Free Convection past an Isothermal Vertical Plate

Prandtl number

2

option

temperature profile

velocity profile

Consider the free convection past a vertical flat plate or wall maintained at a constant temperature

T

w

with

T

w

>

T

∞

, where

T

∞

is the fluid temperature far from the wall. W. M. Deen derived the governing equations for such a problem (see the reference):

'''

F

+3F



F

-2

2





F



+Θ=0

and

''

Θ

+3PrF



Θ

=0

with

F(0)=



F

(0)=



F

(∞)=Θ(∞)=0

and

Θ(0)=1

,

where

Θ=

T-

T

∞

T

w

-

T

∞

is the dimensionless temperature,

Pr=

ν

α

is the Prandtl number (a dimensionless number that gives the rate of viscous momentum transfer relative to heat conduction), and



F

is proportional to the fluid velocity (

F

is a modified stream function). One has to use the shooting technique to solve this split boundary value problem. This Demonstration displays the temperature and velocity profiles for various values of the Prandtl number versus a similarity variable

η=

1/4

Gr

x

4

y

x

, where

Gr

x

=

g

3

x

βΔT

2

ν

is the Grashof number. When

Pr

is reduced, the dimensionless temperature variations extend farther from the wall (an indication of higher rates of heat conduction).