# 数学代写|数值方法作业代写numerical methods代考|CS3513

## 数学代写|数值方法作业代写numerical methods代考|Stability Analysis

The idea of the stability analysis came from Weidman et al. [44]. In their study, they noticed that there exists more than one solution called dual solutions. It is important to note that this analysis is introduced to determine which solution provides a good physical meaning to the flow (stable solution). Since we obtained the dual solutions, thus we are encouraged to determine which solutions are stable. To carry out this analysis, Equations (2)-(4) must be in unsteady case. Hence, the new dimensionless time variable is taken as $\tau=2 U t / x$. Thus, we have
$$\begin{gathered} \frac{\partial u}{\partial t}+u \frac{\partial u}{\partial x}+v \frac{\partial u}{\partial r}=\frac{v}{r} \frac{\partial}{\partial r}\left(r \frac{\partial u}{\partial r}\right), \ \frac{\partial T}{\partial t}+u \frac{\partial T}{\partial x}+v \frac{\partial T}{\partial r}=\frac{\alpha}{r} \frac{\partial}{\partial r}\left(r \frac{\partial T}{\partial r}\right)+\kappa\left[D_B \frac{\partial T}{\partial r} \frac{\partial C}{\partial r}+\frac{D_T}{T_{\infty}}\left(\frac{\partial T}{\partial r}\right)^2\right]+\frac{Q^}{\rho C_p}\left(T-T_{\infty}\right), \ \frac{\partial C}{\partial t}+u \frac{\partial C}{\partial x}+v \frac{\partial C}{\partial r}=\frac{D_B}{r} \frac{\partial}{\partial r}\left(r \frac{\partial C}{\partial r}\right)+\frac{D_T}{T_{\infty}} \frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial T}{\partial r}\right)-K^\left(C-C_{\infty}\right), \end{gathered}$$
and the new similarity transformations take the following form
$$\psi=v x f(\eta, \tau), \quad \eta=\frac{U r^2}{v x}, \quad \theta(\eta, \tau)=\frac{T-T_{\infty}}{T_w-T_{\infty}}, \quad \phi(\eta, \tau)=\frac{C-C_{\infty}}{C_w-C_{\infty}}, \quad \tau=\frac{2 U t}{x} .$$
Please note that the use of $\tau$ is related to an initial value problem that is consistent with the solution that will be attained in practice (physically realizable). Afterwards, encorporating Equation (19) into Equations (16)-(18), we obtains
$$\begin{gathered} 2 \frac{\partial}{\partial \eta}\left(\eta \frac{\partial^2 f}{\partial \eta^2}\right)+f \frac{\partial^2 f}{\partial \eta^2}-\frac{\partial^2 f}{\partial \eta \partial \tau}+\tau \frac{\partial f}{\partial \eta} \frac{\partial^2 f}{\partial \eta \partial \tau}-\tau \frac{\partial^2 f}{\partial \eta^2} \frac{\partial f}{\partial \tau}=0, \ \frac{2}{P r} \frac{\partial}{\partial \eta}\left(\eta \frac{\partial \theta}{\partial \eta}\right)+f \frac{\partial \theta}{\partial \eta}+2 \eta\left[N t\left(\frac{\partial \theta}{\partial \eta}\right)^2+N b \frac{\partial \theta}{\partial \eta} \frac{\partial \phi}{\partial \eta}\right]+\frac{1}{2} Q \theta-\frac{\partial \theta}{\partial \tau}+\tau \frac{\partial f}{\partial \eta} \frac{\partial \theta}{\partial \tau}-\tau \frac{\partial \theta}{\partial \eta} \frac{\partial f}{\partial \tau}=0, \ 2 \frac{\partial}{\partial \eta}\left(\eta \frac{\partial \phi}{\partial \eta}\right)+2 \frac{N t}{N b} \frac{\partial}{\partial \eta}\left(\eta \frac{\partial \theta}{\partial \eta}\right)+L e\left(f \frac{\partial \phi}{\partial \eta}-\frac{\partial \phi}{\partial \tau}\right)-\frac{1}{2} L e K \phi+L e \tau \frac{\partial f}{\partial \eta} \frac{\partial \phi}{\partial \tau}-L e \tau \frac{\partial \phi}{\partial \eta} \frac{\partial f}{\partial \tau}=0, \end{gathered}$$
together with the boundary conditions
$$\begin{gathered} f(c, \tau)=\frac{\varepsilon}{2} c+\tau \frac{\partial f}{\partial \tau}(c, \tau), \frac{\partial f}{\partial \eta}(c, \tau)=\frac{\varepsilon}{2}, \theta(c, \tau)=1, \quad \phi(c, \tau)=1, \ \frac{\partial f}{\partial \eta}(\eta, \tau) \rightarrow \frac{1}{2}(1-\varepsilon), \theta(\eta, \tau) \rightarrow 0, \phi(\eta, \tau) \rightarrow 0 \text { as } \eta \rightarrow \infty . \end{gathered}$$

