1. Density y Volume,  Mass, m C Elemental Volume,   Mass, m x z

2. Fluid Shear Force, Fx Velocity u Fluid Element at time, t P P’ Force, Fx Velocity u Fluid Element at time, t Fluid Element at time, t+t y y x y N O

3. Viscosity In some sense measures fluidity of a fluid. Actually it is the resistance offered by a layer of fluid to the motion of an adjacent one. Consider the two-plate experiment. In case of a fluid in between them, we know that the upper plate moves with a speed U whereas the lower plate does not move. This sets up a velocity gradient in a direction normal to flow.

absolute or dynamic viscosity. Its dimensions are ML-1 T-1 4. Newtonian Fluids For a Newtonian fluid ie., In general m is called absolute or dynamic viscosity. Its dimensions are ML-1 T-1 Kinematic viscosity (n) is defined as ( m/r) . Its dimensions are M L-3 Air and water are common examples

5. Non-Newtonian Fluids Shear stress not proportional to deformation rate Toothpaste, paint are common examples Bingham Plastic Shear Stress, t pseudoplastic Dilatant Newtonian Deformation rate

6. Temperature Effect Values of viscosity m and kinematic viscosity n for various fluids are tabulated in handbooks and textbooks. For air viscosity may be calculated using the Sutherland formula, where T is in Kelvin and m is in kg/s m. It is observed that viscosity of a liquid decreases with temperature where as that of a gas increases with temperature. Find out why.

7. Velocity Field Velocity at a point may be defined as the instantaneous velocity of a fluid particle passing through that point. For a steady flow the properties do not change with time - If S is any property,

8. 1, 2, 3 Dimensional Flows r One Dimensional flow. u = u(r) x One, Two, Three Dimensional Flows -- One, Two, Three Space Coordinates required to specify Velocity Field r R One Dimensional flow. u = u(r) u x umax y Two Dimensional flow. u = u(x,y) u=u(x,y) u=u(x,y) x

9. Surface Tension q>90 0 q<90 0 It is the apparent interfacial stress that acts when a liquid has a density interface like liquid-gas, liquid-solid, liquid-liquid 2R Dh q 2R q>90 0 Dh q q<90 0

10. Surface Shapes water water wax soap Wetting Non-wetting

11. Forces on half a fluid drop

12. Continuum Flow For most engineering applications we consider fluid to be continuous. But we do know that matter consists of molecules. To be considered continuous a fluid must have a large number of molecules in a tiny place which is small compared to the body dimensions. Under ordinary conditions this is true. For eg., A cubic metre of Air at STP contains 2.5 x 10 25 molecules. Its mean free path is like 6.6 x 10-8 m.

13. Rarefied Flows At great heights from the sea level it is not possible to consider air to be continuous. The molecular mean free path may be of the same order of magnitude as the body dimensions. Eg., at an altitude of 130 km the mean free path of air is 10.2m. Then it becomes important to consider individual or groups of molecules. This leads to the discipline of Rarefied Gas Dynamics.

14. Bulk Modulus of Elasticity Compressibility of a fluid may be expressed in terms of Bulk Modulus of Elasticity. For air k is equal to g (adiabatic conditions) and p (isothermal) For water, k =2.2 Gpa, meaning that when a pressure of 0.1Mpa acts upon a cubic metre of water, the change in volume resulting is 1/22000 m 3.