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chpt_4_suggested_qns

# chpt_4_suggested_qns - Chapter 4 Differential Relations for...

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Unformatted text preview: Chapter 4 Differential Relations for Fluid Flow P4.1 Solution: (a) The flow is unsteady because time t appears explicitly in the components. (b) The flow is three-dimensional because all three velocity components are nonzero. (c) Evaluate, by laborious differentiation, the acceleration vector at (x, y, z) ( 1, 1, 0). 2 2 4 du u u u u u v w 4x 4tx(4t) 2t y(0) 4xz(0) 4x 16t x dt t x y z 2 2 2 2 dv v v v v u v w 4ty 4tx(0) 2t y( 2t ) 4xz(0) 4ty 4t y dt t x y z d w w w w w u v w 4tx(4z) 2t y(0) 4xz(4x) 16txz 16x z dt t x y z 2 4 d or: (4x 16t x) ( 4ty 4t y) (16txz 16x z) dt V i 2 j k at (x, y, z) ( 1, 1, 0), we obtain 2 3 d 4(1 4t ) 4t(1 t ) (c) dt Ans. V i j k (d) At (–1, 1, 0) there are many unit vectors normal to d V /dt. One obvious one is k . Ans. P4.2 Solution: Here we have only the single ‘one-dimensional’ convective acceleration: 2 2 1 . o o V du u x u V A n s dt x L L o 2V x 1 L L (a) 2 2 2(10) 2 6 10 , 1 400(1 4 ), 6 /12 6 /12 o ft du x For L and V x with x in feet s d t At x 0, du/dt 400 ft/s 2 (12 g’s); at x L 0.5 ft, du/dt 1200 ft/s 2 (37 g’s). Ans. (b) P4.3 Solution: (a) Do each component of acceleration: 2 2 x y du u u u v (x y x)(2x 1) ( 2xy y)( 2y) a dt x y 2 2 dt x y dv v v u v (x y x)( 2y) ( 2xy y)( 2x 1) a 212 Solutions Manual Fluid Mechanics, Seventh Edition At (x, y) (1, 2), we obtain ax 18 i and ay 26 j Ans. (a) (b) At (x, y) (1, 2), V –2 i – 6 j . A unit vector along a 40 line would be n cos40 i sin40 j . Then the velocity component along a 40 line is 40 V ( 2 6 ) (cos40 sin 40 ) Ans 40 V n i j i j 5.39 units . (b) (c) The maximum acceleration is amax [18 2 26 2 ] 1/2 31.6 units at 55.3 Ans. (c, d) _______________________________________________________________________ P4.5 Solution: (a) For two-dimensional steady flow, the acceleration components are 2 o o o o 2 U U du u u x y u v U U ( ) x y 2 o o o o 2 dt x y L L L L U U dv v v x y u v U ( ) U dt x y L L L L Therefore the resultant 2 2 2 2 o o (U /L )(x y ) (U /L (purely radial) (a) Ans. a i j ) r (b) For the given resultant acceleration of 25 m/s...
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chpt_4_suggested_qns - Chapter 4 Differential Relations for...

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