Group Method of Uniform Surface Heat Flux from a Vertical Cone Using Laminar Free Convection

Group Method of Uniform Surface Heat Flux from a Vertical Cone Using Laminar Free Convection

Tirupathi Maheshwaran Bapuji Pullepu* Sandra Pinelas

Department of Mathematics, SRMIST, Kattankulathur 603203, India

Department of Statistics, Dwaraka Doss Goverdhan Doss Vaishnav College, Chennai 600106, India

Departamento de Ciências Exatas e Engenharia, Academia Militar, 2720-113 Amadora and Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, Aveiro 3810193, Portugal

Corresponding Author Email:
bapujip@yahoo.com
Page:
749-756
|
DOI:
https://doi.org/10.18280/mmep.100303
19 December 2022
|
Revised:
28 February 2023
|
Accepted:
13 March 2023
|
Available online:
30 June 2023
| Citation

OPEN ACCESS

Abstract:

In this study, the Group Transformation method is employed to simulate the laminar free convection problem involving a viscous incompressible fluid on a vertically oriented cone with a uniform heat flux on its surface. The non-dimensional governing partial differential equations (PDEs) and their boundary conditions are reduced to ordinary differential equations (ODEs) with corresponding appropriate conditions. To solve the resulting non-linear ODEs, the Range-Kutta shooting method is applied. The temperature and velocity fields are graphically presented for various parameters, such as the semi-vertical angle and Prandtl number. Furthermore, the local Nusselt number and skin fraction are analyzed numerically.

Keywords:

group method, heat transfer, natural convection, ordinary differential equations, partial differential equations, vertical cone

1. Introduction

2. Mathematical Formulation

The present analysis, a consistent axisymmetric free convection of incompressible viscous flow is studied on a vertical cone having uniformly applied heat flux on its surface. It is assumed that viscous dissipation effects and pressure gradient along the boundary layer are negligible.

1.png

Figure 1. Physical model and co-ordinate system

The co-ordinate system is as shown in Figure 1, x signifying along the cone surface from the apex (x=0) and y signifying the distance normal to a cone surface. Temperature T'w on a cone surface is kept same, where temperature value is uniform, far from the temperature of cone surface. With the thermal buoyancy effect, an upward flow is created for T'w>T'. Other than the fluctuations in the densities in buoyancy force, where fluid properties are kept constant. For energy, momentum and continuity, their governing equations of boundary layer with Boussinesq approximations are as follows:

${{(ru)}_{x}}+{{(rv)}_{y}}=0\,$                                        (1)

$u{{u}_{x}}+v{{u}_{y}}=g\beta ({{T}^{'}}-{{T}^{'}}_{\infty })\cos \phi +\vartheta {{u}_{yy}}$                                   (2)

$u{{T}^{'}}_{x}+v{{T}^{'}}_{y}=\alpha {{T}^{'}}_{yy}$                             (3)

The basic and restrictive conditions are as follows:

\begin{align} & u(x,0)=\,v(x,0)=0,\,\,{{T}^{'}}_{y}=\frac{-{{q}_{w}}(x)}{k}\,\,\,\,\,\,at\,\,\,\,y=0 \\ & u(0,y)=0,\,\,{{T}^{'}}(0,\infty )={{T}_{\infty }}^{'}\,\,\,\,\,\,\,\,\,at\,\,\,\,x=0\,\, \\ & u(x,\infty )=0,\,\,\,\,{{T}^{'}}(x,\infty )=T\,\,\,\,\,\,\,at\,\,\,\,y\to \infty \\\end{align}                                (4)

Local Nusselt number Nux and Local skin friction τx are given by:

\begin{align} & {{\tau }_{x}}=\mu {{\left( {{u}_{y}} \right)}_{y=0}}\,\,\,\,\,\, \\ & N{{u}_{x}}=\frac{x}{{{T}^{'}}_{w}-{{T}^{'}}_{\infty }}{{\left( -{{T}^{'}}_{y} \right)}_{y=0}} \\\end{align}                        (4a)

