Consider a 2D steady incompressible laminar channel flow bet

Consider a 2-D, steady, incompressible, laminar channel flow between two parallel plates with gap h = 1 cm. Assume that the flow is fully developed over the entire channel. In order to measure the pressures in the flow, two manometers with L = 10 m are installed at the ends of the upper plate. The fluid is SAE 10W oil (SG = 0.92, µ = 0.1 N•s/m2 ).

(a) When the difference between the manometer heights (h1 – h2) is 20 cm, what is the pressure gradient !\" dp/dx in the channel flow?

(b) What is the volume flow rate Q through the channel? The width of the channel is 10 cm.

(c) Is the channel flow laminar or turbulent?

(d) What is the frictional force acting on the bottom plate?

Solution

a) dp / dx = rho * g * (h1-h2) / L

                 = 0.92 * 1000 * 9.81 * 0.2 / 10

                 = 180.504 Pa/m

b) volumetric flow rate for laminar flow = b*h^3 * dp/dx / 12*mu

                                                         = 0.1 * 0.01^3 * 180.504 / (12 * 0.1) = 0.000015042 m^3/s

c) Re = rho * Q / b * mu    = 920 * 0.000015042 / (0.1 * 0.1)   < 2300 flow is laminar

d) force   = shear stress * L * b

               = mu * du/dy * L * b

               = dp/dx * h * l * b / 2   = 0.90252 N