MATH 6750
Homework # 1
Due Date: 09/25/2013
1. Show that ∇ · ( T × x ) = − x × ( ∇ · T ) + : T . Here T is a rank 2 tensor, x is a rank 1 tensor and is the permutation symbol.
2. Consider the flow field u ( x , t ) = ( xy )
2 e x
+ ze
− αt e y
+ cos(2 xz ) e z
.
Calculate
(a) the fluid acceleration in an Eulerian frame;
(b) the fluid acceleration in a Lagrangian frame;
(c) the velocity gradient tensor ∇ u ;
(d) the curl of the velocity field ∇ × u .
3. Consider cylindrical coordinates: x = r cos θ, y = r sin θ, z = z .
(a) Find an expression for the rate-of-strain tensor E in cylindrical coordinates.
HINT: ∇ = e r
∂ r
+ e
θ
1 r
∂
θ
+ e z
∂ z
, ∂
θ e r
= e
θ and ∂
θ e
θ
= − e r
.
(b) Consider a 2-D incompressible Newtonian flow u = u r
( r, θ ) e r
. Find E .
(c) Find the traction (stress vector) in the directions e r and e
θ for the above Newtonian flow.
4. A velocity field is given by u = f ( r ) x , with r = | x | = p x 2 + y 2 + z 2 . What is the most general form of the scalar function f ( r ) such that the flow field is incompressible?
5. Convert the continuity equation written in Cartesian coordinates, ρ t
+ ∇ · ( ρ u ) = 0, into polar cylindrical coordinates, x = r cos( θ ) e x
+ r sin( θ ) e y
+ z e z
, where u = u r e r
+ u
θ e
θ
+ u z e z
.
6. In one dimension, the Eulerian velocity is given to be u ( x, t ) = 2 x/ (1 + t ).
(a) Find the Lagrangian coordinate x ( α, t ).
(b) Find the Lagrangian velocity as a function of α, t .
(c) Find the Jacobian J = ∂x/∂α as a function of α, t .
(d) If the density satisfies ρ ( α, 0) = α and mass is conserved, find ρ ( α, t ) using the Lagrangian form of mass conservation.
(e) From (a) and (d) evaluate ρ as a function of x, t , and verify that the Eulerian conservation of mass equation is satisfied by ρ ( x, t ) and u ( x, t ).