Governing equations for natural convection on a vertical plate

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Figure 1 Natural convection over a vertical flat plate (Pr >1).

For external natural convection near a vertical flat plate as shown in Fig. 1, the boundary layer assumption can be applied to simplify the above generalized governing equations. The boundary layer treatment for the case of natural convection is very similar to that for the case of forced convection. The difference between the natural convection problem shown in Fig. 1 and forced convection over a flat plate is that the free stream velocity in the outside of the velocity boundary layer in natural convection is zero for natural convection. In addition, the pressure outside the boundary layer is hydrostatic for the case of natural convection, instead of being externally imposed as in the case of forced convection. For 2-D external convection of an incompressible fluid as shown in Fig. 1, the continuity equation becomes

$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$	(1)

If one assumes that the fluid is single component so that the natural convection is driven by the density difference induced by the temperature gradient, the momentum equation becomes:

$\rho \frac{D\mathbf{V}}{Dt}=\left( -\nabla p+{{\rho }_{\infty }}\mathbf{g} \right)-{{\rho }_{\infty }}\mathbf{g}\beta (T-{{T}_{\infty }})+\nabla \cdot (\mu \nabla \mathbf{V})$	(2)

Applying the boundary layer assumption and assuming steady flow with constant thermophysical properties, the momentum equation becomes:

$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=-\frac{1}{{{\rho }_{\infty }}}\frac{\partial p}{\partial x}-g+g\beta (T-{{T}_{\infty }})+\nu \frac{{{\partial }^{2}}u}{\partial {{y}^{2}}}$	(3)

The pressure in the boundary layer, p, is independent of y ( $\partial p/\partial y=0$ ) and equals that outside the boundary layer at the same longitudinal position, p_∞, i.e.,

$\frac{\partial p}{\partial x}=\frac{dp}{dx}=\frac{d{{p}_{\infty }}}{dx}$

The hydrostatic pressure, p_∞, is dictated by the density and the longitudinal position:

$\frac{d{{p}_{\infty }}}{dx}=-{{\rho }_{\infty }}g$

Substituting the above two equations into eq. (3), the momentum equation becomes:

$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\nu \frac{{{\partial }^{2}}u}{\partial {{y}^{2}}}+g\beta (T-{{T}_{\infty }})$	(4)

After applying the boundary layer assumption and assuming the viscous dissipation is negligible, the energy equation becomes:

$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\alpha \frac{{{\partial }^{2}}T}{\partial {{y}^{2}}}$	(5)

At the heated wall, the non-slip and impermeable conditions yield the following boundary condition for the momentum equation:

$u = v = 0, at y = 0$	(6)

The temperature at the heated wall is specified, i.e.,

$T = T w, at y = 0$	(7)

Since the quiescent fluid far away from the heated plate is not disturbed by the existence of the heated plate, the velocity at the locations away from the flat plate should be zero:

$u=v=0,\text{ }y\to \infty$	(8)

Also, the temperature of the fluid outside the thermal boundary layer is not affected by the heated wall:

$T={{T}_{\infty }},\text{ }y\to \infty$	(9)

References

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

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Governing equations for natural convection on a vertical plate

From Thermal-FluidsPedia

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