Learning Goal: $=\frac{g({}_s-{}_f)D^2}{18v},$ where ${}_s$ is the sphere density

and ${}_f$ is the fluid density.

Figure

$1$ of $1$

Part B $-$ Determine the viscosity using a rotational viscometer

You have a rotational viscometer for measuring fluids with densities similar to the density of water $({}_w).$ Its dimensions are

$r_i=4cm,t=0\mathrm{.}5mm,$ and $h=4cm.$ The angular velocity of the cylinder can be varied from $=0\mathrm{.}0105ra\frac{d}{s}$ to $=$

$2\mathrm{.}094ra\frac{d}{s}.$ What is the smallest viscosity that can be measured, if the torque sensor requires a torque of at least $M=$

$1{10}^{-5}N*m?$

Express your answer in $\frac{N*s}{m^2}$ to three significant figures.

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$=$ Part C $-$ Another kind of viscometer

Now consider Stokes flow with a steel sphere $({}_s=7800k\frac{g}{m^3})$ falling through honey $({}_f=1350k\frac{g}{m^3}).$ To use the

Stokes flow approximation, the ratio $\frac{{}_{f}vD}{}$ should be no greater than $0\mathrm{.}5.$ What is the minimum viscosity the honey must

have in order to measure it using this method if the diameter of the sphere is $D=2cm?$

Express your answer in $\frac{N*s}{m^2}$ to three significant figures.

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$=$

$\frac{N*s}{m^2}$

To understand how the viscosity of a fluid can be

measured and how errors in the process affect the

measurement.

The viscosity of a fluid can be measured in different

ways. For some fluids, a rotational viscometer $($

Figure $1)$ can be used. A stationary solid cylinder is

suspended centered in a larger outer cylinder,

creating a thin gap of the fluid to be measured

between the cylinders. The outer cylinder rotates at

a known angular velocity $$ that is low enough that

the flow in the gap is orderly and does not change

with time $($known as laminar flow$).$ Assuming a

linear velocity profile, and neglecting the gap at the

bottom of the cylinders, the viscosity of the fluid is

$=\frac{Mt}{2r_i^2{r}_{o}h},$ where $M$ is the torque on the

inner cylinder, $r_i$ and $r_o$ are the inner and outer

radii, $h$ is the height of the cylinders, and $t$ is the

gap thickness.

For transparent, very high viscosity liquids, another

method can be used. When the viscosity is high

enough, a small sphere dropped through the liquid

will fall very slowly. This kind of flow is called

Stokes Flow. The drag force on the sphere is

$F_d=3Dv,$ where $D$ is the radius of the sphere

and $v$ is the velocity of the sphere. Rearranging this

equation and balancing the drag force with the

weight of the sphere the viscosity is

$=\frac{g({}_s-{}_f)D^2}{18v},$ where ${}_s$ is the sphere density

and ${}_f$ is the fluid density.

Part A $-$ Rotational viscometer assumptions

There are several assumptions used to develop the relationship between torque and viscosity of a fluid in a rotational

viscometer. Two key assumptions are:

The contribution from the submerged end of the cylinder is small.

The velocity profile is linear.

Consider how these assumptions affect the calculated viscosity. Each quantity used in the calculations has some

measured value $(h_m,t_m,$ etc.$),$ which may be different from the exact value $(h_e,t_e,$ etc.$).$ For each condition given below,

determine if the calculated viscosity using the original assumptions and measurements is too large, too small, or

unchanged.

Drag the appropriate items to their respective bins.

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