In Example 1.2 we replaced a cylindrical problem with a linear approximation. The velocity distribution for this flow, taking the cylindrical character into account V₁ = ∞ (₁ ²²₂ ²) · ( ²² - , ) where R is the radius of the outer cylinder, r is the local radius, k = rinner cylinder / R, and @ is the angular velocity of the inner cylinder. (a) Verify that this distribution shows a zero velocity at the radius of the outer, non-moving cylinder and shows V₂ = kR at the surface of the inner, rotating cylinder. (b) The shear rate in cylindrical coordinates, for a fluid whose velocity depends only on r (equivalent to dV/ dy in rectangular coordinates), is given by 6= shear rate cylindrical coordinates rical) = r 2 " ( 1₁ ²= R²) 6 = w d dr Show that for the above velocity distribution, the shear rate at the surface of the inner cylinder is given by Ve (-;-) (c) Show that the shear rate is 12.26/s, which is 1.15 times the value for the flat approximation in Example 1.2. The

1) Pavithran Ravindran explains to a group of inspiring CBE students that the Reynolds number is a dimensionless group defined for a fluid flowing in a pipe as Re = Dvp/µ where D = pipe diameter, v = average fluid velocity, p = fluid density, and u = fluid viscosity. When the value of the Reynolds number is less than about 2100, the flow is laminar -- the fluid flows in smooth streamlines. For Reynolds numbers above 2100, flow is turbulent -- a great deal of internal agitation takes place. %3D Methyl ethyl ketone (MEK) flows through a 2.067-in. ID pipe at 20°C, at which temperature its density is 0.805 g/cm³ and its viscosity is 0.43 centipoise (1 centipoise = 1.00 x 10-3 kg/m.s). The average fluid velocity is 0.048 ft/s. Determine whether the flow is laminar or turbulent. %3D

Consider the steady, isothermal, irrotational flow of an incompressible frictionless fluid subject only to a body force characterized by a potential O, where F,=- V@ (in units of force/volume). Since the fluid is frictionless, the only surface force is due to pressure, F =-pn. By completing a momentum balance on a material volume of fluid subject to the constraints listed above, derive Bernouilli's law, P + ½pv² + O = constant along a fluid streamline.

1. FA Newtonian fluid is flowing between two large parallel plates in the z direction. The top plate is moving in the positive z direction at a constant velocity Vo. The bottom plate is stationary. The velocity profile this flow is given by V(y) = V₁ (a) (b) (77) where 2H is the distance between the two plates. y is the vertical coordinate from the center plane, and is the velocity profile as a function of y. Draw a schematic of the flow system with the appropriate coordinate system. For plates with a total width W in the x-direction and length L in the z-direction, derive an expression for the volumetric flow rate Q through the plates.