Question

7. Consider flow of a viscous fluid in an annulus between two rotating cylinders as shown in the figure. The inner cylinder is rotating with an angular velocity of Na, and the outer cylinders rotating in the same direction with an angular velocity of . Let a and b be the radii of-the inner and outer cylinders. Let the flow be steady, laminar, and assume no axial or radial flow. (a) Using the appropriate form of the Navier-Stokes equations, find an equation for the velocity v. Clearly define the boundary conditions for this equation. Solve the equation to obtain an expression for v9. (b) Find an expression for the viscous shear stress along the cylinder wall at r = a. Assuming L as the length of the cylinder, calculate the torque acting on the inner cylinder. (c) Find an expression for the viscous shear stress along the cylinder wall at r = b. Assuming L as the length of the cylinder, calculate the torque acting on the outer cylinder.

(d) This configuration can be used as a viscometer to measure the viscosity (µ) of the fluid.In that case, the inner cylinder is held fixed N₁ = 0 and the outer cylinder is rotating.By measuring the torque magnitude on the outer cylinder, it is possible to obtain the viscosity. If the viscometer measures a torque of 0.01N.m, and the diameter of the inner cylinder is 100mm, the annular gaps is 1mm, the length is 120 mm, and the outer cylinder rotates at 40 rpm, calculate the viscosity of the fluid contained within the viscometer.

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