Velocity Field#

\(v_r\)

1

v_r

\(v_\theta\)

2

v_theta

\(v_\phi\)

3

v_phi

\(v_r'\)

4

vp_r

\(v_\theta'\)

5

vp_theta

\(v_\phi'\)

6

vp_phi

\(\overline{v_r}\)

7

vm_r

\(\overline{v_\theta}\)

8

vm_theta

\(\overline{v_\phi}\)

9

vm_phi

\(\frac{\partial v_r}{\partial r}\)

10

dv_r_dr

\(\frac{\partial v_\theta}{\partial r}\)

11

dv_theta_dr

\(\frac{\partial v_\phi}{\partial r}\)

12

dv_phi_dr

\(\frac{\partial v_r'}{\partial r}\)

13

dvp_r_dr

\(\frac{\partial v_\theta'}{\partial r}\)

14

dvp_theta_dr

\(\frac{\partial v_\phi'}{\partial r}\)

15

dvp_phi_dr

\(\frac{\partial \overline{v_r}}{\partial r}\)

16

dvm_r_dr

\(\frac{\partial \overline{v_\theta}}{\partial r}\)

17

dvm_theta_dr

\(\frac{\partial \overline{v_\phi}}{\partial r}\)

18

dvm_phi_dr

\(\frac{\partial v_r}{\partial \theta}\)

19

dv_r_dt

\(\frac{\partial v_\theta}{\partial \theta}\)

20

dv_theta_dt

\(\frac{\partial v_\phi}{\partial \theta}\)

21

dv_phi_dt

\(\frac{\partial v_r'}{\partial \theta}\)

22

dvp_r_dt

\(\frac{\partial v_\theta'}{\partial \theta}\)

23

dvp_theta_dt

\(\frac{\partial v_\phi'}{\partial \theta}\)

24

dvp_phi_dt

\(\frac{\partial \overline{v_r}}{\partial \theta}\)

25

dvm_r_dt

\(\frac{\partial \overline{v_\theta}}{\partial \theta}\)

26

dvm_theta_dt

\(\frac{\partial \overline{v_\phi}}{\partial \theta}\)

27

dvm_phi_dt

\(\frac{\partial v_r}{\partial \phi}\)

28

dv_r_dp

\(\frac{\partial v_\theta}{\partial \phi}\)

29

dv_theta_dp

\(\frac{\partial v_\phi}{\partial \phi}\)

30

dv_phi_dp

\(\frac{\partial v_r'}{\partial \phi}\)

31

dvp_r_dp

\(\frac{\partial v_\theta'}{\partial \phi}\)

32

dvp_theta_dp

\(\frac{\partial v_\phi'}{\partial \phi}\)

33

dvp_phi_dp

\(\frac{\partial \overline{v_r}}{\partial \phi}\)

34

dvm_r_dp

\(\frac{\partial \overline{v_\theta}}{\partial \phi}\)

35

dvm_theta_dp

\(\frac{\partial \overline{v_\phi}}{\partial \phi}\)

36

dvm_phi_dp

\(\frac{1}{r}\frac{\partial v_r}{\partial \theta}\)

37

dv_r_dtr

\(\frac{1}{r}\frac{\partial v_\theta}{\partial \theta}\)

38

dv_theta_dtr

\(\frac{1}{r}\frac{\partial v_\phi}{\partial \theta}\)

39

dv_phi_dtr

\(\frac{1}{r}\frac{\partial v_r'}{\partial \theta}\)

40

dvp_r_dtr

\(\frac{1}{r}\frac{\partial v_r'}{\partial \theta}\)

41

dvp_theta_dtr

\(\frac{1}{r}\frac{\partial v_r'}{\partial \theta}\)

42

dvp_phi_dtr

\(\frac{1}{r}\frac{\partial \overline{v_r}}{\partial \theta}\)

43

dvm_r_dtr

\(\frac{1}{r}\frac{\partial \overline{v_\theta}}{\partial \theta}\)

