---
# System prepended metadata

title: 2.2 Analysis of Continua
tags: [Mathematical modeling and optimization]

---

---
tags: Mathematical modeling and optimization
title: 2.2 Analysis in Continuum Systems
---

## 2.2 Analysis of Continua

The **material derivative** is a fundamental concept in continuum mechanics that describes the rate of change of a physical quantity as the continuum deforms. It accounts for both the local rate of change (illustrated by the transition from a black line to a gray dashed line) and the transport of the quantity due to fluid motion.

<div style="text-align:center"><img src="https://live.staticflickr.com/65535/53576036970_3f6106c161.jpg" width=30%></div>

</br>
</br>

Consider a physical property $\psi$ within a continuum, which depends on position $x_i$ and time $t$, denoted as $\psi(x_i, t)$. Using the chain rule of differentiation, the rate of change of $\psi(x_i, t)$ incorporates both spatial and temporal variations:

$$
\begin{align}
{\displaystyle \frac{d}{dt}\psi(x_i\ , t)} = &{\frac{\partial \psi}{\partial t}\ \frac{\partial t}{\partial t} + \frac{\partial \psi}{\partial x_i}\ \frac{\partial x_i}{\partial t}}\\
\\
 = & {\frac{\partial \psi}{\partial t}\  + u_i\  \frac{\partial \psi}{\partial x_i}}
\end{align}
$$

where $u_i$ represents the velocity component. The material derivative is applicable to any physical property, such as density, temperature, or pressure, as long as it is part of the continuum.



### 2.2.1 Control Volume (CV) and Control Surface (CS):

The concept of **control volume (CV)** and **control surface (CS)** are central to fluid mechanics and thermodynamics, where the analysis focuses on a defined region as an bystanding observer. It is called [__Eulerian representation__](https://en.wikipedia.org/wiki/Lagrangian_and_Eulerian_specification_of_the_flow_field)

A __**control volume**__ is a spatial region which is selected to analyze a system. It can be an arbitrary volume that can move and deform with time. The control volume can encompass fluid, a solid, or a combination of both. Within this volume, various physical properties such as mass, momentum, and energy are studied to analyze the behavior of the system under consideration.

Characteristics of a Control Volume are:
- **Arbitrary Shape and Size**: The control volume can have any shape and size depending on the specific problem being analyzed.
- **Deformable and Movable**: The control volume can deform and move with time, allowing for the study of systems undergoing changes in shape or position.
- **Fixed or Moving Boundaries**: The boundaries of the control volume can be either fixed or moving depending on the problem. For instance, in the case of a flow through a pipe, the control volume boundaries may remain stationary, while in the case of a moving piston in a cylinder, the boundaries move with time.
- **Conservation Laws**: The conservation laws of mass, momentum, and energy are applied to the control volume to analyze the behavior of the system within it.

The __control surface__ is the boundary surface of the control volume. It separates the control volume from the surrounding environment. The control surface can be real or imaginary, and it serves as the interface through which mass, momentum, and energy may enter or leave and contributes to changes within the control volume. 
   
As an example, consider the water flows through a pipe. Here, the pipe itself serves as the control volume, and its internal surface are the control surface. By applying conservation laws within this control volume, we can analyze properties such as flow rate, pressure distribution, and energy losses.

<img src="https://live.staticflickr.com/65535/53562960132_684880ae87_z.jpg">

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<br/>
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Furthermore, multiple CVs and CSs can be employed to analyze complex phenomena, such as heat transfer between a pipe and its surroundings.
 


<img src="https://live.staticflickr.com/65535/53565912866_d5f745976b_c.jpg">

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Considering a dynamic continuum within a stationary control volume regime. As the continuum progresses, two distinct phenomena occur: firstly, internal elements within the continuum undergo movement, and secondly, these elements traverse the control surface.

Mathematically speaking, the dynamics of a property $\psi$ in a control volume is described:

$$\displaystyle  \frac{D}{Dt}\int\limits_{CV} \psi \ dV  = \int\limits_{CV} \frac{d}{d t}\psi\ dV + \int\limits_{CV} \psi \ \frac{\partial u_i}{\partial x_i} \ dV\qquad$$, where 


- $\displaystyle \frac{D}{D t}\int\limits_{CV} \psi \ dV$ is the change of property in a control volume
- $\displaystyle \frac{d}{d t}\psi$ is the material derivative, describing the changes inside of the contineous body.
- $\displaystyle\frac{\partial u_i}{\partial x_i}$ denotes the local change of velocity in the control volume.

