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

(d) Consider a fully developed pipe flow, describe the differences in time-mean velocity structure (e.g. velocity across the pipe) between laminar and turbulent flow regimes. (5 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) 

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