Editing Grashof number (section)

==Derivation==
The first step to deriving the Grashof number is manipulating the volume expansion coefficient, '''<math>\mathrm{\beta}</math>''' as follows.

<math display="block">\beta = \frac{1}{v}\left(\frac{\partial v}{\partial T}\right)_p =\frac{-1}{\rho}\left(\frac{\partial\rho}{\partial T}\right)_p</math>

The <math>v</math> in the equation above, which represents [[specific volume]], is not the same as the <math>v</math> in the subsequent sections of this derivation, which will represent a velocity.  This partial relation of the volume expansion coefficient, '''<math>\mathrm{\beta}</math>''', with respect to fluid density, '''<math>\mathrm{\rho}</math>''', given constant pressure, can be rewritten as

<math display="block">\rho=\rho_0 (1-\beta \Delta T)</math>

where:
* <math>\rho_0</math> is the bulk fluid density
* <math>\rho</math> is the boundary layer density
* <math>\Delta T = (T - T_0)</math>, the temperature difference between boundary layer and bulk fluid.

There are two different ways to find the Grashof number from this point. One involves the energy equation while the other incorporates the buoyant force due to the difference in density between the boundary layer and bulk fluid.

=== 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>

===Buckingham π theorem===
Another form of dimensional analysis that will result in the Grashof number is known as the [[Buckingham π theorem]]. This method takes into account the buoyancy force per unit volume, <math>F_b</math> due to the density difference in the boundary layer and the bulk fluid.

<math display="block">F_b = (\rho - \rho_0) g</math>

This equation can be manipulated to give,

<math display="block">F_b = -\beta g \rho_0 \Delta T.</math>

The list of variables that are used in the Buckingham π method is listed below, along with their symbols and dimensions.

{| class="wikitable"
|-
! Variable
! Symbol
! Dimensions
|-
| Significant length
| <math>L</math>
| <math>\mathrm{L}</math>
|-
| Fluid viscosity
| <math>\mu</math>
| <math>\mathrm{\frac{M}{L t}}</math>
|-
| Fluid heat capacity
| <math>c_p</math>
| <math>\mathrm{\frac{Q}{M T}}</math>
|-
| Fluid thermal conductivity
| <math>k</math>
| <math>\mathrm{\frac{Q}{L t T}}</math>
|-
| Volume expansion coefficient
| <math>\beta</math>
| <math>\mathrm{\frac{1}{T}}</math>
|-
| Gravitational acceleration
| <math>g</math>
| <math>\mathrm{\frac{L}{t^2}}</math>
|-
| Temperature difference
| <math>\Delta T</math>
| <math>\mathrm{T}</math>
|-
| Heat transfer coefficient
| <math>h</math>
| <math>\mathrm{\frac{Q}{L^2 t T}}</math>
|}

With reference to the [[Buckingham π theorem]] there are {{math|9 – 5 {{=}} 4}} dimensionless groups. Choose {{mvar|L}}, <math>\mu,</math> {{mvar|k}}, {{mvar|g}} and <math>\beta</math> as the reference variables. Thus the <math>\pi</math> groups are as follows:

:<math>\pi_1 = L^a \mu^b k^c \beta^d g^e c_p</math>,

:<math>\pi_2 = L^f \mu^g k^h \beta^i g^j \rho</math>,

:<math> \pi_3 = L^k \mu^l k^m \beta^n g^o \Delta T</math>,

:<math> \pi_4 = L^q \mu^r k^s \beta^t g^u h</math>.

Solving these <math>\pi </math> groups gives:

:<math> \pi_1 = \frac{\mu(c_p)}{k} = \mathrm{Pr}</math>,

:<math> \pi_2 =\frac{l^3 g \rho^2}{\mu^2}</math>,

:<math> \pi_3 =\beta \Delta T</math>,

:<math> \pi_4 =\frac{h L}{k} = \mathrm{Nu}</math>

From the two groups <math>\pi_2</math> and <math>\pi_3,</math> the product forms the Grashof number:

:<math>\pi_2 \pi_3=\frac{\beta g \rho^2 \Delta T L^3}{\mu^2} = \mathrm{Gr}.</math>

Taking <math>\nu = \frac{\mu}{\rho}</math> and <math>\Delta T = (T_s - T_0)</math> the preceding equation can be rendered as the same result from deriving the Grashof number from the energy equation.

:<math>\mathrm{Gr} = \frac{\beta g \Delta T L^3}{\nu^2}</math>

In forced convection the [[Reynolds number]] governs the fluid flow. But, in natural convection the Grashof number is the dimensionless parameter that governs the fluid flow. Using the energy equation and the buoyant force combined with dimensional analysis provides two different ways to derive the Grashof number.

=== Physical Reasoning ===
It is also possible to derive the Grashof number by physical definition of the number as follows:

<math display="block">\mathrm{Gr} = \frac{\mathrm{Buoyancy~Force}}{\mathrm{Friction~Force}} 
=\frac{mg}{\tau A}=\frac{L^3 \rho \beta (\Delta T) g }{\mu (V/L) L^2}
=\frac{L^2  \beta (\Delta T) g}{\nu V}</math>

However, above expression, especially the final part at the right hand side, is slightly different from Grashof number appearing in literature. Following dimensionally correct scale in terms of dynamic viscosity can be used to have the final form.

<math display="block">\mathrm{\mu} = \rho V L</math>Writing above scale in Gr gives;

<math display="block">\mathrm{Gr} = \frac{L^3  \beta (\Delta T) g}{\nu^2 }</math>Physical reasoning is helpful to grasp the meaning of the number. On the other hand, following velocity definition can be used as a characteristic velocity value for making certain velocities nondimensional.

<math display="block">\mathrm{V} 
=\frac{L^2  \beta (\Delta T) g}{\nu Gr}</math>