Continuity and momentum in polar plane

Solution of continuity and momentum equations in polar form

1. 1

2. Equations Continuity Equation: Which shows conservation of Mass   2

3. Momentum Equation Which shows conservation of Momentum 3

4. These are the equations which we use only in coordinate system   4

5. 5

6. Question Can we use these equations , when we have fluid in cylindrical channel In pipe or in Veins??   6

7. Topic of presentation Derivation of Continuity equation in coordinate system Derivation of Momentum Equation in coordinate system   7

8. Cartesian cylindrical 8

9. 9

10. Results from graph 10

11. Direction of   11

12. Conversion of unit vectors in terms of (r )   12

13. velocity components in (r )   As we know that velocity field in system is And the velocity field in cylindrical system is   13

14. Conversion of Del operator in ( r )   As we know that Del operator in ( ) systems is   14

15. Solutio n As we know that 15

16. And also we have By taking derivatives we have 16

17. As so that , hence we have   17

18. Using above values we will get 18

19. After simplification we will get “Del operator” in (r )   19

20. Conversion of in   20

21. For this we will use previous results , we have After simplification we will get , 21

22. Required continuity equation in   22

23. Special case (Incompressible fluid) 23

24.   24

25. Derivation of Momentum Equation for any fluid in Cylindrical Coordinate System 25

26. As we know that momentum equation in Cartesian coordinate system is, Where is material derivate is cacuhy stress tensor and is body force 26

27. We will use same procedure as we did in the derivation of continuity equation, we will transform every term in form   27

28. Material derivative in ( ) form   As we know that , 28

29. As we know from previous results that, 29

30. After simplification we will get 30

31. Now 31

32. After simplification we will get   32

33.   33

34. Stress Tensor 34

35. By putting previous results we will get After simplification we will get stress tensor in we have   35 35

36.   36

37. Put these values in Momentum Equation we will get in Momentum Equation in We have   37

38. Component of direction   38

39. Component of   39

40. Navier Stock Equation As we know that N.S equation in   40

41. By using previous results we will get   41

42. Component of Navier stock direction   42

43. Component of Navier stock   43

44. Applications Blood flow in veins Fluid flow through cylindrical pipe (house pipe) 44

45. Any Question 45