Utilization of enclosed dimensionless quantities:

\begin{align} & {{x}^{*}}=\frac{x}{l},\,\,\,\,\,{{y}^{*}}=\frac{y}{l}\,{{({{G}_{{{r}_{l}}}})}^{\frac{1}{5}}},\,\,\,\,{{r}^{*}}=\frac{r}{l},\,\,\text{where}\,r=x\sin \varphi \\& {{u}^{*}}=\frac{ul}{\vartheta }{{({{G}_{{{r}_{l}}}})}^{\frac{-2}{5}}},\,\,\,\,{{v}^{*}}=\frac{vl}{\vartheta }{{({{G}_{{{r}_{l}}}})}^{\frac{-1}{5}}},\,\,\,\,T=\frac{({{T}^{'}}-{{T}^{'}}_{\infty }){{({{G}_{{{r}_{l}}}})}^{\frac{1}{5}}}}{ql/k}\, \\ & {{G}_{{{r}_{l}}}}=\frac{g\beta \,\,\,q\,{{l}^{4}}}{{{\vartheta }^{2}}k},\,\,\,\,\,\,\,\Pr =\frac{\vartheta }{\alpha } \\\end{align}                      (5)

Using (5), the reduction of Eqns. (1) to (3) yields the dimensionless form as follows:

${{({{r}^{*}}{{u}^{*}})}_{{{x}^{*}}}}+{{({{r}^{*}}{{v}^{*}})}_{{{y}^{*}}}}=0$                           (6)

${{u}^{*}}{{u}^{*}}_{{{x}^{*}}}+{{v}^{*}}{{u}^{*}}_{{{y}^{*}}}=T\cos \phi +{{u}^{*}}_{{{y}^{*}}{{y}^{*}}}$                 (7)

${{u}^{*}}{{T}_{{{x}^{*}}}}+{{v}^{*}}{{T}_{{{y}^{*}}}}=\frac{1}{\Pr }{{T}_{{{y}^{*}}{{y}^{*}}}}$                         (8)

Given the following boundary conditions:

\begin{align} & {{u}^{*}}({{x}^{*}},0)=0,\,\,\,\,\,{{v}^{*}}({{x}^{*}},0)=0,\,\,{{T}_{{{y}^{*}}}}=-1\,\,\,at\,\,\,{{y}^{*}}=0 \\ & {{u}^{*}}(0,{{y}^{*}})=0,\,\,\,\,\,T(0,{{y}^{*}})=0\,\,\,\,\,\,\,\,\,at\,\,\,\,{{x}^{*}}=0\, \\ & {{u}^{*}}({{x}^{*}},\infty )=0,\,\,\,\,\,T({{x}^{*}},\infty )=0\,\,\,\,as\,\,\,{{y}^{*}}\to \infty \\\end{align}                   (9)

From Eq. (4a), the local dimensionless local Nusselt number $N u_{x^*}$ and skin-friction $\tau_{x^*}$ becomes:

\begin{align} & {{\tau }_{{{x}^{*}}}}=G{{r}_{l}}^{3/5}{{\left( {{u}^{*}}_{{{y}^{*}}} \right)}_{{{y}^{*}}=0}} \\ & N{{u}_{{{x}^{*}}}}=\frac{{{x}^{*}}G{{r}_{l}}^{1/5}}{{{T}_{{{y}^{*}}=0}}}{{\left( -{{T}_{{{y}^{*}}}} \right)}_{{{y}^{*}}=0}} \\\end{align}                              (9a)

Similarity variables are follows:

\begin{align} & \omega ={{x}^{*}}{{r}^{*}}M({{x}^{*}},{{y}^{*}})\,\,\,\,\,\,\,\, \\ & T({{x}^{*}},{{y}^{*}})={{x}^{*}}\,\,T({{x}^{*}},{{y}^{*}}) \\\end{align}                           (9b)

In order to reduce the number of equations from 3 to 2, we introduce the stream function such that:

${{u}^{*}}=\frac{1}{{{r}^{*}}}{{\omega }_{{{y}^{*}}}}and\,\,{{v}^{*}}=\frac{-1}{{{r}^{*}}}{{\omega }_{{{x}^{*}}}}$                             (9c)

Condition (6) is satisfied via a stream function, and (7) and (8) are switched to accompanying conditions.

