Figure 4 Mass in and out of the box of the fluid [5]
Now let \("\rho u"\) is the mass flow rate per unit area at the center
of the element, then with help of forward finite difference scheme along
the x – direction
Forward FDM along the right surface
\(\frac{\partial\rho u}{\partial x}=\frac{\left.\ \text{ρu}\right\rceil_{x+\delta x}-\left.\ \text{ρu}\right\rceil_{\text{center}}}{\delta x/2}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\)
\(\left.\ \text{ρu}\right\rceil_{x+\delta x}=\left.\ \text{ρu}\right\rceil_{\text{center}}+\frac{\partial\rho u}{\partial x}\frac{\text{δx}}{2}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\)
Backward FDM along the left surface
\(\frac{\partial\rho u}{\partial x}=\frac{\left.\ \text{ρu}\right\rceil_{\text{center}}-\left.\ \text{ρu}\right\rceil_{x-\delta x}}{\delta x/2}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\)
\(\left.\ \text{ρu}\right\rceil_{x-\delta x}=\left.\ \text{ρu}\right\rceil_{\text{center}}-\frac{\partial\rho u}{\partial x}\frac{\text{δx}}{2}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\)
Net rate of mass in x – direction
=\(\left.\ (\rho u\right\rceil_{x+\delta x}-\left.\ \text{ρu}\right\rceil_{x-\delta x})\delta y\delta z\ \)
\(=\left\{\left.\ \text{ρu}\right\rceil_{\text{center}}+\frac{\partial\rho u}{\partial x}\frac{\text{δx}}{2}-\left(\left.\ \text{ρu}\right\rceil_{\text{center}}-\frac{\partial\rho u}{\partial x}\frac{\text{δx}}{2}\right)\right\}\text{δyδz}\)
\(=\left\{\left.\ \text{ρu}\right\rceil_{\text{center}}+\frac{\partial\rho u}{\partial x}\frac{\text{δx}}{2}-\left.\ \text{ρu}\right\rceil_{\text{center}}+\frac{\partial\rho u}{\partial x}\frac{\text{δx}}{2}\right\}\text{δyδz}\)
\(Net\ rate\ of\ mass\ in\ x\ \ direction=\frac{\partial\rho u}{\partial x}\text{δxδyδz\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ }\left(3.8\right)\)
Similarly at y – direction
\(\frac{\partial\rho v}{\partial y}=\frac{\left.\ \text{ρv}\right\rceil_{y+\delta y}-\left.\ \text{ρv}\right\rceil_{\text{center}}}{\delta y/2}\)Forward FDM along the right surface
\(\left.\ \text{ρv}\right\rceil_{y+\delta y}=\left.\ \text{ρv}\right\rceil_{\text{center}}+\frac{\partial\rho v}{\partial y}\frac{\text{δy}}{2}\)
\(\frac{\partial\rho v}{\partial y}=\frac{\left.\ \text{ρv}\right\rceil_{\text{center}}-\left.\ \text{ρv}\right\rceil_{y-\delta y}}{\delta y/2}\)Backward FDM along the left surface
\(\left.\ \text{ρv}\right\rceil_{y-\delta y}=\left.\ \text{ρv}\right\rceil_{\text{center}}-\frac{\partial\rho v}{\partial y}\frac{\text{δy}}{2}\)
Net rate of mass in x – direction
=\(\left.\ (\rho v\right\rceil_{x+\delta x}-\left.\ \text{ρv}\right\rceil_{y-\delta y})\delta x\delta z\ \)
\(=\left\{\left.\ \text{ρv}\right\rceil_{\text{center}}+\frac{\partial\rho v}{\partial y}\frac{\text{δy}}{2}-\left(\left.\ \text{ρv}\right\rceil_{\text{center}}-\frac{\partial\rho v}{\partial y}\frac{\text{δy}}{2}\right)\right\}\text{δxδz}\)
\(=\left\{\left.\ \text{ρv}\right\rceil_{\text{center}}+\frac{\partial\rho v}{\partial y}\frac{\text{δy}}{2}-\left.\ \text{ρv}\right\rceil_{\text{center}}+\frac{\partial\rho v}{\partial y}\frac{\text{δy}}{2}\right\}\text{δxδz}\)
\(Net\ rate\ of\ mass\ in\ y\ \ direction=\frac{\partial\rho v}{\partial y}\text{δxδyδz\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ }\left(3.9\right)\)
