Question 4. [20 Points] Given the velocity components for a certain incompressible, steady-flow field as 1 Vx E(1+b)x Vy = - 8(1 b)y 1 1 vz = z where x, y, and z are the position components in a 3D Cartesian coordinate system, Osb S1, and the parameter : (dimension T-1) is known as the elongation rate, this type of flow is known as simple shearfree flow. Complete the following tasks: (a) [5 Points] Show that the given velocity field satisfies the continuity equation. (b) [10 Points] Applying Newton's law of viscosity for an incompressible fluid of viscosity u, T= -u[5v + (Ov)") Show that the stress tensor becomes T(1 + b) 0 0 T = Tyx Tyy Tyz = 0 (1-5) 0 [Tzx Tzy Tzz] 0 0 -23) (c) [5 Points) For steady simple shearfree flows we define two viscosity function Neil,b) and nezl, b) to describe the two normal stress differences: Tzz Txx = -Nei, b) Tyy - Txx = -Nez(, b) Show that for the special steady-state shear flow known as the uniaxial elongational flow, where b = 0 and > 0, we have nez = 0, and nen = 3u. Here, nei is known as the elongational viscosity, and the ratio of elongational viscosity to shear viscosity is known as the Trouton ratio. Txx Txy Txz