Temperature in laminar flow along a semi-infinite plate

 A common problem in heat transfer is the influence of a constant-temperature semi-infinite plate on a constant free-stream velocity flow. The most straight forward approach to this problem is to introduce the similarity variable \( \eta \)

\[ \eta = \frac{y}{\sqrt{\nu x/u_{\infty}}}\]

and then assume that the nondimensional temperature \(\theta\) is a function of that variable

\[ \theta = \theta(\eta) = \frac{T_s-T}{T_s-T_{\infty}} \]

By applying this coordinate transformation the partial differential equation describing energy conservation simplifies to to the ODE

\[ \theta'' + \frac{\text{Pr}}{2}\zeta\theta' = 0 \]

where \(\zeta=\zeta(\eta)\) is the similarity solution to the hydrodynamic counterpart of this problem as given by the Blasius equation

\[ \zeta'''+\frac{1}{2}\zeta\zeta'' = 0\]

Since the energy equation after the similarity transform is relatively simple it can be directly integrated in order to reveal the solution

\[ \theta(\eta) = \frac{\int_0^\eta \left[ \exp\left(-\frac{\text{Pr}}{2}\int_0^\eta \zeta d\eta \right)\right] d\eta}{\int_0^\infty \left[ \exp\left(-\frac{\text{Pr}}{2}\int_0^\eta \zeta d\eta \right)\right] d\eta} \]

as is shown in (William Kays, 2005). This relation between \( \eta \) and \( \theta\) can be seen for several values of the Prandtl number on figure 1.

A more physical interpretation of the results can be seen on figure 2 where the temperature profile has been plotted as a function of downstream distance at various distances from the plate. In this case the following assumptions are made, \( \text{Pr}=1 \), \( \nu=10^{-6}\text{m}^2/\text{s} \) and \( u_{\infty}=2.56\text{cm/s} \)

The code for generating the figures above can be downloaded here: Matlab code for generating temperature profiles

References

William Kays (2005). Convective Heat and Mass Transfer. New York: McGraw-Hill .^