${{\omega }_{{{y}^{*}}}}{{\left( \frac{1}{{{r}^{*}}}{{\omega }_{{{y}^{*}}}} \right)}_{{{x}^{*}}}}-\frac{1}{{{r}^{*}}}{{\omega }_{{{x}^{*}}}}{{\omega }_{{{y}^{*}}{{y}^{*}}}}=T{{r}^{*}}\cos \phi +{{\omega }_{{{y}^{*}}{{y}^{*}}{{y}^{*}}}}$                     (10)

$\frac{1}{{{r}^{*}}}\left( {{\omega }_{{{y}^{*}}}}{{T}_{{{x}^{*}}}}-{{\omega }_{{{x}^{*}}}}{{T}_{{{y}^{*}}}} \right)=\frac{1}{\Pr }{{T}_{{{y}^{*}}{{y}^{*}}}}$                   (11)

Boundary condition (9) communicated as:

\begin{align} & \underset{{{y}^{*}}\to 0}{\mathop{\lim }}\,{{\omega }_{{{y}^{*}}}}=0\,\,\,\,\,\,\,\underset{{{y}^{*}}\to 0}{\mathop{\lim }}\,{{\omega }_{{{x}^{*}}}}=0\,\,\,\,\underset{{{y}^{*}}\to 0}{\mathop{\lim }}\,{{T}_{{{y}^{*}}}}=-1 \\ & \underset{{{y}^{*}}\to \infty }{\mathop{\lim }}\,{{\omega }_{{{y}^{*}}}}=0\,\,\,\,\,\,\underset{{{y}^{*}}\to \infty }{\mathop{\lim }}\,T=0 \\\end{align}                    (12）

3. Group Formulation of the Problem

Method of solution depend on the application of a one parameter group transformation to the PDE (10) to (11). Under this transformation the two independent variables will be reduced by one and the differential Eqns. (10) and (11) transform into ODE in only one independent variable, which is the similarity variable: $h: \bar{P}=C^p(b) P+k^p(b)$.

3.1 The group systematic formulation

The procedure is initiated with the group G, a class of one-parameter ‘b’ of the form:

\left. \begin{align} & \overline{x}={{C}^{{{x}^{*}}}}(b){{x}^{*}}+{{k}^{{{x}^{*}}}}(b)\,\,\,\,\,\,\,\overline{y}={{C}^{{{y}^{*}}}}(b){{y}^{*}}+{{k}^{{{y}^{*}}}}(b) \\ & \overline{\omega }={{C}^{\omega }}(b)\omega +{{k}^{\omega }}(b)\,\,\,\,\,\,\,\,r={{C}^{{{r}^{*}}}}(b){{r}^{*}}+{{k}^{{{r}^{*}}}}(b) \\ & \overline{T}={{C}^{T}}(b)T+{{k}^{T}}(b) \\\end{align} \right\}                       (13)

\left. \begin{align} & \overline{{{P}_{i}}}=\left( {}^{{{C}^{s}}}/{}_{{{C}^{i}}} \right){{P}_{i}} \\ & \overline{{{P}_{ij}}}=\left( {}^{{{C}^{s}}}/{}_{{{C}^{i}}{{C}^{j}}} \right){{P}_{ij}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,i,\,\,\,j=x,y \\\end{align} \right\}\,                           (14)

3.2 Transformation

To transform the differential equation, transformation of derivatives is obtained from G via chain rule operations.

P stands for ω, r*, T.

Under the (13) and (14), Eqns. (10) and (11) are changed invariantly, becomes:

\begin{align} & {{\overline{\omega }}_{y}}{{\left( \frac{1}{r}{{\overline{\omega }}_{y}} \right)}_{x}}-\frac{1}{r}{{\overline{\omega }}_{x}}{{\overline{\omega }}_{yy}}-r\overline{T}\overline{\cos \varphi }-{{\overline{\omega }}_{yyy}}= \\ & {{E}_{1}}(a)\left[ {{\omega }_{{{y}^{*}}}}{{\left( \frac{1}{{{r}^{*}}}{{\omega }_{{{y}^{*}}}} \right)}_{{{x}^{*}}}}-\frac{1}{{{r}^{*}}}{{\omega }_{{{x}^{*}}}}{{\omega }_{{{y}^{*}}{{y}^{*}}}}-{{r}^{*}}T\cos \varphi -{{\omega }_{{{y}^{*}}{{y}^{*}}{{y}^{*}}}} \right]\, \\\end{align}                      (15)

