AE311_15_Tut_01 - AE311 Aerodynamics(2016-17 Odd Semester...

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AE311 Aerodynamics (2016-17, Odd Semester) Tutorial – 1 1. Prove the following relation ( i, j, k range over 1, 2, 3) [ Refer Levi-Civita tensor and Un- derstanding Viscoelasticity ] (a) δ ij δ jk = δ ik (b) δ ij δ ji = 3 (c) δ ij a i = a j (d) ijk ijm = 2 δ km (e) ijk jki = 6 (f) ijk a j a k = 0 (g) a × b = ijk a i b j 2. Using tensor analysis (index notation), verify the following vector identities (a) a · ( b × c ) = b · ( c × a ) = c · ( a × b ) (b) a × ( b × c ) = ( a · c ) b - ( a · b ) c (c) ( a × b ) · ( c × d ) = ( a · c )( b · d ) - ( a · d ) · ( b · c ) (d) u · ∇ u = ( 1 2 u · u ) - u × ( ∇ × u ) (e) ∇ · ( a × b ) = b · ( ∇ × a ) - a · ( ∇ × b ) (f) ∇ × ( φ ) = 0 (where φ is scalar) (g) ∇ · ( ∇ × a ) = 0 (h) ∇ × ( φ a ) = φ ( ∇ × a ) + φ × a ijk can also be defined as the scalar triple product of unit vectors, e i · ( e j × e k ). 3. Show that the scalar product S ij Ω ji vanishes identically if S ij is a symmetric tensor and Ω ij is skew symmetric. 4. Show that the dyadic product of two vectors, a b , is a second order tensor 5. Express the Navier Stokes equation for incompressible flows in cylindrical polar co-ordinates.
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