Consider an incompressible fluid above an infinite plate located in the ₁-3 plane (at

x2 = 0), in the absence of body forces. The plate oscillates sinusoidally along with a

velocity amplitude to and angular frequency w. The velocity of the liquid is given by

and is the kinematic viscosity (related to the dynamic viscosity and density p according

to v = µ/p).

(a) Verify that the given fluid velocity satisfies the axiom of mass balance.

(b) Evaluate grad v and its symmetric part D.

(c) Evaluate curl v.

(d) Evaluate the acceleration a of the fluid.

(e) For an incompressible Newtonian fluid the stress tensor is given by T = -pI+2μD,

where I is the identity tensor. For the given velocity and pressure, evaluate the stress

T.

(f) Verify that your solution for T satisfies the axiom of linear momentum balance.

(g) Evaluate the traction vector on the oscillating plate.

(h) What are the eigenvalues and eigenvectors of D? Hint: Simplify the notation by

using D₁, for the non-zero components of D instead of the full expressions evaluated

in part (b).

(i) What is the relation between the eigenvalues and eigenvectors of D and T?