Discussion
The velocity in a one-dimensional unsteady flow is given by \((x^{2} - t)\), where x is the position and t is the time. The total acceleration at any x and t is
Answer: D
The total acceleration is \(-1 - 2xt + 2x^{3}\).
The total (material) acceleration in one dimension is the local plus the convective part:
\[ \begin{aligned} a &= \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} \\ &= -1 + (x^{2} - t)(2x) \\ &= -1 + 2x^{3} - 2xt \end{aligned} \]
The other options get the convective term wrong: they drop the factor 2 from \(\partial u/\partial x = 2x\), or the sign of the xt term, or both. The local term -1 is right in all four, so the convective term is what decides the answer.
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