Vector Calculus & Applications

Question 1

Conditions for streamfunctions

Answer 1

incompressible
unsteady
rotational
(assume 2D flow)

Question 2

No vorticity

Answer 2

irrotational flow

Question 3

Steady flow

Answer 3

velocity vector u is Independent of time

partial derivative of u w.r.t. is 0

Question 4

Equation for mass

Answer 4

M=densityxVolume

Question 5

Assumption for Bernouilli’s Eqn

Answer 5

F_ext is conservative

Question 6

Pressure force

Answer 6

• grad ρ dxdydz

- grad ρ V

Question 7

Lagrangian

Answer 7

e.g. pathline

Question 8

Eulerian

Answer 8

e.g. streamline

Question 9

solenoidal

Answer 9

grad • u = 0
divergence-free
(u is a vector)

Question 10

irrotational

Answer 10

grad x u = 0

u is a vector

Question 11

Streamline

Answer 11

an instantaneous curve tangent to the velocity vector at any point on the curve
r(s) = (x(s),y(s),z(s))

Question 12

Field line

Answer 12

a curve whose tangent is parallel to the vector field at each point along the curve

Question 13

Tangent vector

Answer 13

r’(t)

Question 14

line element ds

Answer 14

sqrt(dr • dr)
or
|dr/dt| dt

Question 15

divergence measures ..

Answer 15

rate of expansion/stretching of a vector field

Question 16

Curl u relates to…

Answer 16

Curl u is related to the ‘rotation or twisting’ of the vector field

Question 17

surface normal

Answer 17

unit normal vector pointing outwards from the surface

Question 18

dA & dS

Answer 18

area element & surface element

Question 19

Is a streamfunction vector or scalar?

Answer 19

scalar