In a Couette flow, two large flat plates lie one atop

Chapter 5, Problem 21SE

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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}

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


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