Modeling the Removal by Rain of Two Interacting Gases Forming Distinct Particulate Matters in the Atmosphere

Abstract

This paper deals with the removal of two interacting gases (gaseous pollutants) forming two distinct particulate matters, namely smaller and larger particulate matters, by rain from the atmosphere. There are several sources that release these gases and particulate particles into the environment. The conversion of gas to particle in the atmosphere results in the formation of smaller particulate matters and these particulate matters combine to form larger particulate matters. Furthermore, the interaction of two gases leads to the formation of larger particulate matters in the atmosphere. The presumption is that there are six interacting phases in the atmosphere under consideration, comprising of the raindrops phase, two distinct phases of gaseous pollutants, smaller and larger particulate matters phases, and a combined phase of gases absorbed in the raindrops. A system of ordinary differential equations is used to develop the mathematical model. In order to investigate the proposed mathematical model, stability theory of differential equation is applied. The results of the theoretical investigation reveal that the gaseous pollutants and the particulate matters are considerably removed by rain from the atmosphere. To verify the findings of analytical investigation, numerical simulation of the model is also executed.

Appendices

Appendix A

Proof: Taking into consideration the following positive definite function to demonstrate the local stability $E^*$,

$$\begin{aligned} V=\frac{1}{2}(l_1C_{r1}^2+l_2C_{11}^2+l_3C_{21}^2+l_4C_{p11}^2+l_5C_{p21}^2+l_6C_{a1}^2). \end{aligned}$$

(A1)

where $C_{r1},C_{11},C_{21},C_{p11},C_{p21}$ and $C_{a1}$ are the small perturbations about $E^*$ given below,

$$\begin{aligned} C_r &=C_r^*+C_{r1},C_1=C_1^*+C_{11},C_2=C_2^*+C_{21},C_{p1}=C_{p1}^*+C_{p11},C_{p2}=C_{p2}^*+C_{p21},\\ C_a &=C_a^*+C_{a1}. \end{aligned}$$

and $l_1,l_2,l_3,l_4,l_5$ and $l_6$ should be considered specifically as they are positive constants.

Using the linearized system that corresponds to $E^*$, we differentiate (A1) with regard to ‘t’  and we get,

$$\begin{aligned} \dot{V}=&-l_1(r_0+r_1C_1^*+r_2C_2^*)C_{r1}^2-l_2(\delta _1+\beta _1+\gamma _1C_2^*+\alpha _1C_r^*)C_{11}^2\\ {} &-l_3(\delta _2+\beta _2+\gamma _2C_1^*+\alpha _2C_r^*)C_{21}^2 -l_4(\delta _{p1}+\lambda +\alpha _{p1}C_r^*)C_{p11}^2-l_5(\delta _{p2}+\alpha _{p2}C_r^*)C_{p21}^2\\ {} &-l_6(k+\nu C_r^*)C_{a1}^2-(l_1r_1C_r^*+l_2\alpha _1C_1^*)C_{r1}C_{11}-(l_1r_2C_r^*+l_3\alpha _2C_2^*)C_{r1}C_{21}\\ {} &-l_4 \alpha _{p1}C_{p1}^*C_{r1}C_{p11}-l_5\alpha _{p2}C_{p2}^*C_{r1}C_{p21}+l_6(\alpha _1C_1^*+\alpha _2C_2^*-\nu C_a^*)C_{r1}C_{a1}\\ {} &-(l_2\gamma _1C_1^*+l_3\gamma _2C_2^*)C_{11}C_{21}+l_4\beta _1C_{11}C_{p1}+l_4\beta _2C_{21}C_{p11}+l_5\gamma C_2^*C_{11}C_{p21}\\ {} &+l_5\gamma C_1^*C_{21}C_{p21}+l_5\lambda C_{p11}C_{p21} +l_6\alpha _1C_r^*C_{11}C_{a1}+l_6\alpha _2C_r^*C_{21}C_{a1}. \end{aligned}$$

(A2)

