Math Review in Fluid Mechanics - Association of … · Math Review in Fluid Mechanics Relabeling: “Integration of Effective Mathematics Teaching review and ... • Bridging the

B.W.B1

Brett Batson, Ph.D.

Mechanical & Aerospace Engineering

Trine University, Angola, Indiana

Math Review in

Fluid Mechanics

2017 Ohio-PKAL Annual

Conference

B.W.B2

Brett Batson, Ph.D.

Mechanical & Aerospace Engineering

Trine University, Angola, Indiana

Math Review in Fluid Mechanics

? !

B.W.B3

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

“Just

math”Applications

♫♪

Math

review

B.W.B4

“The minute we need a thing, we begin paying

for it whether we buy it or not.Ӡ

Considering the lack of knowledge in, and skill with, algebra,

calculus, and differential equations:

The Problem

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

Press, Columbia, MO.

• Students begin paying by an impaired ability to grasp

concepts in fluid mechanics (and other subjects).

• They continue paying through lost credit on assignments.

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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.”

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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.

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Math Review – The Need

0

20

40

60

80

1 2 3 4

Exams

Before the Math Review

Exam I Exam II Exam III Final

Exam II is

math intensive~48 Students

~17 pts.

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• Properties

• Pressure fields

• Bernoulli equation

• Acceleration fields

• Kinematics

• Integral analysis

▪ Conservation of mass

▪ Newton’s second law

▪ 1st law of thermodynamics

• Differential analysis

▪ Continuity

▪ Navier-Stokes equations

• Potential flow

▪ Stream function

▪ Velocity potential

• Pipe flow

▪ Laminar

▪ Turbulent

• External flow

▪ Lift, drag

Fluid MechanicsCover a lot of material

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Class InversionIn 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

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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!

Inverting Fluid Mechanics

▪ “Math” practice problems included

▪ “Math” problems fair game on quizzes

and exams

B.W.B11

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.

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• Properties

• Pressure fields

• Bernoulli equation

• Reynolds Transport

• Acceleration fields

• Kinematics

Math Review – Topics

Taylor series

Gradient ( ), logarithms,

1st order ODEs

Partial derivatives, total derivatives

Evaluating determinants, curl

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• Differential analysis

▪ Conservation of mass

o Stream function

▪ Newton’s 2nd law

o Exact solutions

Math Review – Topics

Divergence

Total differential

Exact differential equations

2nd order differential equations

Method of undetermined coefficients

Surface integrals

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Math Review – The Results

0

20

40

60

80

100

1 2 3 4

Exams

Before math review After math review

Exam I Exam II Exam III Final

~144

Students~48

Students

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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.

B.W.B16

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!

B.W.B17

Math Review – Questions?

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