Control volumen and control surface are the fundamental concepts in the Computational Fluid Dynamics (CFD), specifically the [finite volume method](https://en.wikipedia.org/wiki/Finite_volume_method). We will dive into this topic in another context.

---

### 2.2.2 Reynolds Transport Theorem


Now we can combine the concept of material derivative of a moving continuum to a rigid control volume. Following the mathematical formulation in the previous section: 

$$
\begin{align}
\displaystyle \qquad \frac{D}{Dt}\int\limits_{CV} \psi \ dV  =& \int\limits_{CV} \frac{d}{d t}\psi\ dV + \int\limits_{CV} \psi \frac{\partial u_i}{\partial x_i} \ dV \\
\\
= &\int\limits_{CV} {\frac{\partial \psi}{\partial t}\ dV + \int\limits_{CV}u_i\  \frac{\partial \psi}{\partial x_i}} dV + \int\limits_{CV} \psi \frac{\partial u_i}{\partial x_i} \ dV \\
\\
= & \int\limits_{CV} \left({\frac{\partial \psi}{\partial t} + \frac{\partial \ }{\partial x_i}} \ (u_i\  \psi)\right) dV 
\end{align}
$$

As it is seen, the local change of velocity and the local change of property can be combine into a flux concept denoting the property carried by its velocity. With the aid of [Gaussian integration law](https://en.wikipedia.org/wiki/Divergence_theorem), the rate of change in a control volume becomes

$$\displaystyle  \frac{D}{Dt}\int\limits_{CV} \psi \ dV  =\int\limits_{CV}\frac{\partial \psi}{\partial t}\ dV + \oint_\limits{CS} \psi\ (u_i \cdot n_i) \ dA$$

<img src="https://mechcontent.com/wp-content/uploads/2021/11/reynolds-transport-theorem.webp" width="80%">


The term $\displaystyle \oint_\limits{CS} \psi\ (u_i \cdot n_i) \ dA$ denotes the flux normal to the control surface as the convective pheonomon in the control surface. This is also the main idea of [__Reynolds Transport Theorem__](https://en.wikipedia.org/wiki/Reynolds_transport_theorem).

<br/>

$\psi$ can be every kind of phyiscal property. As common starting points, mass and momentum analysis in the control volume are the usually taken for the force evaluation on mechanical systems. Thus

$$
\begin{align}
\displaystyle  \frac{D m}{Dt}\biggr\rvert_{CV} &= \frac{D}{Dt}\int\limits_{CV} \rho \ dV  =\int\limits_{CV}\frac{\partial \rho}{\partial t}\ dV + \oint_\limits{CS} \rho\ (u_i \cdot n_i) \ dA \\
\\
\displaystyle  F_i\biggr\rvert_{CV} = \frac{D (m u_i)}{Dt}\biggr\rvert_{CV} &= \frac{D}{Dt}\int\limits_{CV} \rho u_i \ dV  =\int\limits_{CV}\frac{\partial \rho u_i}{\partial t}\ dV + \oint_\limits{CS} \rho u_i\ (u_i \cdot n_i) \ dA 
\end{align}
$$

<br/>
<br/>
-->


__<ins>Scenario: rocket propulsion__

The [thrust of a rocket engine](https://www.grc.nasa.gov/www/k-12/airplane/rockth.html) is created by the reaction force of the ejecting gas. 

<img style="float: right;" src="https://live.staticflickr.com/65535/54477226272_961c7d9500_o.png" width="40%" >

By selecting the entire rocket as a control volume, the mass and momentum conservation are formulated as:



$$
\begin{align}
&\frac{d \ m}{dt} = - \dot{m}_{out}\\
\\
&\frac{d \ (m \vec{u})}{dt}=\underbrace{\dot{m}_{out}\ \vec{u}_{out} \ + \ (p_0 - p_{out}) A_{out}}_{\text{Thrust force}}
\end{align}
$$
where 
- $A_{out}$ is the nozzle ejecting area, and
- $p_{out}$, $p_{0}$ are the pressure level on the nozzle exit and environment 




The momentum change in selected control volume is then the thrust force of the ejected gas. The exit pressure level is a gas dynamics related characteristics, which depends on nozzle shape, pressure in the rocket and the gas thermodynamic property. Phenomena such as [under and over expansion](https://aerospaceweb.org/question/propulsion/q0220.shtml) are directly followed be the pressure behavior.


