Question

The device shown in Figure OA3 is called a cone-plate viscometer. The angle of the cone is very small, so that sin 0 0, and the gap is filled with the test liquid. The torque M to rotate the cone at the angular velocity Q is measured. Assuming a linear fluid velocity profile along the thickness of the fluid film and non-slip boundary condition at the surfaces of the solid bodies, derive an expression for fluid dynamic viscosity u as a function of (M, R, Q,0) [30 ] R- Fluid Figure QA3

Answer 1

Given that,

The angle of cone is very small sin $\Theta$ $\approx$$\Theta$

Answers - A element strupat a distance of r and the element strip dr do ao-oldy with Consider a strip at a clistance r facom the centowe width dede as the shut height of the conical stoup, and dy as the height of the trip from the wall -> The slant height of the conical storep sin (90 Loja dr do= dr sin(90-0)

=) The height of the strup fouom the wall tan (90-6)= ah dy dr dyz dr tancao-o) coto dig=drdano :-) The area of the staip da Ard J . QA = wardro -) Assume the flow to be lamince, Velocity porofile as livece constant viscosity, and follows no-slip Condition. As the velocity profile is lineas, a velocity varies from zero ad alal (no-slip) to the maximum of the surface of conical disk. lah - angular velocity Ml-linear velocity. Here,

=) Change in velocity, de = -2dr =) Shear stress at the surface of the conical disk. T = le due Herre, - shock stoussacting at the swface, M-viscosity of the fluid, dy - height of the stoup forom the wall. ..thdr drtano = un - tano The 109que guequired to rotate the conice stoup. - dM - T (dAS Herve, DM-toegue required to rotate the conical .. . strip dra - M.2 Catr do solt tang - 2ne12 rear > defal too Sing byal torque requised for the conical disk M=Som

- atuh pk a sino , = 2.Turr3 R. sino 310 2012R² 3 sing to find find out the =) Rearrange the expression viscosity of the fluid. l= 3M sino - 202R3 There toor, the viscosity of the flued is. 3 Msing ZARR3