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Chapter 4: Problem 61

From calculus, one obtains the following formula (Leibnitz rule) for the time derivative of an integral that contains time in both the integrand and the limits of the integration: \\[ \frac{d}{d t} \int_{x_{1}(t)}^{x_{2}(t)} f(x, t) d x=\int_{x_{1}}^{x_{2}} \frac{\partial f}{\partial t} d x+f\left(x_{2}, t\right) \frac{d x_{2}}{d t}-f\left(x_{1}, t\right) \frac{d x_{1}}{d t} \\] Discuss how this formula is related to the time derivative of the total amount of a property in a system and to the Reynolds transport theorem.

Short Answer

Expert verified
The Leibniz Rule is similar to the Reynolds Transport Theorem by splitting into a local change term (change inside the system) and a convective transport term (change by flowing in and out of the system).

Step by step solution

01

Break Down the Leibniz Rule

The Leibniz Rule involves differentiating an integral depending on two variables, in this case \(x\) and \(t\). The overall formula is: \[\frac{d}{d t} \int_{x_{1}(t)}^{x_{2}(t)} f(x, t) d x=\int_{x_{1}}^{x_{2}} \frac{\partial f}{\partial t} d x+f\left(x_{2}, t\right) \frac{d x_{2}}{d t}-f\left(x_{1}, t\right) \frac{d x_{1}}{d t} \]Here, the left-hand side describes the time derivative of an integral. The right-hand side sums three portions - the integral of the function's partial time derivative, the product of the function at the upper limit, and the time derivative of the upper limit, and the product of the function at the lower limit and the time derivative of the lower limit.
02

Relate to Property Change in a System

The Leibniz Rule demonstrates how the total amount of a property in a system (represented as an integral of a function) changes with time. The change depends on: how the property itself changes over time inside the system (first term), how much property is flowing into the system (third term, negative), and how much property is flowing out of the system (second term).
03

Connect with Reynolds Transport Theorem

Reynolds Transport Theorem also describes changes to a system over time. It splits into two terms: a local change term (determined by changes inside the system), and a convective transport term (defined by changes at the system boundary). The Leibniz rule performs a similar split, thus showing a strong relationship with the Reynolds Transport Theorem.

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Most popular questions from this chapter

A car accelerates from rest to a final constant velocity \(V_{f}\) and a police officer records the following velocities at various locations \(x\) along the highway: $$\begin{array}{ll} x=0 & V=0 \mathrm{mph} \\ x=100 \mathrm{ft} & V=34.8 \mathrm{mph} \\ x=200 \mathrm{ft} & V=47.6 \mathrm{mph} \\ x=300 \mathrm{ft} & V=52.3 \mathrm{mph} \\ x=400 \mathrm{ft} & V=54.0 \mathrm{mph} \\ x=1000 \mathrm{ft} & V=55.0 \mathrm{mph}=V_{f} \end{array}$$ Find a mathematical Eulerian expression for the velocity \(V\) traveled by the car as a function of the final velocity \(V_{f}\) of the car and \(x\) if \(t=0\) at \(V=0 .[\) Hint: Try an exponential fit to the data.

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A fluid flows along the \(x\) axis with a velscity given by \(\mathbf{V}=(x / t) \hat{\mathbf{i}},\) where \(x\) is in feet and \(t\) in seconds. (a) Plot the speed for \(0 \leq x \leq 10\) ft and \(t=3\) s. (b) Plot the speed for \(x=7\) ft and \(2 \leq t \leq 4 \mathrm{s},\) (c) Determine the local and convective acceleration. (d) Show that the acceleration of any fluid particle in the flow is zero. (e) Explain physically how the velocity of a particle in this unsteady flow remains constant throughout its motion.
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