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?  Fig: 1  Fig: 2