Fluid statics Part - I, Basic equations of fluid statics

Pascal's Law of Hydrostatics; Basic equations of fluid statics

1. Fluid statics Part - I Hydrostatics or Fluid Statics is the study of fluids at rest. It's practical applications are numerous. Some of which are Fluid Manometers, buoyancy measurements, density calculations. 

2. Fluid Elements - Definition: Fluid element - Body Force : Surface Force

3. Fluid Elements - Definition: Fluid element can be defined as an infinitesimal region of the fluid continuum in isolation from its surroundings. Two types of forces exist on fluid elements Body Force : distributed over the entire mass or volume of the element. It is usually expressed per unit mass of the element or medium upon which the forces act. Example: Gravitational Force, Electromagnetic force fields etc. 

4. Fluid Elements - Definition: Surface Force : Forces exerted on the fluid element by its surroundings through direct contact at the surface. Surface force has two components: Normal Force: along the normal to the area Shear Force: along the plane of the area. The ratios of these forces and the elemental area in the limit of the area tending to zero are called the normal and shear stresses respectively.

5. Pascal's Law of Hydrostatics The states that the normal stresses at any point in a fluid element at rest are directed towards the point from all directions and they are of the equal magnitude.

6. Pascal's Law of Hydrostatics Let us prove this law by considering the equilibrium of a small fluid element shown in Figure below.

7. Pascal's Law of Hydrostatics

8. Pascal's Law of Hydrostatics Also, Summing the forces yields

9. Pascal's Law of Hydrostatics…. Since the volume of the fluids    is very small, the weight of the element is negligible in comparison with other force terms. So the above Equation becomes                     P y  =  P  n                                                          Hence,  P  n   =  P  x   =  P  y                                       Similar relation can be derived for the z-axis direction. This law is valid for the cases of fluid flow where shear stresses do not exist. The cases are Fluid at rest. No relative motion exists between different fluid layers. For example, fluid at a constant linear acceleration in a container. Ideal fluid flow where viscous force is negligible.

10. Basic equations of fluid statics

11. Basic equations of fluid statics The body force is given by The surface force

12. Basic equations of fluid statics When  only the first two terms become significant. The above equation becomes Similarly, pressures at the center of all the faces can be derived in terms of  P (x, y, z)  and its gradient.

13. Basic equations of fluid statics… When  only the first two terms become significant. The above equation becomes Similarly, pressures at the center of all the faces can be derived in terms of  P (x, y, z)  and its gradient….Note

14. Basic equations of fluid statics… Similarly the surface forces on the other two directions (x and z) will be The surface force which is the vectorical sum of the force scalar components.

15. Basic equations of fluid statics… The total force acting on the fluid is The total force per unit volume is. For a static fluid,  dF =0. Then,   

16. Basic equations of fluid statics…

17. Basic equations of fluid statics… If acceleration due to gravity   is expressed as , the components in the x, y and z directions will be ….* …...** …..*** The above equations (*,**,***) are the basic equation for a fluid at rest.

18. Basic equations of fluid statics… If the gravity is aligned with one of the co-ordinate axis, for example z- axis, then The component equations are reduced to                                                                                                                

19. Basic equations of fluid statics… This simplification is valid under the following conditions. Static fluid Gravity is the only body force. The z-axis is vertical and upward.