\begin{align} & \frac{1}{r}\left[ {{\overline{\omega }}_{y}}{{\overline{T}}_{x}}-{{\overline{\omega }}_{x}}{{\overline{T}}_{y}} \right]-\frac{1}{\Pr }{{\overline{T}}_{yy}} \\ & ={{E}_{2}}(a)\left[ \frac{1}{{{r}^{*}}}\left( {{\omega }_{{{y}^{*}}}}{{T}_{{{x}^{*}}}}-{{\omega }_{{{x}^{*}}}}{{T}_{{{y}^{*}}}} \right)-\frac{1}{\Pr }{{T}_{{{y}^{*}}{{y}^{*}}}} \right]\,\, \\\end{align}                  (16)

where, E1(a), E2(a) are functions or Constant.

\begin{align} & \frac{{{C}^{\omega }}{{C}^{T}}}{{{C}^{{{r}^{*}}}}{{C}^{{{x}^{*}}}}{{C}^{{{y}^{*}}}}}\left[ \frac{1}{{{r}^{*}}}{{\omega }_{{{y}^{*}}}}{{\omega }_{{{x}^{*}}{{y}^{*}}}}-\frac{1}{{{r}^{*}}^{2}}{{\left( {{\omega }_{{{y}^{*}}}} \right)}^{2}}{{r}^{*}}_{{{x}^{*}}}-\frac{1}{{{r}^{*}}}{{\omega }_{{{y}^{*}}{{y}^{*}}}} \right] \\ & -{{r}^{*}}{{C}^{{{r}^{*}}}}{{C}^{T}}T\cos \phi -\frac{{{C}^{\omega }}}{{{({{C}^{{{y}^{*}}}})}^{2}}}{{\omega }_{{{y}^{*}}{{y}^{*}}{{y}^{*}}}}+{{A}_{1}}(a) \\ & ={{V}_{1}}(a)\left[ {{\omega }_{{{y}^{*}}}}{{\left( \frac{1}{{{r}^{*}}}{{\omega }_{{{y}^{*}}}} \right)}_{{{x}^{*}}}}-\frac{1}{{{r}^{*}}}{{\omega }_{{{x}^{*}}}}{{\omega }_{{{y}^{*}}{{y}^{*}}}}-T{{r}^{*}}\cos \phi -{{\omega }_{{{y}^{*}}{{y}^{*}}{{y}^{*}}}} \right] \\\end{align}                      (17)

\begin{align} & \frac{{{C}^{\omega }}{{C}^{T}}}{{{C}^{{{r}^{*}}}}{{C}^{x}}{{C}^{y}}}\frac{1}{{{r}^{*}}}\left( {{\omega }_{{{y}^{*}}}}{{T}_{{{x}^{*}}}}-{{\omega }_{{{x}^{*}}}}{{T}_{{{y}^{*}}}} \right)-\frac{1}{\Pr }\frac{{{C}^{T}}}{{{({{C}^{{{y}^{*}}}})}^{2}}}{{T}_{{{y}^{*}}{{y}^{*}}}}+{{A}_{2}}(a) \\& ={{V}_{2}}(a)\left[ \frac{1}{{{r}^{*}}}\left( {{\omega }_{{{y}^{*}}}}{{T}_{{{x}^{*}}}}-{{\omega }_{{{x}^{*}}}}{{T}_{{{y}^{*}}}} \right)-\frac{1}{\Pr }{{T}_{{{y}^{*}}{{y}^{*}}}} \right] \\\end{align}                         (18)