44

dvm_theta_dtr

\(\frac{1}{r}\frac{\partial \overline{v_\phi}}{\partial \theta}\)

45

dvm_phi_dtr

\(\frac{1}{r\mathrm{sin}\theta}\frac{\partial v_r}{\partial \phi}\)

46

dv_r_dprs

\(\frac{1}{r\mathrm{sin}\theta}\frac{\partial v_\theta}{\partial \phi}\)

47

dv_theta_dprs

\(\frac{1}{r\mathrm{sin}\theta}\frac{\partial v_\phi}{\partial \phi}\)

48

dv_phi_dprs

\(\frac{1}{r\mathrm{sin}\theta}\frac{\partial v_r'}{\partial \phi}\)

49

dvp_r_dprs

\(\frac{1}{r\mathrm{sin}\theta}\frac{\partial v_\theta'}{\partial \phi}\)

50

dvp_theta_dprs

\(\frac{1}{r\mathrm{sin}\theta}\frac{\partial v_\phi'}{\partial \phi}\)

51

dvp_phi_dprs

\(\frac{1}{r\mathrm{sin}\theta}\frac{\partial \overline{v_r}}{\partial \phi}\)

52

dvm_r_dprs

\(\frac{1}{r\mathrm{sin}\theta}\frac{\partial \overline{v_\theta}}{\partial \phi}\)

53

dvm_theta_dprs

\(\frac{1}{r\mathrm{sin}\theta}\frac{\partial \overline{v_\phi}}{\partial \phi}\)

54

dvm_phi_dprs

\(\frac{\partial^2 v_r}{\partial r^2}\)

55

dv_r_d2r

\(\frac{\partial^2 v_\theta}{\partial r^2}\)

56

dv_theta_d2r

\(\frac{\partial^2 v_\phi}{\partial r^2}\)

57

dv_phi_d2r

\(\frac{\partial^2 v_r'}{\partial r^2}\)

58

dvp_r_d2r

\(\frac{\partial^2 v_\theta'}{\partial r^2}\)

59

dvp_theta_d2r

\(\frac{\partial^2 v_\phi'}{\partial r^2}\)

60

dvp_phi_d2r

\(\frac{\partial^2 \overline{v_r}}{\partial r^2}\)

61

dvm_r_d2r

\(\frac{\partial^2 \overline{v_\theta}}{\partial r^2}\)

62

dvm_theta_d2r

\(\frac{\partial^2 \overline{v_\phi}}{\partial r^2}\)

63

dvm_phi_d2r

\(\frac{\partial^2 v_r}{\partial \theta^2}\)

64

dv_r_d2t

\(\frac{\partial^2 v_\theta}{\partial \theta^2}\)

65

dv_theta_d2t

\(\frac{\partial^2 v_\phi}{\partial \theta^2}\)

66

dv_phi_d2t

\(\frac{\partial^2 v_r'}{\partial \theta^2}\)

67

dvp_r_d2t

\(\frac{\partial^2 v_\theta'}{\partial \theta^2}\)

68

dvp_theta_d2t

\(\frac{\partial^2 v_\phi'}{\partial \theta^2}\)

69

dvp_phi_d2t

\(\frac{\partial^2 \overline{v_r}}{\partial \theta^2}\)

70

dvm_r_d2t

\(\frac{\partial^2 \overline{v_\theta}}{\partial \theta^2}\)

71

dvm_theta_d2t

\(\frac{\partial^2 \overline{v_\phi}}{\partial \theta^2}\)

72

dvm_phi_d2t

\(\frac{\partial^2 v_r}{\partial \phi^2}\)

73

dv_r_d2p

\(\frac{\partial^2 v_\theta}{\partial \phi^2}\)

74

dv_theta_d2p

\(\frac{\partial^2 v_\phi}{\partial \phi^2}\)

75

dv_phi_d2p

\(\frac{\partial^2 v_r'}{\partial \phi^2}\)

