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.

B.W.B5

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.”

B.W.B6

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.

B.W.B7

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.

B.W.B8

• 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

B.W.B9

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

B.W.B10

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.

B.W.B12

• 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

B.W.B13

• 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

B.W.B14

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

B.W.B15

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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