H Circular disk Rigid clamp R Liquid r The lower disk is rotated with a steady angular velocity w Circular disk e direction iAxis of rotation
The following figure shows the diametral cross-section of a viscometer, which
consists of two opposed circular horizontal disks, each of radius R, spaced by a vertical
distance H; the intervening gap is filled by a liquid of constant viscosity μ and constant
density. The upper disk is stationary, and the lower disk is rotated at a steady angular velocity
ω in the θ direction.
(a) There is only one nonzero velocity component, vθ, so the liquid everywhere
moves in circles. Simplify the general continuity equation in cylindrical coordinates and
hence deduce those coordinates (r, θ, z) on which vθ may depend.
(b) Now consider the θ-momentum equation and simplify it by eliminating all zero
terms. Explain briefly why you would expect ∂P/∂θ to be zero and why you cannot
neglect the term ∂2vθ /∂z2
(c) Suppose that the velocity in the θ direction is of the form vθ = ωrf(z), where the
function f(z) is yet to be determined. Substitute it into the simplified θ-momentum
balance and determine f(z), using the boundary conditions that vθ is zero on the upper disk
and vθ = ωr on the lower disk.
(d) Why would you designate the shear stress exerted by the liquid on the lower disk
as τzθ? Evaluate this stress as a function of radius.
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