# 数学代写|数值方法作业代写numerical methods代考|MATH131

## 数学代写|数值方法作业代写numerical methods代考|Governing Formula and Modeling

A steady nanofluid flow past a horizontal thin needle is examined. The geometry of the problem is illustrated in Figure 1 with $u$ and $v$ denoting $x$ and $r$ components of velocity, respectively, and $r=$ $R(x)=(v c x / U)^{1 / 2}$ represents the needle radius. The needle is considered to move with uniform velocity $U_w$ in the same or reverse direction of the external flow of constant velocity $U_{\infty}$ with the composite velocity $U=U_w+U_{\infty}$. It is assumed that $T_w$ and $C_w$ are the constant wall temperature and nanoparticle concentration and as $r \rightarrow \infty$, the ambient temperature and nanoparticle fraction are $T_{\infty}$ and $C_{\infty}$ such that $T_w>T_{\infty}$ and $C_w>C_{\infty}$. In view of small needle size, the pressure gradient is ignored, however, the transverse curvature effect is required.

By using Buongiorno’s nanofluid model, the relevant governing boundary layer systems for the flow are $[6,42]$
$$\begin{gathered} \frac{\partial}{\partial x}(r u)+\frac{\partial}{\partial r}(r v)=0, \ u \frac{\partial u}{\partial x}+v \frac{\partial u}{\partial r}=\frac{v}{r} \frac{\partial}{\partial r}\left(r \frac{\partial u}{\partial r}\right), \ u \frac{\partial T}{\partial x}+v \frac{\partial T}{\partial r}=\frac{\alpha}{r} \frac{\partial}{\partial r}\left(r \frac{\partial T}{\partial r}\right)+\kappa\left[D_B \frac{\partial T}{\partial r} \frac{\partial C}{\partial r}+\frac{D_T}{T_{\infty}}\left(\frac{\partial T}{\partial r}\right)^2\right]+\frac{Q^}{\rho C_p}\left(T-T_{\infty}\right), \ u \frac{\partial C}{\partial x}+v \frac{\partial C}{\partial r}=\frac{D_B}{r} \frac{\partial}{\partial r}\left(r \frac{\partial C}{\partial r}\right)+\frac{D_T}{T_{\infty}} \frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial T}{\partial r}\right)-K^\left(C-C_{\infty}\right) . \end{gathered}$$
The physical boundary restrictions are
$$u=U_w, v=0, T=T_w, \quad C=C_w \text { at } r=R(x),$$
$$u \rightarrow U_{\infty}, T \rightarrow T_{\infty}, C \rightarrow C_{\infty} \text { as } r \rightarrow \infty,$$
in which $v$ is kinematic viscosity, $T$ is the temperature of nanofluid, $C$ is the concentration of nangrarlicles, $a$ is the heremal diffusivilys $p$ is the densily, $C_p$ is he heal capacily at uniform pressure, $\kappa=\left(\rho C_p\right)_s /\left(\rho C_p\right)_f$ is the proportion of effectual heat capacity of nanofluid in which subscripts ‘ $s$ ‘ and ‘ $\mathrm{f}^{\prime}$ ‘ refer to solid nanoparticle and base fluid, $Q^=Q_0 / x$ is the dimensionless heat generation, $K^=K_0 / x$ is the dimensionless reaction rate, $Q_0$ is the heat generation coefficient and $K_0$ is the chemical reaction coefficient. It is worth mentioning that the dimensionless parameters $Q^$ and $K^$ are the function of $x$ and its value varies locally throughout the flow motion. Besides, $D_B$ and $D_T$ are Brownian and thermophoresis diffusion coefficients, respectively.

## 数学代写|数值方法作业代写numerical methods代考|The similarity transformation technique

The similarity transformation technique has been used for obtaining the ordinary differential equations. Hence, the following non-dimensional parameters are introduced
$$\psi=v x f(\eta), \quad \eta=\frac{U r^2}{v x}, \quad \theta(\eta)=\frac{T-T_{\infty}}{T_w-T_{\infty}}, \quad \phi(\eta)=\frac{C-C_{\infty}}{C_w-C_{\infty}},$$
where the stream functions are given as
$$u=r^{-1} \frac{\partial \psi}{\partial r}, \quad v=-r^{-1} \frac{\partial \psi}{\partial x} .$$
The stream functions (7) satisfies the continuity Equation (1). Using Equations (6) and (7), we obtain the following equations
$$\begin{gathered} 2 \eta f^{\prime \prime \prime}+2 f^{\prime \prime}+f f^{\prime \prime}=0 \ \frac{2}{P r}\left(\eta \theta^{\prime}\right)^{\prime}+f \theta^{\prime}+2 \eta\left(N t \theta^{\prime 2}+N b \theta^{\prime} \phi^{\prime}\right)+\frac{1}{2} Q \theta=0, \ 2\left(\eta \phi^{\prime}\right)^{\prime}+2 \frac{N t}{N b}\left(\eta \theta^{\prime}\right)^{\prime}+\text { Lef } \phi^{\prime}-\frac{1}{2} L e K \phi=0 . \end{gathered}$$
Also, the boundary condition could be rewritten as
$$\begin{gathered} f(c)=\frac{\varepsilon}{2} c, f^{\prime}(c)=\frac{\varepsilon}{2}, \theta(c)=1, \phi(c)=1, \ f^{\prime}(\eta) \rightarrow \frac{1}{2}(1-\varepsilon), \theta(\eta) \rightarrow 0, \phi(\eta) \rightarrow 0 \text { as } \eta \rightarrow \infty, \end{gathered}$$
where prime denotes the differentiation with regard to similarity variable $\eta$. Besides, assume $\eta=c$ to represent size or thickness of the needle.

Here, Pr, $N t, N b, Q, L e$ and $\varepsilon$ represent the Prandtl number, thermophoresis parameter, Brownian motion parameter, heat generation parameter, Lewis number and velocity ratio parameter. $K$ is the chemical reaction parameter with $K>0$ represents a destructive reaction, and $K<0$ represents generative reaction.

# 数值方法代考

## 数学代写|数值方法作业代写numerical methods代考|Governing Formula and Modeling

$$u=U_w, v=0, T=T_w, \quad C=C_w \text { at } r=R(x),$$
$u \rightarrow U_{\infty}, T \rightarrow T_{\infty}, C \rightarrow C_{\infty}$ as $r \rightarrow \infty$

## 数学代写|数值方法作业代写numerical methods代考|The similarity transformation technique

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

$$u=r^{-1} \frac{\partial \psi}{\partial r}, \quad v=-r^{-1} \frac{\partial \psi}{\partial x}$$

$$2 \eta f^{\prime \prime \prime}+2 f^{\prime \prime}+f f^{\prime \prime}=0 \frac{2}{P r}\left(\eta \theta^{\prime}\right)^{\prime}+f \theta^{\prime}+2 \eta\left(N t \theta^{\prime 2}+N b \theta^{\prime} \phi^{\prime}\right)+\frac{1}{2} Q \theta=0,2\left(\eta \phi^{\prime}\right)^{\prime}+2 \frac{N t}{N b}\left(\eta \theta^{\prime}\right)^{\prime}$$

$$f(c)=\frac{\varepsilon}{2} c, f^{\prime}(c)=\frac{\varepsilon}{2}, \theta(c)=1, \phi(c)=1, f^{\prime}(\eta) \rightarrow \frac{1}{2}(1-\varepsilon), \theta(\eta) \rightarrow 0, \phi(\eta) \rightarrow 0 \text { as } \eta \rightarrow \infty,$$

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