JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech II FLUID MECHANICS Previous Question Paper (March - 2021 - R13/R15)

Consider a plane surface of arbitrary shape submerged in a static liquid at an angle θ with the free surface. Let A be the total area of the surface, G be the centroid, and P be the center of pressure. Let h_G be the depth of the centroid and h_P be the depth of the center of pressure from the free surface.

The total pressure force F acting on the surface is given by:

To find the center of pressure, we equate the moment of the total force about the free surface to the sum of the moments of the elementary forces acting on the surface. Let an elementary area dA be at a distance y from the axis O-O (the intersection of the plane of the surface and the free surface). The depth of this element is h = y sin θ.

The force on the element is dF = ρg(y sin θ) dA. The moment of this force about O-O is:

Integrating over the entire area, the total moment M is:

where I_O is the second moment of area about the axis O-O. Equating this to the moment of the resultant force F acting at the center of pressure (at distance y_P from O-O):

Substituting F = ρgA y_G sin θ (where y_G is the distance of the centroid from O-O):

Using the parallel axis theorem, I_O = I_G + A y_G², where I_G is the moment of inertia about the centroidal axis:

Converting back to vertical depths using h_P = y_P sin θ and h_G = y_G sin θ:

Pascal's Law states that the pressure applied to any part of an enclosed fluid at rest is transmitted equally and undiminished to every portion of the fluid and to the walls of the container, acting at right angles to the surface.

To prove this, consider a small wedge-shaped element of fluid of unit thickness in a static fluid. Let the pressure intensities on the faces be Px, Py, and Ps, and the lengths of the sides be dx, dy, and ds. The forces acting on the faces are the product of pressure and area.

Resolving forces in the x and y directions:

• Sum of forces in x-direction: Px dy - Ps ds sin(θ) = 0. Since ds sin(θ) = dy, it follows that Px = Ps.
• Sum of forces in y-direction: Py dx - Ps ds cos(θ) - (ρg × 0.5 × dx × dy) = 0. Since ds cos(θ) = dx and the weight term (ρg × 0.5 × dx × dy) vanishes as the element size approaches zero, it follows that Py = Ps.

Thus, Px = Py = Ps, proving that pressure at a point in a static fluid is equal in all directions.

Applications of Pascal's Law include:

• Hydraulic Press: Used for lifting heavy loads by applying a small force on a small piston to generate a large force on a larger piston.
• Hydraulic Brakes: Used in automobiles where pressure applied to the brake pedal is transmitted through fluid to the brake pads at the wheels.
• Hydraulic Jacks: Used in garages and workshops for lifting vehicles.
• Hydraulic Accumulators: Used to store fluid under pressure for operating heavy machinery.