ECH 3264 (CWP)

Dimensionless equations * Incompressible Newtonian fluids:

Φ+∇=

+∇+−∇=

⋅∇

equation)energy convective(DtDT

)equation Stokes-Navier(Dt

uDequation) continuity(0=u

2

2

µρ

ρµρ

TkC

gup

p

equations) 5 and unknowns (5 T ,u p, :Unknowns * Dimensionless quantities:

u * =

u U ∞

i.e., ui* =

ui

U∞

(x*, y*,z*) =

xL

,yL

,zL

or xi

* =xi

L

t* =t

L / U ∞

, p* =p

ρU ∞2 , θ =

T − T0

T∞ − T0

* Dimensionless equations: (* dropped for simplicity)

∇⋅ u = 0Du Dt

= −∇p +1

Re∇2u +

1Fr

k DθDt

=1

Pe∇2θ +

BrPe

Φ

=∆

≡

=⋅=⋅=

==≡

=≡

==≡

∞

∞

∞∞

∞

∞∞

number)(Brinkman conductionby transfer

ndissipatio by viscous productionenergy UBr

number Prandtl is Pr whereRePrPe

number)(Peclet conductionby nsfer energy traconvectionby nsfer energy traPe

number) (Froude forcegravity force inertialFr

number) (Reynolds force viscousforce inertialRe

2

2

energyTk

LU

LUk

CLUgL

U

LULU

p

µαν

ναν

α

ρ

νµρ

U∞, T∞

ρ, µ, Cp, k T0

L

ECH 3264 (CWP)

* Equations of Energy

)conduction oflaw s(Fourier'

n)dissipatio viscousnegligible withequationenergy e(Convectiv2

Tk

TkDtDTC

p

∇−=

∇=

q

ρ

In Cartesian coordinates (x, y, z):

∂∂

+∂∂

+∂∂

=

∂∂

+∂∂

+∂∂

+∂∂

2

2

2

2

2

2

zT

yT

xTk

zTu

yTu

xTu

tTC zyxpρ

T∇−= kq : zTkq

yTkq

xTkq zyx ∂

∂−=

∂∂

−=∂∂

−=

In cylindrical coordinates (r,θ, z):

∂∂

+∂∂

+

∂∂

∂∂

=

∂∂

+∂∂

+∂∂

+∂∂

2

2

2

2

211

zTT

rrTr

rrk

zTuT

ru

rTu

tTC zrp θθ

ρ θ

T∇−= kq : zTkqT

rkq

rTkq zr ∂

∂−=

∂∂

−=∂∂

−=θθ

1

In spherical coordinates (r, θ, φ):

∂∂

+

∂∂

∂∂

+

∂∂

∂∂

=

∂∂

+∂∂

+∂∂

+∂∂

2

2

2222

2 sin1sin

sin11

sin

φθθθ

θθ

φθθρ φθ

Tr

Trr

Trrr

k

Tr

uTr

urTu

tTC rp

T∇−= kq : φθθ φθ ∂

∂−=

∂∂

−=∂∂

−=T

rkqT

rkq

rTkqr sin

11