1. Who Causes the Deformation/Rotation inViscous Fluid Flow
P M V Subbarao
Professor
Mechanical Engineering Department
I I T Delhi
Scalar Product of  and a Tensor ……

2. Cartesian Fluid Element

3. Cartesian Fluid Element

4. Fluid element under a general three-dimensional stress condition.
The first index i refers to the axis, to which the fluid element surface is perpendicular.
The second index j indicates the direction of the force component.

5. Generation of Extensive Nature to Force

6. The Culprit

7. Force per Unit Area
Divide the individual components of the above force vectors by their corresponding area of the fluid element side.
The results of these divisions exhibit the components of a second order stress tensor.

8. Force per unit area : An Important Tensor in Viscous Fluid Flows
The Stress tensor

9. General Stress Condition

10. Stresses on Two Opposite Faces

11. Force Balance on A Plane
Force Balance on Plane 1 in x1 direction:
Force Balance on Plane 1 in x2 direction:
Force Balance on Plane 1 in x1 direction:

12. The Resultant Force on A Plane
The resulting Force on Plane 1:
The resulting Force on Plane 2:
The resulting Force on Plane 3:

13. Force Balance per Unit Volume
The total resulting forces acting on the entire surface of the element are obtained by adding the nine components.
Defining the volume element , we divide the results by and obtain the resulting force vector that is acting on the volume element.

14. Transpose : A Natural Thinking

15. Stress is the Mother of Force
The stress is A tensor
It can be easily shown that
The above expression is a scalar differentiation of the second order stress tensor and is called the divergence of the tensor field.
We conclude that the force acting on the surface of a fluid element is the divergence of its stress tensor.
The stress tensor is usually divided into its normal and shear stress parts.

16. Stress Tensor in an Incompressible Viscous Fluid Flow
How do they act on fluid elements entering/ Leaving
a Control Volume (Viscous Fluid Flow)????

17. Format of Examinations
Minor Examinations: each of one hour duration.
Major Examination: Two hours duration.
All the examinations are open – hand written notes examinations.
All kinds of your own hand written notes are allowed to be used during examinations. (Class notes, Tutorial notes, etc.)
No printed or photocopied material is allowed.
Printed copies of my lectures are also not allowed.

18. Weightage of examinations
A Grand total of 200 points will be considered.
Following distribution is sued for individual evaluations:
Minor
Minor –
Major 70
Assignment 40
Course Project 20

19. Outcome of Viscous Fluid Flow Analysis

20. The Art of Using Principle of Conservation
The conservation laws in integral form are, strictly speaking, valid for closed systems.
In fluid flows, the mass flow continuously crosses the system boundary.
To apply the conservation laws to open systems, special mathematical tools are essential.
A treatment for the volume integral of an arbitrary field quantity f(x,t) demands a special engineering law.