Chapter 3: Problem 26
If we have a perfect gas of mass density \(\rho_{m} \mathrm{~kg} / \mathrm{m}^{3}\), and we assign a velocity \(\mathbf{U} \mathrm{m} / \mathrm{s}\) to each differential element, then the mass flow rate is \(\rho_{m} \mathbf{U} \mathrm{kg} /\left(\mathrm{m}^{2}-\mathrm{s}\right)\). Physical reasoning then leads to the continuity equation, \(\nabla \cdot\left(\rho_{m} \mathbf{U}\right)=-\partial \rho_{m} / \partial t .(a)\) Explain in words the physical interpretation of this equation. (b) Show that \(\oint_{s} \rho_{m} \mathbf{U} \cdot d \mathbf{S}=-d M / d t\), where \(M\) is the total mass of the gas within the constant closed surface \(S\), and explain the physical significance of the equation.
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