Similarly at z – direction
\(\frac{\partial\rho w}{\partial z}=\frac{\left.\ \text{ρw}\right\rceil_{z+\delta z}-\left.\ \text{ρw}\right\rceil_{\text{center}}}{\delta z/2}\)Forward FDM along the right surface
\(\left.\ \text{ρw}\right\rceil_{z+\delta z}=\left.\ \text{ρw}\right\rceil_{\text{center}}+\frac{\partial\rho w}{\partial z}\frac{\text{δz}}{2}\)
\(\frac{\partial\rho z}{\partial z}=\frac{\left.\ \text{ρw}\right\rceil_{\text{center}}-\left.\ \text{ρw}\right\rceil_{z-\delta z}}{\delta z/2}\)Backward FDM along the left surface
\(\left.\ \text{ρw}\right\rceil_{z-\delta z}=\left.\ \text{ρw}\right\rceil_{\text{center}}-\frac{\partial\rho w}{\partial z}\frac{\text{δz}}{2}\)
Net rate of mass in z – direction
=\(\left.\ (\rho w\right\rceil_{z+\delta z}-\left.\ \text{ρw}\right\rceil_{z-\delta z})\delta x\delta y\ \)
\(=\left\{\left.\ \text{ρw}\right\rceil_{c\text{enter}}+\frac{\partial\rho w}{\partial z}\frac{\text{δz}}{2}-\left(\left.\ \text{ρw}\right\rceil_{\text{center}}-\frac{\partial\rho w}{\partial z}\frac{\text{δz}}{2}\right)\right\}\text{δxδy}\)
\(=\left\{\left.\ \text{ρw}\right\rceil_{\text{center}}+\frac{\partial\rho w}{\partial z}\frac{\text{δz}}{2}-\left.\ \text{ρw}\right\rceil_{\text{center}}+\frac{\partial\rho w}{\partial z}\frac{\text{δz}}{2}\right\}\text{δxδy}\)
\(Net\ rate\ of\ mass\ in\ z\ \ direction=\frac{\partial\rho w}{\partial z}\text{δxδyδz\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ }\text{\ \ \ }\left(3.10\right)\)
Thus;
Net flow rate of mass flow
rate\(\ =\left[\frac{\partial\rho u}{\partial x}+\frac{\partial\rho v}{\partial y}+\frac{\partial\rho w}{\partial z}\right]\text{δxδyδz}\)
\(\frac{\partial\rho}{\partial t}\text{δxδyδz}+\left[\frac{\partial\rho u}{\partial x}+\frac{\partial\rho v}{\partial y}+\frac{\partial\rho w}{\partial z}\right]\delta x\delta y\delta z=0\)
\(\frac{\partial\rho}{\partial t}+\frac{\partial\rho u}{\partial x}+\frac{\partial\rho v}{\partial y}+\frac{\partial\rho w}{\partial z}=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{\ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ }\ \left(3.11\right)\)
The above equation represents the continuity equation which is one of
the fundamental equations of fluid mechanics. It is valid for transient,
compressible or incompressible fluid flow. It can be written in vector
form as follow;
\(\frac{\partial\rho}{\partial t}+\mathbf{\nabla}\bullet\rho\mathbf{V=}0\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ }\ \left(3.12\right)\)
Two special cases that the researchers interested on them due to wide
range of their applications in engineering and industry like solar
collectors, internal pipe flow, nanofluid enclosure along with natural
convection which they are;
- For steady, compressible fluid flow whick makes the density as a
function of space only;\(\mathbf{\nabla}\bullet\rho\mathbf{V}=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(3.13\right)\)
- For incompressible flow;\(\mathbf{\nabla}\bullet\mathbf{V}=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(3.14\right)\)\(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(3.15\right)\)
- Momentum Equation
Before derivatives the momentum and energy equations, it is very
important to full – understanding the following concept which is called
Material Derivative.
