transport-iitb-tut
1
CL601: Mahesh S Tirumkudulu 1 CL203: Introduction to Transport Phenomena Tutorial 1: (July 28, 2015) These problems have been taken from Appendix A of the textbook (BSL). 1. A vector filed v(x, y, z ) is said to be irrotational if, ∇× v = 0. Which of the following fields are irrotational? (a) v x = by, v y =0,v z = 0; (b) v x = -by, v y = bx, v z =0 2. Evaluate ∇.v, ∇v, and ∇.vv for the two velocity fields in Q1. 3. If r is the position vector with components (x 1 ,x 2 ,x 3 ) and magnitude r, verify that (a) ∇( 1 r )= - r r 3 ; (b) ∇(a.r)= a. 4. If r is the instantaneous position vector for a particle, show that the velocity and acceleration of the particle in cylindrical coordinates are given by (use A.7-2, p829): v = dr dt =ˆ e r dr dt +ˆ e θ r dθ dt +ˆ e z dz dt a =ˆ e r d 2 r dt 2 - r dθ dt 2 ! +ˆ e θ r d 2 θ dt 2 +2 dθ dt dr dt +ˆ e z d 2 z dt 2 5. Use A.7-2, p829 to write in cylindrical coordinates, (a) ∇v +(∇v) T , (b) v.[∇v] θ , i.e., the θ component of the expression.
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CL601: Mahesh S Tirumkudulu 1
CL203: Introduction to Transport PhenomenaTutorial 1: (July 28, 2015)
These problems have been taken from Appendix A of the textbook (BSL).
1. A vector filed v(x, y, z) is said to be irrotational if, v = 0. Which of the following fieldsare irrotational? (a) vx = by, vy = 0, vz = 0; (b) vx = by, vy = bx, vz = 0
2. Evaluate .v, v, and .vv for the two velocity fields in Q1.
3. If r is the position vector with components (x1, x2, x3) and magnitude r, verify that (a)(1r ) =
rr3
; (b) (a.r) = a.
4. If r is the instantaneous position vector for a particle, show that the velocity and accelerationof the particle in cylindrical coordinates are given by (use A.7-2, p829):
v =dr
dt= er
dr
dt+ er
d
dt+ ez
dz
dt
a = er
(d2r
dt2 r
(d
dt
)2)+ e
[rd2
dt2+ 2
(d
dt
)(dr
dt
)]+ ez
d2z
dt2
5. Use A.7-2, p829 to write in cylindrical coordinates, (a) v + (v)T , (b) v.[v], i.e., the component of the expression.
