ME 3700 – Fluid Mechanics

Mathematics Review

Vector Review

θcosABBABABABA zzyyxx =++=•1. Dot Product

kAjAiAA zyxˆˆˆ ++=

r

kBjBiBB zyxˆˆˆ ++=

r

( ) ( ) ( )kBABAjBABAiBABABA xyyxzxxzyzzyˆˆˆ −+−+−=×

2. Cross Product

θsinABBA =×

x

y

Ar B

r

xBi

j

yB

θ

Right Hand Convention

Integral Calculus Review

( )dSAndAS∫∫ •=∀•∇

∀

rrˆ

1. Gauss Theorem – Divergence theorem

x

S

y z

n

∀

dS

dSnSd ˆ=r

( )∫∫∫∫∫ •=∀•∇∀ S

dSAndArr

ˆ

Flux of A across S in the n directionDivergence of A in

The volume V

S1

S2

S3

Integral Review

∀∂∂=∀

∂∂

∫∫∀∀

dFt

dFt

2. Leibnitz Rule

Differential Calculus Review

kz

jy

ix

ˆˆˆ∂∂+

∂∂+

∂∂=∇

1. Gradient Operator

kz

jy

ix

ˆˆˆ∂∂+

∂∂+

∂∂=∇ φφφφ

Gradient of a Scalar (i.e., density, temperature,etc)

zA

yA

xAA zyx

∂∂+

∂∂

+∂∂=•∇

r2. Divergence

kyA

xA

jxA

zAi

zA

yAA xyzxyz ˆˆˆ

∂∂−

∂∂

+

∂∂−

∂∂+

∂

∂−

∂∂=×∇

r

3. Curl

x

y z

φ∇

2

2

2

2

2

22

zyx ∂∂+

∂∂+

∂∂=∇•∇=∇ φφφφφ

4. Laplacian -

dzzVdy

yVdx

xVVd

∂∂+

∂∂+

∂∂=

rrrr

5. Total Differential

Differential Calculus Review

Error Propagation- Functions of more than one variable

Apply Taylor Series to functions of multiple variables, I.e., f(x,y,z)

( ) ( ) ( ) ...),,(),,( 111111 TOHzzzfyy

yfxx

xfzyxfzyxf ii

iii

iii

iiiiiii +−

∂∂+−

∂∂+−

∂∂+= ++++++

Neglecting 2nd order and higher terms, the error in f is:

( ) zzfy

yfx

xfzyxf ~~~~,~,~ ∆

∂∂+∆

∂∂+∆

∂∂=∆

Where are estimates of the error in x, y and z

( ) nn

n xxfx

xfx

xfx

xfxxxxf ~...~~~~,...,~,~,~

33

22

11

321 ∆∂∂++∆

∂∂+∆

∂∂+∆

∂∂=∆

In general, 1st order approximation of the error in f is:

x~∆ y~∆ z~∆

Error Propagation- Reynolds Number Example

νUD=Re

Where: V = average fluid velocity (m/s)D = pipe diameter (m)ν = kinematic viscosity (m2/s) water

Estimate the error in Re for the Given data:

6100.1~1.0~5.0~

−×==

=

νD

U

610005.0~001.0~01.0~

−×=∆=∆

=∆

νD

U

( ) νν

ν ~Re~Re~Re~,~,~Re ∆∂∂+∆

∂∂+∆

∂∂=∆ D

DU

UDU

( ) ( )( )( ) ( )6

2666 10005.0101

1.5.001.1015.01.

1011.~,~,~Re −

−−− ××

+×

+×

=∆ νDU

( ) νννν

ν ~~~~,~,~Re 2 ∆−+∆+∆=∆ UDDUUDDU

m/sm

m2/s

( ) 2505001000~,~,~Re ++=∆ νDU

%5.3000,50Re1750000,50Re

±=±=

( )( ) 000,50101

1.5.Re 6 =×

== −νUD