where,

\begin{aligned} & \mathrm{A}_1(a)=\sum_{n=1}^{\infty}\left(\begin{array}{c}-1 \\ n\end{array}\right)\left(\frac{k^{r^*}}{C^{r^*} r^*}\right)^m \frac{\left(C^\omega\right)^2}{C^{r^*} C^{x^*}\left(C^{y^*}\right)^2} \frac{1}{r^*}\left(\omega_{y^*} \omega_{x^* y^*}-\omega_{x^*} \omega_{y^* y^*}\right) \\ & -\sum_{n=1}^{\infty}(-2)\left(\frac{k^{r^*}}{C^{r^*} r^*}\right)^m \frac{\left(C^\omega\right)^2}{C^{r^*} C^{x^*}\left(C^{y^*}\right)^2} \frac{1}{r^{* 2}}\left(\omega_{y^*}\right)^2 r_{x^*}^* \\ & -\left(C^* k^{r^*} r^*+k^{r^*} C^{r^*} T\right)\left(C^{\cos \phi} \cos \phi+k^{\cos \phi}\right)\end{aligned}                 (19)

\begin{aligned} & A_2(a)=\sum_{n=1}^{\infty}\left(\begin{array}{l}-1 \\ n\end{array}\right)\left(\frac{k^{r^*}}{C^{r^*} r^*}\right)^n \\ & \frac{C^\omega C^T}{C^{r^*} C^{x^*} C^{y^*}} \frac{1}{r^*}\left(\omega_{y^*} T_{x^*}-\omega_{x^*} T_{y^*}\right)\end{aligned}             (20)

Invariance of Eqns. (17) and (18) $\Rightarrow A_1(b)=0=A_2(b)$.

By substituting, the above equations are satisfied.

${{k}^{{{r}^{*}}}}={{k}^{T}}={{k}^{{{y}^{*}}}}=0$                           (21)

$\frac{{{({{C}^{\omega }})}^{2}}}{{{C}^{{{r}^{*}}}}{{C}^{{{x}^{*}}}}{{({{C}^{{{y}^{*}}}})}^{2}}}=\frac{{{C}^{\omega }}}{{{\left( {{C}^{{{y}^{*}}}} \right)}^{3}}}=\frac{{{C}^{\omega }}}{{{C}^{{{y}^{*}}}}}={{C}^{{{r}^{*}}}}{{C}^{T}}={{V}_{1}}(b)$                        (22)

$\frac{{{C}^{\omega }}{{C}^{T}}}{{{C}^{{{r}^{*}}}}{{C}^{{{x}^{*}}}}{{C}^{{{y}^{*}}}}}=\frac{{{C}^{T}}}{{{\left( {{C}^{{{y}^{*}}}} \right)}^{2}}}={{V}_{2}}(b)$                      (23)

These yields:

${{C}^{{{x}^{*}}}}={{\left( {{C}^{{{y}^{*}}}} \right)}^{2}},\,\,\,{{C}^{{{r}^{*}}}}=\frac{1}{{{\left( {{C}^{{{y}^{*}}}} \right)}^{2}}},\,\,\,\,{{C}^{\omega }}={{C}^{{{y}^{*}}}}$     (24)

Boundary Eqns. (19) and (20) are also invariant:

${{k}^{{{r}^{*}}}}={{k}^{T}}=0,\,\,\,\,\,\,{{C}^{T}}=1\,$                       (25)

Finally, a limited, exhaustive G that is constantly changing, (17) and (18) conditions and the most extreme conditions (19) and (20) We get G from the above conditions.

G=\left\{ \begin{align} & \overline{x}={{({{C}^{y}})}^{2}}{{x}^{*}}+{{k}^{{{x}^{*}}}} \\ & \overline{y}={{C}^{{{y}^{*}}}}{{y}^{*}} \\ & r=\frac{{{r}^{*}}}{{{\left( {{C}^{{{y}^{*}}}} \right)}^{2}}} \\ & \overline{\omega }={{C}^{{{y}^{*}}}}\omega +{{k}^{\omega }} \\ & \overline{T}=T \\\end{align} \right.                                    (26)

3.3 Group transformation of the boundary layer flow equations

Our aim is to make use of group methods ro represent the problem in the form of an ODE in a single independent variable. Then we have to proceed in our analysis to obtain a complete set of absolute invariants.

If $v=v\left(x^*, y^*\right)$ is the absolute invariants of the independent variables x* and y*, then:

${{m}_{j}}({{x}^{*}},{{y}^{*}},\,\omega ,\,\,\phi ,{{r}^{*}},T)={{M}_{j}}(\upsilon ({{x}^{*}},{{y}^{*}}))\,\,\,\,j=1,2,3$                       (27)

In group theory, using central assumptions is that if function mj (x*, y*, ω, ϕ, r*, T) satisfies the even derivative state of the first prompt, then one parameter group is levelled and unchanging. It states that:

$\begin{array}{ll}\sum_{i=1}^5\left(\chi_i w_i+\delta_i\right) \frac{\partial m}{\partial w_i}=0, & w_i=x^*, y^*, \omega, r^*, T \\ \text { where } \quad \chi_i=\frac{\partial C^{w_i}}{\partial b}(b) & \delta_i=\frac{\partial k^{w_i}}{\partial b}\left(b^0\right)\end{array}$                          (28)

Since $k^{r^*}=k^T=k^{y^*}=0$.

From Eq. (23) and using (22) we get:

\begin{aligned} & \delta_2=\frac{\partial k^y}{\partial b}\left(b^0\right)=0, \delta_4=\frac{\partial k^{r^*}}{\partial b}\left(b^0\right)=0, \delta_5=\frac{\partial k^T}{\partial b}\left(b^0\right)=0 \\ & \delta_3=\frac{\partial k^\omega}{\partial b}\left(b^0\right)=0 \quad \text { (ie) } \delta_2=\delta_3=\delta_4=\delta_5=0\end{aligned}

By satisfies the first order linear PDE, υ(x*, y*) is an invariant by Eq. (22):

$({{\chi }_{1}}{{x}^{*}}+{{\delta }_{1}})\frac{\partial \upsilon }{\partial {{x}^{*}}}+{{\chi }_{2}}{{y}^{*}}\frac{\partial \upsilon }{\partial {{y}^{*}}}=0\,$                      (29)

From the above equation we get:

$\frac{\partial \upsilon }{\partial {{x}^{*}}}=0$                           (30)

Therefore equation:

$\upsilon ={{y}^{*}}$                        (31)

Similarly, not changing the analysis of ω, r*, T dependent variables:

$\begin{gathered}\omega\left(x^*, y^*\right)=\Gamma_1\left(\mathrm{x}^*\right) \mathrm{M}(\iota), \\ r^*\left(x^*, y^*\right)=\Gamma_2\left(\mathrm{x}^*\right) \mathrm{E}(\iota), \\ T\left(x^*, y^*\right)=\mathrm{T}(v)\end{gathered}$                                           (32)

where, $\Gamma_1\left(\mathrm{x}^*\right), \Gamma_2\left(\mathrm{x}^*\right), \mathrm{M}(\iota), \& \mathrm{E}(\iota)$ are functions that are computed. Since r*(x*, y*) is independent of y*:

${{r}^{*}}({{x}^{*}},{{y}^{*}})={{r}^{*}}_{0}$                      (33)

4. The ODE Reduction

As the general analysis proceeds, the established form of the dependent and independent absolute invariant is used to obtain ODEs. Generally, the absolute invariant υ(x*, y*) has the form given in condition (31).

Substitute Eq. (32) into Eq. (17) and after dividing Г1(x*), we get:

\begin{aligned} & (17) \Rightarrow \frac{1}{r^*} \omega_{y^*} \omega_{x^* y^*}-\frac{1}{r^{* 2}}\left(\omega_{y^*}\right)^2 r_{x^*}^*-\frac{1}{r^*} \omega_{x^*} \omega_{y^* y^*} -T r^* \cos \phi-\omega_{y^* y^* y^*}=0\end{aligned}   \begin{aligned} & M^{\prime \prime \prime}+\frac{1}{r_0^* \Gamma_2} M M^{\prime \prime} \frac{\partial \Gamma_1}{\partial x^*}-\left(\frac{1}{r_0^* \Gamma_2} \frac{\partial \Gamma_1}{\partial x^*}-\frac{\Gamma_1}{r_0^* \Gamma_2^2} \frac{\partial \Gamma_2}{\partial x^*}\right) M^{\prime 2} +\frac{r_0^* \Gamma_2 \Gamma_3}{\Gamma_1} T \cos \phi=0\end{aligned}                       (34)

