Viscous Flow

TUTORIAL SHEET -2007-08 ME 332 Fluid Mechanics –II Viscous Flow Exact Solution of Navier Stokes equation 1. Flow between parallel plates is driven by movement of the top surface as well as a prescribed pressure gradient. Develop an expression and sketch the fully developed velocity profile. Determine the magnitude of the adverse pressure gradient that reduces the shear on the stationary wall to zero. 2. Develop an expression for friction factor for flow of a power law fluid through a circular tube. 3. Develop an expression for friction factor for flow of a power law fluid in a circular annulus as a function of Reynolds (modified) number and radius ratio. 4. An infinite plate is pulled in its own plane with a velocity U for time T and is suddenly stopped. Find the time required by the bulk of the fluid to attain subsequently a velocity of 0.01U. 5. When rain flows down an inclined roof top, it forms a layer whose velocity u and volume flow rate Q/W as illustrated in figure 1. a. Find the expression for u and Q/W. b. Consider now the case of wind flow that exerts a shear stress τ on the upper surface (y = h) of the liquid layer. The wind tends to push the rain layer upwards while gravity pulls the layer downwards. Derive an expression for the thickness h of the liquid layer for which there is no net volume Q/W along the roof, express in terms of the parameters τ, ρ, g and φ. y



g sin φ

Wind

h

air Water

φ

φ 

x Figure 1

6. A layer of oil thickness a and viscosity µo floats on top of water of thickness b and viscosity µw. Both layers are contained between two large flat plates, the lower of which is stationary and the upper of which moves at a speed U in the x-direction. Derive expression (a) the speed Vi of the upper water-oil interface and (b) the volumetric flow rates of oil and water, Qo/W and Qw/W, per unit distance normal to the direction of flow. Express your answers in terms of a, b, U, µo and µw.