We report on an investigation of student thinking about steady-state pipe flow of an incompressible fluid. About 250 undergraduate engineering students were given a test consisting of two hydrodynamics questions, combining multiple-choice format with subsequent open-ended explanations. There is substantial evidence that students have difficulty applying and prioritizing the two basic principles of mass conservation (expressed in the continuity equation) and energy conservation (i.e., Bernoulli’s equation). When faced with questions that involve gravity, dissipative effects (“friction”), or a visible pressure drop, a considerable number of students did not invoke the continuity equation in situations where applying it is a necessary step for arriving at the correct answer. Instead, even after lecture instruction on this topic, many of the first-year students based their answers on ill-supported assumptions about local pressures. Some of them used formal arguments from a simplified Bernoulli equation (“lower pressure means higher velocity”), while others based their answer on intuitive arguments (“higher pressure leads to higher velocity”). We also found reasoning based on analogies to single-particle motion (“flow velocity decreases when flowing upwards or friction is present”). Contrary to other researchers, we did not see any evidence for the hypothesis that students think of water as a compressible fluid. Instead, students’ answers often indicate a lack of understanding of the conservation of mass or its implications for incompressible fluids or of the role that this principle plays in the context of fluid flow. In addition, our data indicate that some students have more general difficulties in describing and reasoning about technical situations, such as applying equations containing multiple variables, distinguishing spatial differences in a quantity from its changes with respect to time, or realizing the meaning of idealizations. We also present some evidence that different levels of activation of students during instruction influence the prevalence of these difficulties and discuss some implications for instruction.

中文翻译：

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流体动力学学生推理：伯努利原理与连续性方程

我们报告了一项关于学生思考不可压缩流体稳态管流的调查。大约 250 名本科工程专业学生接受了由两个流体动力学问题组成的测试，将多项选择题格式与随后的开放式解释相结合。有大量证据表明，学生难以应用质量守恒（用连续性方程表示）和能量守恒（即伯努利方程）这两个基本原理并对其进行优先排序。当面临涉及重力、耗散效应（“摩擦”）或可见压降的问题时，相当多的学生在应用连续性方程是获得正确答案的必要步骤的情况下没有援引连续性方程。相反，即使在关于这个主题的讲座指导之后，许多一年级学生的答案是基于对当地压力的缺乏支持的假设。他们中的一些人使用了来自简化伯努利方程的形式论据（“较低的压力意味着较高的速度”），而另一些人则基于直观的论据（“较高的压力导致较高的速度”）。我们还发现了基于单粒子运动类比的推理（“当向上流动或存在摩擦时，流速会降低”）。与其他研究人员相反，我们没有看到任何证据支持学生将水视为可压缩流体的假设。相反，学生的答案通常表明他们缺乏对质量守恒或其对不可压缩流体的影响或该原理在流体流动背景下所起的作用的理解。此外，我们的数据表明，一些学生在描述和推理技术情况时有更多的普遍困难，例如应用包含多个变量的方程、区分一个量的空间差异和它随时间的变化，或者理解理想化的含义。我们还提供了一些证据，表明学生在教学过程中不同程度的活跃度会影响这些困难的普遍性，并讨论对教学的一些影响。