Physically, any property is a function of space and time. The space is
represented by the coordinates (x, y, z) and time which is denoted as t.
so that we can write the total or substantive or material derivative
which is denoted as \(\varnothing\) as indicated below;
\(\frac{D\varnothing}{\text{Dt}}=\frac{\partial\varnothing}{\partial t}+\frac{\partial\varnothing}{\partial x}\frac{\text{dx}}{\text{dt}}+\frac{\partial\varnothing}{\partial y}\frac{\text{dy}}{\text{dt}}+\frac{\partial\varnothing}{\partial z}\frac{\text{dz}}{\text{dt}}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(3.16\right)\)
\(\frac{\text{dx}}{\text{dt}}=u,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{\text{dy}}{\text{dt}}=v,\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\frac{\text{dz}}{\text{dt}}=w\ \ \ \ \ \ \)
\(\frac{D\varnothing}{\text{Dt}}=\frac{\partial\varnothing}{\partial t}+u\frac{\partial\varnothing}{\partial x}+v\frac{\partial\varnothing}{\partial y}+w\frac{\partial\varnothing}{\partial z}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(3.17\right)\)
So that the acceleration can be written as below;
\(a=\frac{D\mathbf{V}}{\text{Dt}}=\frac{\partial\mathbf{V}}{\partial t}+u\frac{\partial\mathbf{V}}{\partial x}+v\frac{\partial\mathbf{V}}{\partial y}+w\frac{\partial\mathbf{V}}{\partial z}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(3.18\right)\)
A shorthand notation for the material derivatives operator is;
\(\frac{D\mathbf{\varnothing}}{\text{Dt}}=\frac{\partial\mathbf{\varnothing}}{\partial t}+\left(\mathbf{V\bullet}\mathbf{\nabla}\right)\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(3.19\right)\)
Where the \(\mathbf{V}\) is the velocity vector and it is given by\(\mathbf{=}\mathbf{u}\hat{\mathbf{\text{\ i\ }}}\mathbf{+v}\hat{\mathbf{\text{\ j\ }}}\mathbf{+w}\hat{\mathbf{\text{\ k}}}\), the velocity gradient is denoted as\(\mathbf{\ \nabla}\) and it is
given by\(\mathbf{\nabla}\mathbf{\varnothing=}\frac{\mathbf{\partial\varnothing}}{\mathbf{\partial x}}\hat{\mathbf{\text{\ i\ }}}\mathbf{+}\frac{\mathbf{\partial\varnothing}}{\mathbf{\partial y}}\hat{\mathbf{\text{\ j\ }}}\mathbf{+}\frac{\mathbf{\partial\varnothing}}{\mathbf{\partial z}}\hat{\mathbf{\text{\ k}}}\)
As an example; acceleration – components will be;
\(a_{x}=\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(3.19a\right)\)
\(a_{y}=\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(3.19b\right)\)
\(a_{z}=\frac{\partial w}{\partial t}+u\frac{\partial w}{\partial x}+v\frac{\partial w}{\partial y}+w\frac{\partial w}{\partial z}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(3.19c\right)\)
Newton’s law physically states that ”the rate of change of momentum of a
fluid particle equals to the sum of forces on the particle”.
Mathematically, it could be written as inserted below
\(\sum F=m\ a\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(3.20\right)\)
There are two major forces acting on fluid particle;
Surface forces
- Pressure forces
- Viscous forces
Body forces
- Gravity force
- Centrifugal force
- Coriolis force
- Electromagnetic field
It is commonly on CFD problems related to heat transfer and fluid
mechanics to include two forces which is a surface force as a separated
force and the body force as a source term.
Now let us analysis the surface forces or stresses which contains normal
stress and shear stress. The shear stress is nothing but the force
magnitude divided by the area. The stress opposite in the direction to
the proposed direction as shown in Figure 5.