Refer to Exercise 10 in Section 3.2. Assume that τ = 35.2 ± 0.1 Pa, h = 12.0 ± 0.3 mm, and μ = 1.49 Pa · s with negligible uncertainty. Estimate V, and find the relative uncertainty in the estimate.
Exercise 10: In a Couette flow, two large flat plates lie one on top of another, separated by a thin layer of fluid. If a shear stress is applied to the top plate, the viscosity of the fluid produces motion in the bottom plate as well. The velocity V in the top plate relative to the bottom plate is given by V = τh/μ, where τ is the shear stress applied to the top plate, h is the thickness of the fluid layer, and μ is the viscosity of the fluid. Assume that μ = 1.49 Pa • s and h = 10 mm, both with negligible uncertainty.
a. Suppose that τ = 30.0 ± 0.1 Pa. Estimate V, and find the uncertainty in the estimate.
b. If it is desired to estimate V with an uncertainty of 0.2 mm/s, what must be the uncertainty in τ?
Solution :
Step 1 of 1:
The velocity in the top plate relative to the bottom plate V is
V=
Where is the shear stress applied to the top plate.
Here ,
.
h is the thickness of the fluid layer.
Here ,
and
is the velocity of the fluid.
Here .
Our goal is :
We need to estimate V and the relative uncertainty in the estimate.
Now we need to estimate V and the relative uncertainty in the estimate.
First, we have to estimate the V value.
We know that and
.
The estimate of V is
.
Therefore the estimate of V is 283.4899 mm/s.
Then the relative uncertainty in the logarithm of quantity is
Now computing the partial derivatives of .
Here we are differentiating with respect .
We know that value.
Therefore .
Then,
Here we are differentiating with respect .
We know that h value.
Therefore .
Now we have to find the relative uncertainty in the estimate.
Then,
We substitute all the values in the above equation.
Therefore the relative uncertainty in the estimate is 0.025.
Hence .