Choosing $l_1=l_2=l_3=1,$

• $l_4<\frac{\displaystyle 1}{\displaystyle 5}(\delta _{p1}+\lambda +\alpha _{p1}C_r^*)$ $min\{a_1,a_2,a_3\},$

• $l_5<min\{a_4,a_5,a_6,a_7\}$ and

• $l_6<\frac{\displaystyle 4}{\displaystyle 15}(k+\nu C_r^*)$ $min\{a_8,a_9,a_{10}\}.$

where,

• $a_1=\frac{\displaystyle r_0+r_1C_1^*+r_2C_2^*}{\displaystyle (\alpha _{p1}C_{p1}^*)^2}$ , $a_2=\frac{\displaystyle \delta _1+\beta _1+\gamma _1C_2^*+\alpha _1C_r^*}{\displaystyle (\beta _1)^2}$ , $a_3=\frac{\displaystyle \delta _2+\beta _2+\gamma _2C_1^*+\alpha _2C_r^*}{\displaystyle (\beta _2)^2}$ ,

• $a_4=\frac{\displaystyle (r_0+r_1C_1^*+r_2C_2^*)(\displaystyle \delta _{p2}+\alpha _{p2}C_r^*)}{\displaystyle 5(\alpha _{p2}C_{p2}^*)^2}$ , $a_5=\frac{\displaystyle (\delta _1+\beta _1+\gamma _1C_2^*+\alpha _1C_r^*)(\delta _{p2}+\alpha _{p2}C_r^*)}{\displaystyle 5(\gamma C_2^*)^2}$ ,

• $a_6=\frac{\displaystyle (\delta _2+\beta _2+\gamma _2C_1^*+\alpha _2C_r^*)(\delta _{p2}+\alpha _{p2}C_r^*)}{\displaystyle 5(\gamma C_1^*)^2}$ , $a_7=\frac{\displaystyle (\delta _{p1}+\lambda +\alpha _{p1}C_r^*)(\delta _{p2}+\alpha _{p2}C_r^*)l_4}{\displaystyle 4\lambda ^2}$ ,

• $a_8=\frac{\displaystyle r_0+r_1C_1^*+r_2C_2^*}{\displaystyle (\alpha _1C_1^*+\alpha _2C_2^*-\nu C_a^*)^2}$ , $a_9=\frac{\displaystyle \delta _1+\beta _1+\gamma _1C_2^*+\alpha _1C_r^*}{\displaystyle (\alpha _1C_r^*)^2}$ ,

• $a_{10}=\frac{\displaystyle \delta _2+\beta _2+\gamma _2C_1^*+\alpha _2C_r^*}{\displaystyle (\alpha _2C_r^*)^2}.$

If the conditions (16)-(17) are fulfilled, $\dot{V}$ will be negative definite, indicating that V is a Lyapunov’s function and hence the theorem.

Appendix B

Proof: Assuming the following positive definite function about $E^*$,

$$\begin{aligned} U&=\frac{1}{2}[m_1(C_r-C_r^*)^2+m_2(C_1-C_1^*)^2+m_3(C_2-C_2^*)^2+m_4(C_{p1}-C_{p1}^*)^2\\ {} & +m_5(C_{p2}-C_{p2}^*)^2+m_6(C_a-C_a^*)^2]. \end{aligned}$$

(B1)

where $m_1 ,m_2 ,m_3 ,m_4,m_5$ and $m_6$ are to considered approximately as they are positive constants.

After some simplifications, we obtain the following by differentiating U with respect to ‘t’  using Eqs. (1–6),

