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In a Couette flow, two large flat plates lie one atop
Chapter 5, Problem 21SE(choose chapter or problem)
In a Couette flow, two large flat plates lie one atop 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 \(V=\tau h / \mu\), 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 \(\mu\), h, and τ are measured independently and that the measurements are unbiased and normally distributed. The measured values are \(\mu\) = 1.6 Pa ·s, h = 15 mm, and τ = 25 Pa. The uncertainties (standard deviations) of these measurements are \(\sigma_{\mu}\) = 0.05, \(\sigma_{n}\) = 1.0, and \(\sigma_{\tau}\) = 1.0.
Use the method of propagation of error (Sec- tion 3.3) to estimate V and its uncertainty \(\sigma_{V}\). Assuming the estimate of V to be normally dis- tributed, find a 95% confidence interval for V. Perform a simulation to determine whether or not the confidence interval found in part (b) is valid.
Equation transcription:
Text transcription:
V=\tau h / \mu
\mu
\sigma_{\mu}
\sigma_{n}
\sigma_{\tau}
\sigma_{V}
Questions & Answers
QUESTION:
In a Couette flow, two large flat plates lie one atop 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 \(V=\tau h / \mu\), 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 \(\mu\), h, and τ are measured independently and that the measurements are unbiased and normally distributed. The measured values are \(\mu\) = 1.6 Pa ·s, h = 15 mm, and τ = 25 Pa. The uncertainties (standard deviations) of these measurements are \(\sigma_{\mu}\) = 0.05, \(\sigma_{n}\) = 1.0, and \(\sigma_{\tau}\) = 1.0.
Use the method of propagation of error (Sec- tion 3.3) to estimate V and its uncertainty \(\sigma_{V}\). Assuming the estimate of V to be normally dis- tributed, find a 95% confidence interval for V. Perform a simulation to determine whether or not the confidence interval found in part (b) is valid.
Equation transcription:
Text transcription:
V=\tau h / \mu
\mu
\sigma_{\mu}
\sigma_{n}
\sigma_{\tau}
\sigma_{V}
ANSWER:Solution
Step 1 of 3
The velocity V is given by
,
mm,
a) We have to find V and uncertainty in the estimate
Now
=25(15)/1.6
=234.375
For finding the uncertainty we have to find differentiation with respect to
=
=
= -146.48
Now differentiate with respect to h then
=
=25/1.6
=15.625
Now differentiate with respect to h then
=
=15/1.6
=9.375
Now
=
=
=
=19.64
The uncertainty is 19.64
The estimate