consider an incompressible fluid above an infinite plate located in th

Question

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?