A Stormer viscometer consists essentially of two concentric cylinders, the inner
ID: 724951 • Letter: A
Question
A Stormer viscometer consists essentially of two concentric cylinders, the inner ( dius kappa R) of which rotates while the outer (radius R) is held stationary. Viscosity is d t rmined by measuring the rate of rotation Ohm. of the inner cylinder under the application of a known torque .Develop an expression for the velocity distribution in this kind of apparatus, as a function of applied torque, for laminar flow of a Newtonian fluid. Use the equations of function of the change. Label each cancelled term with the corresponding assumption.Explanation / Answer
The operation of the viscometer relies on a linear velocity prole in the gap between the rotor and
the xed cylinder. If the velocity prole is linear the viscous shear stress on the surface of the rotor
can be written
= µk ----------------------(1.1)
where is the viscous shear stress, µ is the uid viscosity, k is a constant that depends only on the
geometry of the viscometer, and is the angular velocity of the rotor. Given the shear stress from
Equation (1.1),
the torque exerted by the rotor on the uid is
Tf = ( A)rr --------------------------------------(1.2)
where A is the wetted surface area of the rotor, and rr is the radius of the rotor. The area, A,
accounts for the inner and outer surfaces of the rotor. Since the uid is in contact with both surfaces
rr is an eective radius.
Neglecting any friction in the pulleys and bearings, the power dissipated by viscous stresses in
the uid, Pf , is equal to the power input of the falling weight, Pw.
Pf = Pw = Tf = WV ----------------------------------(1.3)
where W is the magnitude of the weight, and V is the velocity of the falling weight.
Combining
Equations (1.1) through (1.3) gives
µ k ^2 A rr = WV ----------------------(1.4)
The velocity of the weight falling a distance L in time t is
V =L/t=2 rs ns/t---------------------------------(1.5)
where ns is the number of revolutions of the spindle, and rs is the radius of the spindle.
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