Navier-Stokes equations with temperature term

[Zalajbeg]

Homework Statement

C dimensionless solute concentration
C0 constant
Grc solutal Grashof number
Grh thermal Grashof number
Le Lewis number
N buoyancy ratio N = Grc/Grh
Pr Prandtl number
r dimensionless radial coordinate
t dimensionless time coordinate
z dimensionless axial coordinate
θ dimensionless temperature
w dimensionless stream function
x dimensionless vorticity

The governing axisymmetric equations for the Newtonian
and laminar binary fluid neglecting heat generation,
viscous dissipation, chemical reaction and thermal radiation
can be expressed as:

Homework Equations

http://postimage.org/image/4gv0vv657/

The Attempt at a Solution

This is a part from an article from R. Cai and C. Gou. My firs assignment is to start with a differential small element and derive this equations. I think first one is a kind of Navier-Stokes equation, the second one is to define worticity the third one is energy equation and the last one is the mass conversion equation.

I could obtain the last two equations but I cannot do the same for the first equation. Is it really Navier-Stokes equation? If so, how can I add the thermal term. I can just obtain inertial and viscous parts but thermal term. May be it is a part of pressure gradient but I couldn't do anything at all. Could someone possibly give me a hint about the thermal term?

 

[mark726]

it is important to approach problems in a systematic and logical manner. In this case, we are dealing with axisymmetric equations for a binary fluid, neglecting various factors such as heat generation, viscous dissipation, chemical reaction, and thermal radiation. Our goal is to derive these equations from a differential small element.

First, let's define the variables and parameters given in the problem. C is the dimensionless solute concentration, C0 is a constant, Grc is the solutal Grashof number, Grh is the thermal Grashof number, Le is the Lewis number, N is the buoyancy ratio, Pr is the Prandtl number, r, t, and z are dimensionless coordinates, θ is the dimensionless temperature, w is the dimensionless stream function, and x is the dimensionless vorticity.

Now, let's consider a small differential element in the fluid. This element has a volume of ΔV, a surface area of ΔS, and is located at a distance r from the origin. The fluid within this element has a velocity of u and a temperature of T. The concentration of the solute within this element is C.

Using the principle of conservation of mass, we can write the mass balance equation for this element as:

∂(ρCΔV)/∂t + ∂(ρuΔS)/∂z = 0

where ρ is the density of the fluid.

Next, using the Navier-Stokes equations for an incompressible fluid, we can write the momentum balance equation for this element as:

ρ(u∂u/∂r + w∂u/∂z)ΔV = -∂pΔS + ρgNΔV + μ(∂^2u/∂r^2 + ∂^2u/∂z^2)ΔV

where p is the pressure, g is the acceleration due to gravity, and μ is the dynamic viscosity of the fluid.

For the energy balance equation, we can write:

ρCp(u∂T/∂r + w∂T/∂z)ΔV = k(∂^2T/∂r^2 + ∂^2T/∂z^2)ΔV

where Cp is the specific heat at constant pressure and