*Supposed a 4000[kg] rocket engine takes off by ejecting constantly 10[kg/s] gas with exit velocity of 2000[m/s]. Assuming the exit pressure level equal to the ambient condition and no friction effect accounted. What is the velocity of the rocket after 10 secs?*

[<ins> __Sample solution__](https://cocalc.com/share/public_paths/b9134bef2d01ed13dffdb94b09a3973bee14bab8)


__<ins>Scenario: weight of the water tank__
A water tank regulates the flow rate by installing pump in a reservoir.  Irregular water supply comes from the top and fill up the store. Supposed the water tank owns it nett weight $W$ with the initial water height as $h$ sitting on a scale. How would the scale shows if the water supply 
<img src="https://live.staticflickr.com/65535/54471942929_309faa4437_o.png" width=70%>



Selecting the tank as the control volume, the transport of mass and momentum can be realized by considering the flow in/out amount of the CV, i.e. over the CS boundaries. 

<br/>

<img src="https://live.staticflickr.com/65535/54475164417_caff59b3de_o.png" width ="50%"  style="float:left;">


$$
\begin{align}
&\frac{d m}{d t } = \dot{m}_{in} - \dot{m}_{out}\\
&\qquad \Rightarrow  m\biggr\rvert_{t + \delta t} = m\biggr\rvert_{t} + \delta t \left(\dot{m}_{in} - \dot{m}_{out}\right)
\\
\\
&\frac{d (m \vec{u})}{d t } = \dot{m}_{in} \vec{u}_{in}  + \dot{m}_{out} \vec{u}_{out}\\
&\qquad \Rightarrow  m \vec{u}\biggr\rvert_{t + \delta t} = m \vec{u}\biggr\rvert_{t} + \delta t \left(  \dot{m}_{in} \vec{u}_{in}  + \dot{m}_{out} \vec{u}_{out}  \right)
\end{align}
$$ 

<br/>

[<ins>__Sample Solution__](https://cocalc.com/share/public_paths/65e888bd5d210f7fbb4f3fc323bcab74b8c87f95)

---

### 2.2.3 Universal law of equilibrium

The **universal conservation laws** in continuum mechanics describe the behavior of physical systems by combining mass, momentum, and energy conservation. These principles enable the analysis of dynamic responses to external forces and system design.


__Material Transport__

The general form of a transport equation based on [Eulerian observation method](https://en.wikipedia.org/wiki/Lagrangian_and_Eulerian_specification_of_the_flow_field) reads:
$$
\begin{align}
\iiint\frac{D}{Dt}\psi\  dV &= \underbrace{\iiint \frac{\partial}{\partial t} \psi \ dV}_{\text{time derivative}}+ \underbrace{\iiint \frac{\partial}{\partial x_j} u_j \psi \ dV}_{\text{Convection}} \\
&=\underbrace{\iiint\mathit{P} \ dV}_{\text{Production}}+ \underbrace{\iiint \epsilon \ dV}_{\text{Dissipation}}+ \underbrace{\iiint\mathit{D}\ dV}_{\text{Diffusion}} + \underbrace{\iiint \mathit{S}\ dV}_{\text{Source}}
\end{align}
$$

Where __Production__$(\mathit{P}\ )$, __Dissipation__$(\mathit{\epsilon}\ )$, __Diffusion__$(\mathit{D}\ )$ and __Source__$(\mathit{S}\ )$ are the phyiscal effects summarized in the general transport. 

__Production__$(\mathit{P}\ )$ refers to the generating mechanism during the transport. Taking the chemical reaction process for example, the concentration of species can be set as $\psi$, where the $\mathit{P}$ is the products of checmical process.

In contrary, __Dissipation__$(\mathit{\epsilon}\ )$ denotes the phenomenon that something is gradually vanishing. In the field turbulence, dissipation refers to the smallest eddies disappering into form of heat due to the viscosity effects.  