## 数学代写|数值方法作业代写numerical methods代考|Graphical Results and Discussion

In this section, the graphical outputs of our problem are interpreted for various effects of the involved parameters. All the computations have been carried out for a wide range of values of the governing parameters; $c(0.1 \leq c \leq 0.2), \varepsilon(-4.3 \leq \varepsilon \leq 0.8), K(-0.1 \leq K \leq 0.2), Q(0 \leq Q \leq 0.4)$, $N b(0.1 \leq N b \leq 0.5), N t(0.1 \leq N t \leq 0.5)$ and for a fixed values of $P r=2$ and $L e=1$. Equations (8)-(10) along with the conditions (11) are computed numerically via bvp4c function that implemented in MATLAB software. Besides the shooting method, there is a new effective method for solving the boundary value problem for ordinary differential equations that is bvp4c package. Mathematically, this package uses the finite difference methods, in which the output is attained using an initial guess provided at the starting mesh point and resize the step to obtain the particular certainty. Nevertheless, to use this package, the boundary value problem must reduce to first order system of ordinary differential equations. To validate the accuracy of the present results, we have initially compared our results to those of Ahmad et al. [42] and Salleh et al. [43]. In this respect, Table 1 shows a comparison value of shear stress $f^{\prime \prime}(c)$ for $\varepsilon=L e=Q=K=0$ for some of the thickness of the needle $c$ when $\operatorname{Pr}=1$. An excellent agreement is observed in these studies.

The effect of needle thickness $c$ on the velocity, temperature and concentration profiles are graphically presented in Figure $2 a-c$. It is noticed from the plots that the velocity, temperature and concentration profiles for upper branch solution increase with the increasing value of needle thickness. Similar observation is found for momentum, thermal and concentration boundary layer thicknesses for the upper branch as the $c$ increase. Mathematically, the shape of graphs obtained in these profiles has asymptote hehaviors and it fulfills the requirement of houmdary eondition (11). One ean see that an increment in the needle thickness decreases the numerical values of surface shear stress $f^{\prime \prime}(c)$, heat flux $-\theta^{\prime}(c)$ and also mass flux $-\phi^{\prime}(c)$. These phenomena are clearly shown in Figure $3 a-c$. This situation occurs due to an increase in the momentum, thermal and concentration boundary layer thicknesses on the surface, and consequently decline the shear stress and slow down the heat and mass transfers from the surface to the flow. Physically, the slender surface of the needle makes heat and mass to diffuse through it quickly compared to thick surface. In addition, the critical values of $\varepsilon$, by which the upper and lower branch solutions connected, are noticed to decrease as the needle thickness reduces. In other words, we can say that the needle thickness has a significant effect on the existence of the dual solutions.

# 数值方法代考

## 数学代写|数值方法作业代写numerical methods代考|Stability Analysis

(稳定的解决方案) 。由于我们获得了对偶解，因此我们鼓励确定哪些解是稳定的。为了进行这种分 析，等式 (2) – (4) 必须在不稳定的情况下。因此，新的无量纲时间变量取为 $\tau=2 U t / x$. 因此，我们 有

$$\psi=v x f(\eta, \tau), \quad \eta=\frac{U r^2}{v x}, \quad \theta(\eta, \tau)=\frac{T-T_{\infty}}{T_w-T_{\infty}}, \quad \phi(\eta, \tau)=\frac{C-C_{\infty}}{C_w-C_{\infty}}, \quad \tau=\frac{2 U t}{x} .$$

$$2 \frac{\partial}{\partial \eta}\left(\eta \frac{\partial^2 f}{\partial \eta^2}\right)+f \frac{\partial^2 f}{\partial \eta^2}-\frac{\partial^2 f}{\partial \eta \partial \tau}+\tau \frac{\partial f}{\partial \eta} \frac{\partial^2 f}{\partial \eta \partial \tau}-\tau \frac{\partial^2 f}{\partial \eta^2} \frac{\partial f}{\partial \tau}=0, \frac{2}{\operatorname{Pr}} \frac{\partial}{\partial \eta}\left(\eta \frac{\partial \theta}{\partial \eta}\right)+f \frac{\partial \theta}{\partial \eta}+2 \eta\left[N t\left(\frac{\partial \theta}{\partial \eta}\right)\right.$$

$$f(c, \tau)=\frac{\varepsilon}{2} c+\tau \frac{\partial f}{\partial \tau}(c, \tau), \frac{\partial f}{\partial \eta}(c, \tau)=\frac{\varepsilon}{2}, \theta(c, \tau)=1, \quad \phi(c, \tau)=1, \frac{\partial f}{\partial \eta}(\eta, \tau) \rightarrow \frac{1}{2}(1-\varepsilon), \theta(\eta, \tau) \rightarrow 0$$

## 数学代写|数值方法作业代写numerical methods代考|Graphical Results and Discussion

(11) 的要求。可以看出，针厚度的增加会降低表面剪切应力的数值 $f^{\prime \prime}(c)$, 热通量 $-\theta^{\prime}(c)$ 还有质量通量 $-\phi^{\prime}(c)$. 这些现象如图所示 $3 a-c$. 这种情况的发生是由于表面上的动量、热和浓度边界层厚度的增加， 从而降低了勖切应力并减慢了从表面到流动的热量和质量传递。在物理上，与厚表面相比，针的细长表 面使热量和质量通过它快速扩散。此外，临界值 $\varepsilon$ ，通过连接上下分支解决方案，注意到随着针厚度的减 小而减小。换句话说，我们可以说针的粗细对对偶解的存在有显着的影响。

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