center of gravity (COG): –Stable: COG is below COB –Unstable: COG is above COB For floating objects, stability is complicated by the fact that the COB changes with rotation

CE Pressure in a Fluid with Rigid-Body Motion From before, Consider an example of linear motion - an open container of liquid accelerating along a straight line in the y-z plane (Figure 2.29); we then have:

CE Pressure in a Fluid with Rigid-Body Motion Due to the imbalance of forces on the liquid in the y and z directions, the slope of the liquid surface will change to produce a pressure gradient that offsets the acceleration; it can be shown that this slope is: