Q2. 25 marks A viscous fluid of constant density p and dynamic viscosity u falls due to gravity between two plates a distance 2h apart, as shown in Fig.2. The flow is fully developed, with a single velocity component w=w(x). There are no applied pressure gradients, and gravity is the only body force. (a) Given the Navier-Stokes equation in the vertical direction Hawa 2 + aw aw 1 + a w ) + f + at ax ay 1 ap az paz plax ay az where (u, v, w) is the flow velocity and (fx,fy, fz) is the body force along the (x, y, z) directions respectively, explain how, after applying the information given to each term separately, it can be reduced to the equation + u aw + V aw + W 2w pax [4 marks] (b) Integrate the derived differential equation, and write the general solution. [6 marks] + g = 0. (c) State the boundary conditions of the derived differential equation, and explain their physical meaning. [2 marks] (d) Apply the boundary conditions to find one of the integration constants. [6 marks] (e) Apply the boundary conditions to find the other integration constant. [6 marks] (f) Write the particular solution that corresponds to the boundary conditions. h Z, W h X [1 mark] Figure 2: Viscous liquid falls due to gravity between two plates a distance 2h apart.