NFYB21003U-Fluid-Mechanics/Lecture1BSlide7.html at main · team-ocean/NFYB21003U-Fluid-Mechanics · GitHub

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    <title>From Leibniz's Theorem to Reynolds Transport Theorem</title>
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        <div class="title">From Leibniz's Theorem to the Reynolds Transport Theorem</div>

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            <p>For \( \Delta t \to 0 \), approximate the integral: \( \int_{V^*(t+\Delta t)} F(\mathbf{x},t+\Delta t) dV \approx \int_{V^*(t)} F(\mathbf{x},t) dV + \int_{V^*(t)} \Delta t \frac{\partial F(\mathbf{x},t)}{\partial t} dV + \int_{\Delta V} F(\mathbf{x},t) dV + \int_{\Delta V} \Delta t \frac{\partial F(\mathbf{x},t)}{\partial t} dV\)</p>
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            <p>This simplifies to:
                \[
                \frac{d}{dt} \int_{V^*(t)} F(\mathbf{x},t) \, dV =
                \lim_{\Delta t \to 0} \frac{1}{\Delta t}
                \left\{ \int_{V^*(t)} \Delta t \frac{\partial F(\mathbf{x},t)}{\partial t} \, dV
                + \int_{\Delta V} F(\mathbf{x},t) \, dV \right\}
                \]</p>
                <p>   and this limit may be taken once the relationship between  \( \Delta V\) and  \( \Delta t\) is known.</p>
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        <p>Substituting into the time derivative equation: \( \frac{d}{dt} \int_{V^*(t)} F(\mathbf{x},t) dV = \lim\limits_{\Delta t \to 0} \frac{1}{\Delta t} \left( \int_{V^*(t+\Delta t)} F(\mathbf{x},t+\Delta t) dV - \int_{V^*(t)} F(\mathbf{x},t) dV \right)\)</p>
        <p>Rearranging:</p>
        <p style="font-size: 13px; line-height: 1.1;"> \( \frac{d}{dt} \int_{V^*(t)} F(\mathbf{x},t) dV = \lim\limits_{\Delta t \to 0} \frac{1}{\Delta t} \bigg( \cancel{\int_{V^*(t)} F(\mathbf{x},t) dV} + \int_{V^*(t)} \Delta t \frac{\partial F(\mathbf{x},t)}{\partial t} dV + \int_{\Delta V} F(\mathbf{x},t) dV + \int_{\Delta V} \Delta t \frac{\partial F(\mathbf{x},t)}{\partial t} dV - \cancel{\int_{V^*(t)} F(\mathbf{x},t) dV} \bigg)\)</p>
        <p>Since \( \Delta t \) is a constant factor: \( \frac{d}{dt} \int_{V^*(t)} F(\mathbf{x},t) dV = \lim\limits_{\Delta t \to 0} \left( \int_{V^*(t)} \frac{\partial F(\mathbf{x},t)}{\partial t} dV + \frac{1}{\Delta t} \int_{\Delta V} F(\mathbf{x},t) dV + \int_{\Delta V} \frac{\partial F(\mathbf{x},t)}{\partial t} dV \right)\)</p>
        <p>Since \( \int_{\Delta V} \frac{\partial F(\mathbf{x},t)}{\partial t} dV \) is of order \( \mathcal{O}(\Delta t) \), it vanishes as \( \Delta t \to 0 \), leading to the final equation: \( \frac{d}{dt} \int_{V^*(t)} F(\mathbf{x},t) dV = \int_{V^*(t)} \frac{\partial F(\mathbf{x},t)}{\partial t} dV + \lim\limits_{\Delta t \to 0} \frac{1}{\Delta t} \int_{\Delta V} F(\mathbf{x},t) dV\)</p>
        <p>This simplifies to: \( \frac{d}{dt} \int_{V^*(t)} F(\mathbf{x},t) dV = \int_{V^*(t)} \frac{\partial F(\mathbf{x},t)}{\partial t} dV + \int_{\Delta V} F(\mathbf{x},t) dV\)</p>
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