The __Diffusion__$(\mathit{D}\ )$ always has a form of $\displaystyle \frac{\partial}{\partial x_j}(\xi \frac{\partial\psi}{\partial x_j})$, showing the property is redistributed in the space around. The mathematical form is described in the [Fick's law.](https://en.wikipedia.org/wiki/Fick%27s_laws_of_diffusion) 

 __Source__$(\mathit{S}\ )$ are the terms which do not affect over surface. Something as gravitational force in the momentum transport or radiation in the energy transport. Sometimes, the source term can also be the non-objective ones derived from the coordinate changes, e.g., Coriolis and centigugal force 


__Conservation of properties__



For the conservative properites such as mass, momentum and energy, there is __no production and dissipation__ obtained in the transport. The transport property $\psi$ and the corresponding terms are

    
| Quantity | $\psi$  | $\mathit{D}$ | $\mathit{S}$ |
| ---  | -------- | -------- | -------- |
| Mass |  $\rho$     | -- | -- |
| Momentum | $\rho u_i$  | $-\frac{\partial P}{\partial x_i}+\frac{\partial}{\partial x_j}(\mu \frac{\partial u_i}{\partial x_j})$ | $\rho g_i$ |
| Energy | $\rho(\frac{1}{2} u_i² +  e)$         | $\frac{\partial}{\partial x_j }(\kappa\frac{\partial T}{\partial x_j}-Pu_i +u_i\frac{\partial}{\partial x_j}(\mu \frac{\partial u_i}{\partial x_j}))$     | $u_i\rho g_i$     |

We can also modify the transport equation for other physical characteristics, e.g. moment $\vec{M} = \rho \vec{r}\times \vec{u}$. The derived property is also conserved.

---

Combining material transport with control volume concept, almost all the phyiscal phenomena in the continuum framework can be described. Here are some examples:
- _Cooling technology with flowing liquid coolant_ is a combination of mass and momentum conservation in describing the flow patterns, with additional energy transport to represent the heat transfer between flow and solid body.
- _Gas species mixture_ is a combination of mass, momentum and energy conservation as well as equation of state in describing the flow patterns with temperature. Besides, additional species transport is needed to illustrate the distribution.
- _Combustion process_ is not only the combination of mass, momentum and energy conservation with species transport, but also the chemical reaction to describe the production and dissipation of species.
- _Aerodynamic effects of moving object_ is the combination purely based on mass and momentum transport. The shear over solid surface boundary is the driving force for all the effects.


Such complexity are addressed in the CFD framework, where all the necessary transport details incorporated with complex three-dimensional geometry are captured using the numerical methods. All we need is to select the corresponding transport equation and define the proper boundary conditions as well as computational domain.

On the other hand, it is possible to simplify the task by focusing on the main flow direction and/or neglecting the irrelevant effects in most of enginnering applications. For example, an zero- or one-dimensional simplified algorithm can be formulated into script for system-wise analysis. 


__<ins>Scenario - 1D Convection__

An [underfloor heating system](https://en.wikipedia.org/wiki/Underfloor_heating) is realized by laying up the water tube under the screed. When the tube is pumped up with hot water, the heat transfers to the screed and warm up the floor and room.


<img src="https://upload.wikimedia.org/wikipedia/commons/6/6b/Underfloor_heating_pipes.jpg" width="34%"> <img src="https://www.red-current.com/images/article_images/underfloor_heating_thermal_.jpg" width="64%">

_Supposed a 100m long tube of 20mm diameter is filled with water (15°C). At the sourcing side, hot water of 40°C is pumped in. The flow rate of water represents 3 kg/s. Say the heat conduct coefficient is 500 W/(m^2 K),_
1. _What is the temperature of the water along the tube?_
2. _What would be th temperature distribution if the tube diameter reduced to 15mm?_ 

Now, we can pull the tube into an one-dimensional domain and apply finite volume method to approach the heat transfer procedure. Control volumes can illustrated as figure below. In each control volume, conservation of mass, momentum and energy are valid to describe the physical properties, where the fluxes (mass, momentum and energy) are the characteristics connecting the neighbouring control volume.

<img src="https://live.staticflickr.com/65535/54510611975_6df31eca55_o.png">


[<ins> __Sample solution__](https://cocalc.com/share/public_paths/6a78cd843e68eea15a682f4db6e8922a3e8b8099)


<img src="https://live.staticflickr.com/65535/54512927080_1e8dd3a7f1_o.png" width="90%">


#### __<ins>Scenario - 1D Diffusion__

Fins are the common geometry applied in passive cooling systems. The heat is conducted through a 1D metallic fin or heat sink attached to the hot component.