$$\begin{aligned} \dot{U}=&-m_1(r_0+r_1C_1+r_2C_2)(C_r-C_r^*)^2-m_2(\delta _1+\beta _1+\gamma _1C_2+\alpha _1C_r)(C_1-C_1^*)^2\\ {} &-m_3(\delta _2+\beta _2+\gamma _2C_1+\alpha _2C_r)(C_2-C_2^*)^2-m_4(\delta _{p1}+\lambda +\alpha _{p1}C_r)(C_{p1}-C_{p1}^*)^2\\ {} &-m_5(\delta _{p2}+\alpha _{p2}C_r)(C_{p2}-C_{p2}^*)^2-m_6(k+\nu C_r)(C_a-C_a^*)^2\\ {} &-(m_1r_1C_r^*+m_2\alpha _1C_1^*)(C_r-C_r^*)(C_1-C_1^*)\\ {} &-(m_1r_2C_r^*+m_3\alpha _2C_2^*)(C_r-C_r^*)(C_2-C_2^*)\\ {} &-m_4\alpha _{p1}C_{p1}^*(C_r-C_r^*)(C_{p1}-C_{p1}^*)-m_5\alpha _{p2}C_{p2}^*(C_r-C_r^*)(C_{p2}-C_{p2}^*)\\ {} &+m_6(\alpha _1C_1^*+\alpha _2C_2^*-\nu C_a^*)(C_r-C_r^*)(C_a-C_a^*)\\ {} &-(m_2\gamma _1C_1^*+m_3\gamma _2C_2^*)(C_1-C_1^*)(C_2-C_2^*)+m_4\beta _1(C_1-C_1^*)(C_{p1}-C_{p1}^*)\\ {} &+m_4\beta _2(C_2-C_2^*)(C_{p1}-C_{p1}^*)+ m_5\gamma C_2^*(C_1-C_1^*)(C_{p2}-C_{p2}^*)\\ {} &+m_5\gamma C_1(C_2-C_2^*)(C_{p2}-C_{p2}^*)+m_5\lambda (C_{p1}-C_{p1}^*)(C_{p2}-C_{p2}^*)\\ {} &+m_6\alpha _1C_r(C_1-C_1^*)(C_a-C_a^*)+m_6\alpha _2C_r(C_2-C_2^*)(C_a-C_a^*). \end{aligned}$$

(B2)

Using minimum and maximum values of the variables involved in the model system and some manipulation, after choosing the following

• $m_1=m_2=m_3=1$,

• $m_4<\frac{\displaystyle 1}{\displaystyle 5}(\delta _{p1}+\lambda )$ $min\{b_1,b_2,b_3\},$

• $m_5<min\{b_4,b_5,b_6,b_7\}$ and

• $m_6<\frac{\displaystyle 4}{\displaystyle 15}k$ $min\{b_8,b_9,b_{10}\},$

where,

• $b_1=\frac{\displaystyle r_0}{\displaystyle (\alpha _{p1}C_{p1}^*)^2}$ , $b_2=\frac{\displaystyle \delta _1+\beta _1}{\displaystyle \beta _1^2}$ , $b_3=\frac{\displaystyle \delta _2+\beta _2}{\displaystyle \beta _2^2}$ , $b_4=\frac{\displaystyle r_0\delta _{p2}}{\displaystyle 5(\alpha _{p2}C_{p2}^*)^2}$ , $b_5=\frac{\displaystyle (\delta _1+\beta _1)\delta _{p2}}{\displaystyle 5(\gamma C_2^*)^2}$ ,

• $b_6=\frac{\displaystyle (\delta _2+\beta _2)\delta _{p2}}{ \displaystyle 5(\gamma Q_1 / (\delta _1+\beta _1))^2}$ , $b_7=\frac{\displaystyle (\delta _{p1}+\lambda )\delta _{p2}m_4}{\displaystyle 4\lambda ^2}$  , $b_8=\frac{\displaystyle r_0}{\displaystyle (\alpha _1C_1^*+\alpha _2C_2^*-\nu C_a^*)^2}$ ,

• $b_9=\frac{\displaystyle \delta _1+\beta _1}{\displaystyle (\alpha _1q /r_0)^2}$  , $b_{10}=\frac{\displaystyle \delta _2+\beta _2}{\displaystyle (\alpha _2q /r_0)^2}.$

If the conditions (19) – (21) are satisfied in the specified region of attraction $\varOmega$, $\dot{U}$ will be negative definite, indicating that U is a Lyapunov’s function and proving the theorem.