76

dvp_r_d2p

\(\frac{\partial^2 v_\theta'}{\partial \phi^2}\)

77

dvp_theta_d2p

\(\frac{\partial^2 v_\phi'}{\partial \phi^2}\)

78

dvp_phi_d2p

\(\frac{\partial^2 \overline{v_r}}{\partial \phi^2}\)

79

dvm_r_d2p

\(\frac{\partial^2 \overline{v_\theta}}{\partial \phi^2}\)

80

dvm_theta_d2p

\(\frac{\partial^2 \overline{v_\phi}}{\partial \phi^2}\)

81

dvm_phi_d2p

\(\frac{\partial^2 v_r}{\partial r \partial \theta}\)

82

dv_r_d2rt

\(\frac{\partial^2 v_\theta}{\partial r \partial \theta}\)

83

dv_theta_d2rt

\(\frac{\partial^2 v_\phi}{\partial r \partial \theta}\)

84

dv_phi_d2rt

\(\frac{\partial^2 v_r'}{\partial r \partial \theta}\)

85

dvp_r_d2rt

\(\frac{\partial^2 v_\theta'}{\partial r \partial \theta}\)

86

dvp_theta_d2rt

\(\frac{\partial^2 v_\phi'}{\partial r \partial \theta}\)

87

dvp_phi_d2rt

\(\frac{\partial^2 \overline{v_r}}{\partial r \partial \theta}\)

88

dvm_r_d2rt

\(\frac{\partial^2 \overline{v_\theta}}{\partial r \partial \theta}\)

89

dvm_theta_d2rt

\(\frac{\partial^2 \overline{v_\phi}}{\partial r \partial \theta}\)

90

dvm_phi_d2rt

\(\frac{\partial^2 v_r}{\partial r \partial \phi}\)

91

dv_r_d2rp

\(\frac{\partial^2 v_\theta}{\partial r \partial \phi}\)

92

dv_theta_d2rp

\(\frac{\partial^2 v_\phi}{\partial r \partial \phi}\)

93

dv_phi_d2rp

\(\frac{\partial^2 v_r'}{\partial r \partial \phi}\)

94

dvp_r_d2rp

\(\frac{\partial^2 v_\theta'}{\partial r \partial \phi}\)

95

dvp_theta_d2rp

\(\frac{\partial^2 v_\phi'}{\partial r \partial \phi}\)

96

dvp_phi_d2rp

\(\frac{\partial^2 \overline{v_r}}{\partial r \partial \phi}\)

97

dvm_r_d2rp

\(\frac{\partial^2 \overline{v_\theta}}{\partial r \partial \phi}\)

98

dvm_theta_d2rp

\(\frac{\partial^2 \overline{v_\phi}}{\partial r \partial \phi}\)

99

dvm_phi_d2rp

\(\frac{\partial^2 v_r}{\partial \theta \partial \phi}\)

100

dv_r_d2tp

\(\frac{\partial^2 v_\theta}{\partial \theta \partial \phi}\)

101

dv_theta_d2tp

\(\frac{\partial^2 v_\phi}{\partial \theta \partial \phi}\)

102

dv_phi_d2tp

\(\frac{\partial^2 v_r'}{\partial \theta \partial \phi}\)

103

dvp_r_d2tp

\(\frac{\partial^2 v_\theta'}{\partial \theta \partial \phi}\)

104

dvp_theta_d2tp

\(\frac{\partial^2 v_\phi'}{\partial \theta \partial \phi}\)

105

dvp_phi_d2tp

\(\frac{\partial^2 \overline{v_r}}{\partial \theta \partial \phi}\)

106

dvm_r_d2tp

\(\frac{\partial^2 \overline{v_\theta}}{\partial \theta \partial \phi}\)

107

dvm_theta_d2tp

\(\frac{\partial^2 \overline{v_\phi}}{\partial \theta \partial \phi}\)

108

dvm_phi_d2tp