Exam paper Jan 2023

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1. Consider the transient parallel flow generated by an infinite flat plate suddenly decelerated at time 𝑡 = 0+. The flat plate coincides with the x-axis. The problem is governed by: 𝑑𝑢 𝑑𝑥 + 𝑑𝑣 𝑑𝑦 = 0 𝜌 (𝑑𝑢 𝑑𝑡 + 𝑢 𝑑𝑢 𝑑𝑥 + 𝑣 𝑑𝑢 𝑑𝑦) = − 1 𝜌 𝑑𝑝 𝑑𝑥 + 𝜇(𝑑2𝑢 𝑑𝑥2 + 𝑑2𝑢 𝑑𝑦2) 𝜌 (𝑑𝑢 𝑑𝑡 + 𝑢 𝑑𝑣 𝑑𝑥 + 𝑣 𝑑𝑣 𝑑𝑦) = − 1 𝜌 𝑑𝑝 𝑑𝑦+ 𝜇(𝑑2𝑣 𝑑𝑥2 + 𝑑2𝑣 𝑑𝑦2) With boundary conditions 𝑢(𝑦 = 0, 𝑡) = 𝑈 √𝑡 𝑢(𝑦 → ∞, 𝑡) = 0 where U is a constant with dimensions [ 𝐿 √(𝑡)] and initial conditions 𝑢(𝑦, 𝑡 = 0) = 0 Assume viscosity 𝜇 density 𝜌 and pressure p are constant. The layers of fluid are set in motion by viscous friction thus laminar flow. a) Using these assumptions, simplify the above problem and show this equation is governed by diffusion equation with appropriate boundary and initial conditions. b) Since there is no geometric length scale, a similarity solution can be sought in the form 𝜂 = 𝑦𝑞 √(4𝜈𝑡) 𝑓(𝜂) = 𝑢(𝑦,𝑡)∗√𝑡 𝑈0 Derive the necessary variables and show that the similarity equation has the form 𝑓′′ + 2(𝜂𝑓′ + 𝑓) = 0 c) Integrate the similarity equation. To determine the constants of integration, assume 𝑓′(𝜂 = 0) = 0 and impose similarity boundary conditions. d) Sketch u and 𝜏 at different times and explain the time evolution of the wall shear stress 𝜏 = 𝜇 𝑑𝑢 𝑑𝑦 in terms of plate velocity 2. Model the motion and effects of a tornado near a wall. Model with a source, consider the flow incompressible, inviscid induced by a source m and vortex of circulation Γ located at 𝑧 = 𝑎 in the presence of a wall represented by the imaginary axis. a) Construct the complex potential for this problem. b) Verify that the imaginary axis is a streamline (no penetration condition) c) Compute the velocity distribution along the imaginary axis induced by a tornado and sketch the velocity m positive (a source) and…