<img src="https://cdn.prod.website-files.com/66a9fa240521ccee763070bb/66aa1d7770a04d0936127f40_3.png" width="90%">

_Dimension of single fin are 10cm height with cross sectional perimeter of 4cm and area of 0.1 cm². The fin is made of copper, for which the thermal conductivity is ~400 W/(m K) and the convection coefficient is set for 100 (W/m²·K)._

_Supposed there is no laternal heat transfer on the fin. The temperature of the bottom side is 80°C and the temperature along the fin height can achieve the steady state solution_. 

- How is the temperature distribution along the fin height?


#### Governing Equation
    
The temperature distribution $T(x)$ along the 1-D fin is modeled by:
$$\underbrace{\rho \ Cp \frac{dT}{dt}}_{\text{time derivative}} =\underbrace{\frac{d}{dx}(k\frac{dT}{dx})}_{\text{diffusion}}−\underbrace{\frac{hP}{A_c}(T−T_\infty)}_{\text{heat loss}}$$

- $\rho$ : density of the fin material (kg/m³).
- $Cp$ : specific heat capacity (J/kg·K)
- $k$ : Thermal conductivity of the fin (W/m·K).
- $h$ : Convective heat transfer coefficient (W/m²·K).
- $P$ : Perimeter of the fin (m).
- $A_c$​: Cross-sectional area (m²).
- $T_\infty$​: Ambient temperature.

*Steady state* is a special situation in this energy transport, meaning the heat comes into the control volume (rod, via heat diffusion) is in equilibrium with heat loss. Thus,
    
$$\frac{d}{dx}(k\frac{dT}{dx}) - \frac{hP}{A_c}(T−T_\infty)  = 0$$
    
An analytical solution can be achieved by solving the 2nd order differential equation. which yields:
$$ T(x) = T_\infty + (T_{\text{base}} - T_\infty) \frac{\cosh(m (L-x))}{\cosh (mL)}, \qquad m = \sqrt{\frac{h P}{k A_c}}$$

#### Boundary Conditions

Base ($x=0$): Fixed temperature (Dirichlet BC). 
    $$\qquad T(0)=T_\text{base}\qquad \text{(e.g., 80°C)}$$ 
Tip ($x=L$): Insulated (Neumann BC). 
$$\qquad \frac{dT}{dx}∣_{x=L}=0$$
    
#### Applying Finite Difference Method (FDM)
Following the concept of FDM, the equation can be discretized and for cell $i$:
    $$\frac{k}{\Delta x^2} T_{i-1}+ \left(-\frac{hP}{A_c}-\frac{2k}{\Delta x^2}\right) T_i +\frac{k}{\Delta x^2} T_{i+1}= -\frac{hP}{A_c} T_\infty$$
which can be easily implemented in a maxtrix form. $A\cdot T_{FDM} = b$

The Dirichlet ($T(0)=T_{\text{base}}$) and Neumann boundary ($\frac{dT}{dx}∣_{x=L}=0$) conditions have to be skilled passed to the maxtrim representation. 

#### Applying Finite Volume Method (FVM)
Integrating the governing equation over a control volume (cell), 
$$\int_{\Delta V}​\frac{d}{dx}(k\frac{dT}{dx}) dV−\int_{\Delta V}\frac{hP}{A_c}(T−T_\infty)dV=0$$
Applying Gaussian integration law for flux, the equation yields

$$(k A_c\frac{dT}{dx})_e - (k A_c\frac{dT}{dx})_w -  hP \Delta x(T_P−T_\infty) = 0$$   

with $\Delta x$ as the length of control volume, and $T_P$ is the cell-center temperature. The supscript $e$ and $w$ indicate to the east and west side of the cell surface.

Further the $\frac{dT}{dx}$ term can be descritized to
    $$\left(\frac{dT}{dx}\right)_e \approx \frac{T_E - T_P}{\Delta x};\quad \quad \left(\frac{dT}{dx}\right)_w \approx \frac{T_P - T_W}{\Delta x},$$
    
which refers to the neighboring cell $W$ and $E$.

<img src="https://live.staticflickr.com/65535/54616354194_a269dbfbd5_b.jpg">

And the governing equation in descrete form shows:
    $$(\frac{kA_c}{\Delta x})T_W + (-2 \frac{kA_c}{\Delta x} - h P \Delta x)T_P  + (\frac{kA_c}{\Delta x})T_E  = −hP\Delta x T_\infty$$

[<ins> __Sample solution__](https://cocalc.com/share/public_paths/97b46586e9d3e652acfdc27579b81c3f616ee937)

<img src="https://live.staticflickr.com/65535/54614275520_609c3dace0.jpg">

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