rning: Academic dishonesty includes such things as cheating, inventing false information or citations, plagiarism, and helping someone commit an act of academic dishonesty. Students who violate university standards of academic integrity are subject to disciplinary sanctions, including failure in the course and suspension from the university. Since dishonesty in any form harms the individual, other students and the university, policies on academic integrity are strictly enforced. You should familiarize yourself with the academic integrity guidelines found in the current student handbook at http://www.fullerton.edu/handbook/. I read, understood the above statements, and completed this exam without anyone’s help. Name: Student ID : Signature: 1. Determine whether the following equation is dimensionally correct? (10 points) , where ρ u=  wvu ∇= ⎣⎢⎢⎢⎢⎢⎡ 𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕 𝜕𝜕𝑦𝑦 𝜕𝜕 𝜕𝜕𝑧𝑧⎦⎥⎥⎥⎥⎥⎤ t p 𝑇𝑇𝐷𝐷 f Density 3D Velocity Vector gradient time pressure Stress tensor (pressure) force 2. Dimensional Analysis: Suppose V , the speed of a liquid flowing through a horizontal pipe, depends on the pressure drop ∆P , the length of the pipe l , the diameter D , density of the liquid ρ , and the viscosity of the liquid µ . Show that V can be modeled using the Buckingham Π theorem as ( , ). 2 2 µρ ρ µ P D lD D V ∆ = ⋅Φ (10 points) 3. Similitude: Suppose the torque τ (N m) required to rotate a disk of diameter D (cm) submersed in a fluid with density ρ (kg/m3 ), and the bulk viscosity ν (kg/m/s) at the rotation rate ω (Radians/s) can be modeled, using the Buckingham ∏-theorem as, ( ) 2 5 2 ν ω τ ρω D = D ⋅Φ [Do not show this!!]. If a scaled model, 50 times smaller, experiences a 5 N-m torque in water with the rotation rate 10 sec 3π rad ω = and the real-sized model is tested in oil. Assuming the two models are similar and water kinematic viscosity = 1.2 oil kinematic viscosity and water density = 3 oil density, a) What is the rotation rate for the real-size model? (5 points) b) At that rotation rate, what is the real-sized torque? (5 points) 4) Modeling with data: Given the data set populations versus the mean velocities over a 50-foot course for 15 locations on Table 4.5 of page138, find two non-polynomial models that can describe the relation between P and V with 𝑅𝑅2 ≥ 92%. For each model, provided the mathematical equation with the parameters specified along with the graph. You must include the R-square values. (10 pts for each model) 5) Prescribing Drug Dosage: Doctors may advise some patients to take a low dose of aspirin daily to prevent blood clots and reduce the risk of stroke and heart attack. Suppose the concentration of a medication administered in a patient follows the exponential decay model C(t) = C0e −kt , where C is the concentration , C0 is the initial dosage, k is the absorbing rate, and t is time. a) Derive the concentration residual in the body after administering n consecutive dosages. (10 points) b) Suppose aspirin is absorbed in the body at the rate k=0.075/hr and each dosage produces 2.5 mg/dl of aspirin concentration in the body. With such prescription, what is the highest aspirin concentration in the body? (2pts) c) How many days does it take to get the aspirin concentration to reach 90% of the highest level? (3pts) 6) Given the differential equation 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 = (3 − 𝑢𝑢)(1 − 𝑢𝑢)2. a) Find the equilibria (3 points) b) Determine the stability at the equilibria (3 points), c) Draw the solution trajectories in phase space (4 points) d) Start with u(0)=0.99 and use the Euler method with ∆𝑡𝑡 = 0.1 to get and graph u(t) for the first 5 seconds. (5 points) 7) For each first order system of differential equations, find its equilibrium point(s), draw the phase space near the equilibrium points, and describe the corresponding stability. a) x y dt dy y x dt dx (3 2 ) (4 ) = − = − (10 points) b) x y dt dy y x dt dx ( 3 2 ) (4 ) = − + = − (10 points)