If we substitute the conditions (26) to (28) into condition (13), we get:

\begin{aligned} & \frac{1}{r^*}\left(\omega_{y^*} T_{x^*}-\omega_{x^*} T_{y^*}\right)-\frac{1}{\operatorname{Pr}} T_{y^* y^*}=0 \\ & \frac{1}{r^*{ }_0 \Gamma_2} \frac{\partial \Gamma_1}{\partial x^*} M(v) \frac{d T}{d y^*}+\frac{1}{\operatorname{Pr}} \frac{d^2 T}{d y^{* 2}}=0\end{aligned}                      (35)

${{C}_{1}}=\frac{1}{{{r}^{*}}_{0}{{\Gamma }_{2}}}\frac{\partial {{\Gamma }_{1}}}{\partial {{x}^{*}}},\,\,\,\,\,{{C}_{2}}=\frac{{{\Gamma }_{1}}}{{{r}^{*}}_{0}{{\Gamma }_{2}}^{2}}\frac{\partial {{\Gamma }_{2}}}{\partial {{x}^{*}}},\,\,\,\,\,{{C}_{3}}=\frac{{{r}^{*}}_{0}{{\Gamma }_{2}}{{\Gamma }_{3}}}{{{\Gamma }_{1}}}$                                 (36)

The above are random coefficients.

By utilizing above documentations of condition (36), the conditions (34) and (35) lessens to:

${{M}^{'''}}+{{C}_{1}}M\,{{M}^{''}}-({{C}_{1}}-{{C}_{2}}){{M}^{{{'}^{2}}}}+{{C}_{3}}\cos \phi T=0$                      (37)

${{C}_{1}}M\,\frac{dT}{d{{y}^{*}}}+\frac{1}{\Pr }\frac{{{d}^{2}}T}{d{{y}^{*}}^{2}}=0$                    (38)

There are corresponding boundary conditions.

${{M}^{'}}(0)=0,\,\,\,\,\,\,\frac{dT}{d{{y}^{*}}}\left( 0 \right)=-1,\,\,{{M}^{'}}(\infty )=T(\infty )=0$                     (39)

Case (i) use values, ${{C}_{1}}=\frac{3}{4},\,\,\,\,\,{{C}_{2}}=\frac{1}{4}\,\,\,\,\And {{C}_{3}}=1$ in Eqns. (37) and (38) we get:

${{M}^{'''}}+\frac{3}{4}M\,{{M}^{''}}-\frac{1}{2}{{M}^{{{'}^{2}}}}+\cos \phi T=0$                      (40)

$\frac{3}{4}M\,\frac{dT}{d{{y}^{*}}}+\frac{1}{\Pr }\frac{{{d}^{2}}T}{d{{y}^{*}}^{2}}=0\,\,\,\,\,(or)\frac{{{d}^{2}}T}{d{{y}^{*}}^{2}}+\frac{3}{4}\Pr M\,\frac{dT}{d{{y}^{*}}}=0$                         (41)

Case (ii) put ${{C}_{1}}=\frac{7}{4},\,\,\,\,\,{{C}_{2}}=\frac{5}{4}\,\,\,\,\And {{C}_{3}}=1$ in Eqns. (37) and (38) we obtain:

${{M}^{'''}}+\frac{7}{4}M\,{{M}^{''}}-\frac{1}{2}{{M}^{{{'}^{2}}}}+\cos \phi T=0$                   (42)

$\frac{7}{4}M\,\frac{dT}{d{{y}^{*}}}+\frac{1}{\Pr }\frac{{{d}^{2}}T}{d{{y}^{*}}^{2}}=0\,\,\,\,\,(or)\frac{{{d}^{2}}T}{d{{y}^{*}}^{2}}+\frac{7}{4}\Pr M\,\frac{dT}{d{{y}^{*}}}=0$                  (43)

With boundary conditions:

${{M}^{'}}(0)=0,\,\,\,\,\,\,\,\,\,\frac{dT}{d{{y}^{*}}}\left( 0 \right)=-1,\,\,\,\,\,{{M}^{'}}(\infty )=T(\infty )=0\,$                       (44)

Using Eqns. 9(b) and 9(c) into Eq. 9(a), the local dimensionless Nusselt number and skin friction becomes:

