(a) What is the definition of vorticity? What fluid properties does vorticity represent measure? (7 points)

(a) As illustrated in Fig. 1a, consider a steady, incompressible. laminar flow (with a constant viscosity) confined by two infinite parallel plates. One plate is moving at speed U, while the other plate is held fixed. Assuming that the flow is parallel (y = 0) and is purely two-dimensional (in x and y direction). The boundary conditions are (1) at y= 0, น = v=w = 0. (2) at y = Ly, น = U, y=w=0. Based on the above assumptions, develop the simplified continuity and momentum equations (please provide reasons for why certain terms can be neglected) (7 points)

(b) Following (a), solve for the velocity field and estimate the shear force per unit area acting on the fixed plate. (8 points)

(c) Let's flip the dimension to x-z plane and remove the moving plate so that the upper boundary is now free surface (Fig. 1b). Wind forcing is applied at the free surface. Assuming that the wind-driven flow is steady, laminar, two dimensional, and fully developed in x direction (ie. no gradients in x). The flow is subject to the tollowing boundary conditions: (1) at z = 0, u = v=w = 0. (2) at z = H, the viscous stress (τ) is only non-zero in x direction, with Express the momentum balance and solve for the velocity profile (u(z) ) (9 points) 

(d) Following (e), express and sketch the profile of stress τ(z). Interpret the force balance. (6 points)

(b) What is irrotational flow? (5 points)

(c) Consider the following steady, two-dimensional (in x-y plane), circular flows. The velocity components are expressed in cylindrical coordinate (in 2D r - θ plane; r is radial distance, e is the angle from the x-axis) (1) Flow in solid-body rotation: ur =0 u0, = ωr (ω is a constant) (2) Line vortex:ur, =0   uθ -=k/r (k is a constant) Sketch the velocity field and calculate the vorticity. (8 points)

(d) Are the circular flows described above irotational? Follow a fluid parcel in these flow fields, comment on how the parcel rotates. (5 points)[ Transformations of velocity components between the Cartesian and polar coordinates]

一、 偏鄉教育需要將個別化教學議題融入班級教學中，請舉一個個別化教學的 實例，並說明在班級教學中如何進行個別化教學？

二、 教師活化教學的實施，需要從學科教學知識與學科學習知識著手，請說明 學科教學知識的意義與學科學習知識的意義？並說明在教學活動中如何顧 及 12 種的理念？

三、 回顧過往認知學派學者 蓋聶（R. M. Gagne)曾提倡「學習階層論」（Hierarchy of learning)，強調教學應有合理的次序存在，並區分了八個由簡至繁的學習 方式，包括：訊號學習、刺激反應連鎖學習、語言聯結學習、區辨學習、 概念學習、原則學習、問題解決學習。另一方面，因應「新課綱」實施， 素養導向教學成為實務現場教師進行專業增能的熱門議題。請問您認為前 述的「學習階層論」所提及的八種學習方式，是否都符合素養導向教學的 精神？或是部分符合？為什麼？請申論您主張背後的理由。