Math Review in Fluid Mechanics 2017 Ohio-PKAL Annual Conference

Brett Batson, Ph.D. Mechanical & Aerospace Engineering Trine University, Angola, Indiana

1 B.W.B Math Review in Fluid Mechanics

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Brett Batson, Ph.D. Mechanical & Aerospace Engineering Trine University, Angola, Indiana

2 B.W.B Math Review in Fluid Mechanics

Relabeling: “Integration of Effective Mathematics Teaching review and reinforcement across the STEM Curriculum” • Supporting, not replacing, the Math Department • Bridging the gap between “just math” and applications ♪ ♫

Math “Just Applications math” review

3 B.W.B The Problem “The minute we need a thing, we begin paying for it whether we buy it or not.”† Considering the lack of knowledge in, and skill with, algebra, calculus, and differential equations: • Students begin paying by an impaired ability to grasp concepts in fluid mechanics (and other subjects). • They continue paying through lost credit on assignments.

†Hines, S. W. ed. 2007. Laura Ingalls Wilder Farm Journalist. University of Missouri Press, Columbia, MO.

4 B.W.B The Problem Defined • Fluid Mechanics is probably the most math-intensive course in the core curriculum. • Students prove they are poorly prepared, mathematically. • Fall 2010 o Curriculum change o Single fluids course • Fluid Mechanics is very “full.”

5 B.W.B The Problem Defined • A prerequisite for this Fluid Dynamics class is Differential Equations o In order to take Differential Equations, students must have earned at least a C in Calculus I and Calculus II, and passed Calculus III. o These students should be well prepared to learn Fluid Dynamics.

6 B.W.B Math Review – The Need Exam II is ~48 Students Exams math intensive Before the Math Review 80

~17 pts. 60

40

20

0 Exam1 I Exam2 II Exam3 III Final4

7 B.W.B Fluid Mechanics Cover a lot of material • Properties • Potential flow • Pressure fields ▪ Stream function • Bernoulli equation ▪ Velocity potential • Acceleration fields • Pipe flow • Kinematics ▪ Laminar • Integral analysis ▪ Turbulent ▪ Conservation of mass • External flow ▪ Newton’s second law ▪ Lift, drag ▪ 1st law of thermodynamics • Differential analysis ▪ Continuity ▪ Navier-Stokes equations

8 B.W.B Class Inversion In the conventionally-taught (lecture) class, there was inadequate time to review math.

Outside Class View lecture video

In Class Take quizzes Work Practice Problems

Outside Class Finish Practice Problems

9 B.W.B Inverting Fluid Mechanics Advantages • Frequent quizzes ▪ Helps keep students on track • Shorter lectures ▪ ~30 minute average per lecture for a 50 minute class ▪ Now I have time to review mathematics! ▪ “Math” practice problems included ▪ “Math” problems fair game on quizzes and exams

10 B.W.B Math Review – The Timing • A math review may be done all at once and at the start of the mathematical-related course. • By the time these skills are needed for the course, students usually have forgotten them, again. • Importantly, the plurality of review topics should be covered immediately before they are needed in the mathematical-related course.

11 B.W.B Math Review – Topics • Properties Taylor series Gradient ( ), logarithms, • Pressure fields 1st order ODEs • Bernoulli equation

• Reynolds Transport

• Acceleration fields Partial derivatives, total derivatives

• Kinematics Evaluating determinants, curl

12 B.W.B Math Review – Topics • Differential analysis

▪ Conservation of mass Divergence

o Stream function Total differential Exact differential equations ▪ Newton’s 2nd law 2nd order differential equations o Exact solutions Method of undetermined coefficients Surface integrals

13 B.W.B Math Review – The Results Exams 100 ~144 ~48 80 Students Students 60

40

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0 Exam1 I Exam2 II Exam3 III Final4 Before math review After math review

14 B.W.B Math Review – Student Comments

Anonymous Course Evaluations: • Math review was also very helpful getting the concepts back into our minds and thinking about math again. • Dr. Batson has done a wonderful job with the math review and I found it to be very helpful.

Personal conversations (unprompted) revealed the same feeling.

15 B.W.B Math Review – Important Points • If your course is too full to review math, consider flipping. • Integrate math review when it will be needed. • Put teeth into the review. o Quizzes • Don’t be afraid to review math not necessarily needed for your class: o “…very helpful getting the concepts back into our minds and thinking about math again.” • The results could be astonishing!

16 B.W.B Math Review – Questions?

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17 B.W.B