Pascals law FLUID MECHANICS

Pascals law FLUID MECHANICS

1. Pascal’s Law According to Pascal’s Law, Pressure or intensity of pressure at a point in a static fluid will be equal in all directions. Let us consider one arbitrary fluid element of rectangular shape ABC as displayed here in following figure. Let us assume that width of fluid element ABC perpendicular to the plane of paper is unity. Let us consider the following terms as mentioned here PX = Pressure acting in X- direction over the face AB PY = Pressure acting in Y- direction over the face AC PZ = Pressure acting in Z- direction over the face BC θ = Angle ABC, as displayed above in figure dx, dy and ds : Fluid element dimensions ρ = Density of the fluid

2. Let us analyse here the forces acting on the fluid element ABC Force on the face AB, AC and BC F AB = P X x Area of face AB = PX. dy. 1 = P X . dy F AC = P Y x Area of face AC = PY. dx. 1 = P Y . dx F BC = P Z x Area of face BC = PZ. ds. 1 = P Z . ds Weight of the fluid element, W = Volume x Density of fluid x acceleration due to gravity W = Area x width of fluid element x Density of fluid x acceleration due to gravity W = (AB x AC/2) x 1 x ρ x g = (dy dx/2) x ρ x g

3. C onsidering the forces in X-direction P x . d y - P Z . ds Sin (90- θ) = 0 P X . dy = P Z . ds Cos θ As we can see from above fluid element ABC, dy = ds Cos θ P X . dy = P Z . dy P X = P Z ............................................(1) Considering the forces in Y-direction P Y . dx - P Z . ds Cos (90- θ) - (dy dx/2) x ρ x g = 0 P Y . dx - P Z . ds Sin θ - (dy dx/2) x ρ x g = 0 As fluid element is very small and therefore, we can neglect the weight of fluid element P Y . dx - P Z . ds Sin θ = 0 As we can see from above fluid element ABC, dx = ds Sin θ P Y . dx - P Z dx = 0 P Y = P Z ................................................(2)

4. From last two expressions mentioned in eq (1) AND eq(2) , we can write following equation as mentioned here P X = P Y = P Z .............................prooved We can say from above equation that pressure at any point in X, Y and Z directions will be same. Pascal’s Law provides the base for any hydraulic system or we can say that complete hydraulic system is based on the principle of Pascal’s Law.

5. Let us understand, how hydraulic system is based on Pascal's Law As we know that pressure at every point in enclosed liquid will be same and hence there is no matter about the shape of vessel or container in which liquid is placed.In order to understand how hydraulic system depends over Pascal’s law , we will consider following case. P1= F1/A1 And P2= F2/A2 According to Pascal's law P1= P2 F1/A1 = F2/A2 F1 =F2 [A1/A2] As we may see in above figure, area A2 is larger as compared to area A1 hence we will require less force to lift the heavy load. This is the basic principle which is used by all hydraulic system. For more detailed information about the Pascal's Law, we must have to find the post i.e. Application of fluid power: Hydraulic Jack.