${{\tau }_{{{x}^{*}}}}G{{r}_{l}}^{-3/5}={{x}^{*}}{{M}^{''}}(0\,),\,\,\,\,\,N{{u}_{{{x}^{*}}}}G{{r}_{l}}^{-1/5}=-{{x}^{*}}\frac{{{T}^{'}}(0\,)}{T(0)}$

5. Results and Discussion

Eqns. (40) to (43) with boundary conditions (44) were solved by numerically using the fourth-order R-K method. The velocity and temperature profiles of the cone for Pr=0.72 are displayed in Figure 2 and the numerical values of local skin-friction $\tau_x^*$, temperature T, for different values of prandtl number are shown in Table1 are compared with similarity solution of Lin [25] using suitable transformation. It is observed that the results are in good agreement with each other. It is also noticed that the present result agrees well with those of Lin [25] and Pop and Watanabe [24] (as pointed out in Table 1).

Table 1. Skin-friction and temperature values are Comparison with Lin [25]

 Local skin friction Temperature Lin result [25] Present result Lin result [25] Present result Pr M’’(0) (7/4) M’’(0) $\tau_x^*$ -T(0) -(7/4) T(0) T 0.72 0.87830 1.2250 1.2154 1.53213 1.7896 1.7783 1 0.78336 1.0799 1.0735 1.39071 1.6324 1.6272 2 0.59251 0.8303 0.8295 1.16102 1.3532 1.3576 4 0.46507 0.6573 0.6330 0.97802 1.1490 1.1501 6 0.39700 0.5562 0.5499 0.88910 1.0501 1.0402 8 0.34963 0.4905 0.4849 0.83251 0.9806 0.9764 10 0.32645 0.4495 0.4459 0.78910 0.9291 0.9280 100 0.13071 0.1839 0.1812 0.49013 0.5572 0.5404

2.png

Figure 2. Velocity and temperature profiles comparison

3.png

Figure 3. Different values of f on velocity profiles

4.png

Figure 4. Different values of f on temperature profiles

5.png

Figure 5. Different Pr. values on velocity profiles

6.png

Figure 6. Different Pr. values on temperature profiles

Temperate and Velocity profile of a cone for Pr=0.72 is graphically represented in Figure 2. The temperature and velocity profiles shown in Figures 3-6 with different $\phi$ and Pr parameters. For Pr=0.71, and varying values of $\phi$, the velocity profile is shown in Figure 3. As f increases near a cone’s apex, the momentum force on a cone surface decreases. Therefore, differences in steady state and temporal maximum velocity value lowers as $\phi$ values of a cone (semi-vertical angle) increase. Also, with decreasing velocity, $\phi$ rises. The boundary layer momentum becomes thick; hence it increases the time required to acquire a steady state for rising value of $\phi$. For Pr =0.71, and varying values of $\phi$, temperature profiles are given in Figure 4. Further thickness and temperature of boundary layer also increases.

At ϕ=10°, different Pr values, the temperature and velocity profiles are represented in Figures 5 and 6. Viscosity increases and thermal conductivity decreases, when Pr increases. From the figure, it can be concluded that with higher Pr, variations in the maximum and steady-state value over time decreased and local skin-friction and Nusselt number profiles are against the $\phi$ angle of a cone for different Pr values.

6. Conclusions

The group technique was used to present numerical solutions of steady laminar free convection on a vertical cone with a uniform heat flux imposed on a surface. Using R-K method, the dimensionless boundary layer equation is solved. Following conclusions were made:

·If the parameters $\phi$ and Pr are increased, the velocity will decrease.

·The temperature rises and decreasing Pr with increasing fvalue.

·Increasing $\phi$, increases the impulse boundary layer.

·Decreasing $\phi$ and increasing Pr, thins the thermal boundary layer.

·The cause of $\phi$ on local Nusselt number $N u_{x^*}$ and local skin friction $\tau_{x^*}$ is low approximately equal to the cone’s apex and slowly increases as the distance from apex increases.

·When Pr or fvalue is reduced, local skin-frictions increases.

·Effect of Increasing $\phi$ or decreasing Pr, local Nusselt numbers reduce.

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