Editing Grashof number (section)

=== Energy equation ===
This discussion involving the energy equation is with respect to rotationally symmetric flow. This analysis will take into consideration the effect of gravitational acceleration on flow and heat transfer. The mathematical equations to follow apply both to rotational symmetric flow as well as two-dimensional planar flow.

<math display="block">\frac{\partial}{\partial s}(\rho u r_0^{n})+{\frac{\partial}{\partial y}}(\rho v r_0^{n})=0</math>

where:

* <math>s</math> is the rotational direction, i.e. direction parallel to the surface
* <math>u</math> is the tangential velocity, i.e. velocity parallel to the surface
* <math>y</math> is the planar direction, i.e. direction normal to the surface
* <math>v</math> is the normal velocity, i.e. velocity normal to the surface
* <math>r_0</math> is the radius.

In this equation the superscript {{mvar|n}} is to differentiate between rotationally symmetric flow from planar flow. The following characteristics of this equation hold true.

* <math>n</math> = 1: rotationally symmetric flow
* <math>n</math> = 0: planar, two-dimensional flow
* <math>g</math> is gravitational acceleration

This equation expands to the following with the addition of physical fluid properties:

<math display="block">\rho\left(u \frac{\partial u}{\partial s} + v \frac{\partial u}{\partial y}\right) = \frac{\partial}{\partial y}\left(\mu \frac{\partial u}{\partial y}\right) - \frac{d p}{d s} + \rho g.</math>

From here we can further simplify the momentum equation by setting the bulk fluid velocity to 0 (<math>u = 0</math>).

<math display="block">\frac{d p}{d s}=\rho_0 g</math>

This relation shows that the pressure gradient is simply a product of the bulk fluid density and the gravitational acceleration. The next step is to plug in the pressure gradient into the momentum equation.

<math display="block">\left(u \frac{\partial u}{\partial s}+v \frac{\partial u}{\partial y}\right)=\nu \left(\frac{\partial^2 u}{\partial y^2}\right)+g\frac{\rho-\rho_0}{\rho}=\nu \left(\frac{\partial^2 u}{\partial y^2}\right)-\frac{\rho_0}{\rho} g \beta (T-T_0)</math>

where the volume expansion coefficient to density relationship <math>\rho-\rho_0 = - \rho_0 \beta (T - T_0)</math> found above and the kinematic viscosity relationship <math>\nu = \frac{\mu}{\rho}</math> were substituted into the momentum equation.<math display="block"> u\left(\frac{\partial u}{\partial s}\right)+v \left(\frac{\partial v}{\partial y}\right)=\nu \left(\frac{\partial^2 u}{\partial y^2}\right)-\frac{\rho_0}{\rho}g \beta(T - T_0)</math>

To find the Grashof number from this point, the preceding equation must be non-dimensionalized. This means that every variable in the equation should have no dimension and should instead be a ratio characteristic to the geometry and setup of the problem. This is done by dividing each variable by corresponding constant quantities. Lengths are divided by a characteristic length, <math>L_c</math>. Velocities are divided by appropriate reference velocities, <math>V</math>, which, considering the Reynolds number, gives <math>V=\frac{\mathrm{Re}_L \nu}{L_c}</math>. Temperatures are divided by the appropriate temperature difference, <math>(T_s - T_0)</math>. These dimensionless parameters look like the following:

* <math>s^*=\frac{s}{L_c}</math>,
* <math>y^* =\frac{y}{L_c}</math>,
* <math>u^*=\frac{u}{V}</math>,
* <math>v^* = \frac{v}{V}</math>, and
* <math>T^*=\frac{(T-T_0)}{(T_s - T_0)}</math>.

The asterisks represent dimensionless parameter. Combining these dimensionless equations with the momentum equations gives the following simplified equation.

:<math>u^* \frac{\partial u^*}{\partial s^*} + v^* \frac{\partial u^*}{\partial y^*} =-\left[ \frac{\rho_0 g \beta(T_s - T_0)L_c^{3}}{\rho\nu^2 \mathrm{Re}_L^{2}} \right] T^*+\frac{1}{\mathrm{Re}_L} \frac{\partial^2 u^*}{\partial {y^*}^2} </math>

:<math>=-\left(\frac{\rho_0}{\rho}\right)\left[\frac{g \beta(T_s - T_0)L_c^{3}}{\nu^2} \right] \frac{T^*}{\mathrm{Re}_L^{2}}+\frac{1}{\mathrm{Re}_L} \frac{\partial^2 u^*}{\partial {y^*}^2}</math>

where:

:<math>T_s</math> is the surface temperature
:<math>T_0</math> is the bulk fluid temperature
:<math>L_c</math> is the characteristic length.

The dimensionless parameter enclosed in the brackets in the preceding equation is known as the Grashof number:

:<math>\mathrm{Gr}=\frac{g \beta(T_s-T_0)L_c^{3}}{\nu^2}.</math>