Fluid Statics Pascal’s Law tells us that “Pressure at any point in a fluid is the same in all directions”. This means that any object submerged in a fluid is subjected to an area force distribution across their entire surface. These area force distributions are called pressures. Fluid – Any continuous substance which, when at rest, is unable to support shear force. Liquids and gases are the most common examples of fluids. For the infinitesimal area at the right it can be seen that all of the pressures are equal. Similar results can be obtained for p 4 by rotating the element 90 o around the y-axis.

Pressure in a fluid is a result of the weight of the fluid above the region of space you are examining. The pressure at the bottom of a drinking glass is due to the weight of the fluid in the glass. This suggests that the fluid pressure varies with vertical position (depth) in a fluid. Consider the situation to the right to verify this. Note that the pressure on the sides (x and z directions) of the parcel would be concentrated at the same depth and therefore would sum to zero. This gives the variation of pressure with depth, where P 0 is the pressure at the surface of the fluid and  is the density of the fluid. Note that h is measured from the surface of the fluid.

Let us now consider the hydrostatic pressure on a submerged rectangular surface. We can consider the system as if it was a loaded beam, by looking at a cross-section. Do not forget to include the width of the area! We can examine the load profile as either a composite body or a single pressure distribution. Composite Body: A’ is the area of the load distribution, not the area of the plate. We use h to represent the vertical depth, while y is parallel to the plane of the plate.

Single Pressure Distribution: Area of Plate Area of Pressure Distribution Alternatively, we can use the average pressure and average depth to determine R. The location of R must be determined as above from the centroid expression.

Curved Plate: A curved plate is very similar to a flat plate, except that you must look in multiple directions simultaneously. The pressure profile, for example, has both x and y components no matter your choice of reference frame. We again have two methods we can use. We can (1) analyze by direct integration, and determine R x and R y as well as the centroid. You would next have to determine the location of R x and R y. Or (2) look at the forces exerted on the volume of water above (or below)the plate. This second method is typically easier. Analyze the constant pressure profile (P y ) on top of the water parcel, then analyze the linear pressure profile with depth (P x ). Summing all of the forces for this volume of water will allow you to determine the interaction R between the water and the plate, which is equal and opposite to the net force R on the plate.

Hydrostatic Pressure on Flat Surfaces of Any Shape For any flat surface submerged in a fluid we can examine it using a method similar to what was done for the flat plate. Similarly we could examine the “pressure volume”. Note that h and x must be expressed in terms of y! It is also necessary to determine the location of the centroid. Note that the center of pressure and the centroid of the plate are not at the same location!!

Buoyancy Objects that are at least partially submerged in a fluid have a buoyant force acting on them. The buoyant force is due to the difference in pressure on opposite sides of the object. The buoyant force can also be defined as the weight of water that has been displaced by the object. Similaly,