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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
稳定性分析的想法来自 Weidman 等人。[44]。在他们的研究中，他们注意到存在不止一种解决方案，称
(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
新的相似变换采用以下形式
$$
\psi=v x f(\eta, \tau), \quad \eta=\frac{U r^2}{v x}, \quad \theta(\eta, \tau)=\frac{T-T_{\infty}}{T_w-T_{\infty}}, \quad \phi(\eta, \tau)=\frac{C-C_{\infty}}{C_w-C_{\infty}}, \quad \tau=\frac{2 U t}{x} .
$$
请注意，使用 $\tau$ 与一个初始值问题相关，该初始值问题与将在实践中获得的解决方案（物理上可实现) 致。然后，将方程 (19) 代入方程 (16) – (18)，我们得到
$$
2 \frac{\partial}{\partial \eta}\left(\eta \frac{\partial^2 f}{\partial \eta^2}\right)+f \frac{\partial^2 f}{\partial \eta^2}-\frac{\partial^2 f}{\partial \eta \partial \tau}+\tau \frac{\partial f}{\partial \eta} \frac{\partial^2 f}{\partial \eta \partial \tau}-\tau \frac{\partial^2 f}{\partial \eta^2} \frac{\partial f}{\partial \tau}=0, \frac{2}{\operatorname{Pr}} \frac{\partial}{\partial \eta}\left(\eta \frac{\partial \theta}{\partial \eta}\right)+f \frac{\partial \theta}{\partial \eta}+2 \eta\left[N t\left(\frac{\partial \theta}{\partial \eta}\right)\right.
$$
连同边界条件
$$
f(c, \tau)=\frac{\varepsilon}{2} c+\tau \frac{\partial f}{\partial \tau}(c, \tau), \frac{\partial f}{\partial \eta}(c, \tau)=\frac{\varepsilon}{2}, \theta(c, \tau)=1, \quad \phi(c, \tau)=1, \frac{\partial f}{\partial \eta}(\eta, \tau) \rightarrow \frac{1}{2}(1-\varepsilon), \theta(\eta, \tau) \rightarrow 0
$$
数学代写|数值方法作业代写numerical methods代考|Graphical Results and Discussion
在本节中，我们的问题的图形输出被解释为所涉及参数的各种影响。所有的计算都是针对广泛的控制参 数值进行的; $c(0.1 \leq c \leq 0.2), \varepsilon(-4.3 \leq \varepsilon \leq 0.8), K(-0.1 \leq K \leq 0.2), Q(0 \leq Q \leq 0.4)$ ， $N b(0.1 \leq N b \leq 0.5), N t(0.1 \leq N t \leq 0.5)$ 对于固定值 $P r=2$ 和 $L e=1$. 方程 (8)-(10) 以及条件 (11) 通过在 MATLAB 软件中实现的 bvp4c 函数进行数值计算。除了射击法之外，还有一种新的有效求解常微 分方程边值问题的方法，就是bvp4c包。在数学上，此包使用有限差分方法，其中使用在起始网格点提供 的初始猜测来获得输出，并调整步长以获得特定的确定性。然而，要使用这个包，边值问题必须减少到 常微分方程的一阶系统。为了验证目前结果的准确性，我们最初将我们的结果与 Ahmad 等人的结果进 行了比较。[42] 和 Salleh 等人。[43]。在这方面，表1显示了盼切应力的比较值 $f^{\prime \prime}(c)$ 为了 $\varepsilon=L e=Q=K=0$ 对于一些粗细的针 $c$ 什么时候 $\operatorname{Pr}=1$. 在这些研究中观察到了极好的一致性。
针粗细的影响 $c$ 对速度、温度和浓度曲线的影响如图所示 $2 a-c$. 从图中可以看出，上支溶液的速度、温 度和浓度分布随責针厚度值的增加而增加。对于上分支的动量、热力和浓度边界层厚度也发现了类似的 观察结果，因为 $c$ 增加。在数学上，在这些轮廓中获得的图形形状具有渐近线特征，并且满足人类条件
(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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