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Applications of Bernoulli wavelet collocation method in the analysis of Jeffery–Hamel flow and heat transfer in Eyring–Powell fluid

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Abstract

In this article, we developed the new functional matrix of integration using the Bernoulli wavelet and proposed a novel technique called the Bernoulli wavelet collocation method (BWCM). The main intention of this study is to present a consistent methodology to compute an imprecise solution of Jeffery–Hamel flow and heat transfer in Eyring–Powell fluid in the presence of a magnetic field by using the BWCM. Jeffery–Hamel flows occur in different realistic situations connecting flow between two non-parallel walls. Applications of such fluids in biological and industrial sciences brought great concern to the investigation of flow characteristics in converging and diverging channels. Here, we transform the nonlinear partial differential equations into coupled ordinary differential equations (ODEs) via similarity transformation. Using the BWCM, coupled ODEs are converted into a system of a nonlinear algebraic equation. This technique finds the numerical solution without any restrictive assumptions and avoids round-off errors. The numerical solutions attained by the proposed scheme point out that the approach is easy to implement and computationally very beautiful. The validity of the BWCM is ascertained by comparing our results with the Haar wavelet method and numerical differentiation Solver in Mathematica results. The influence of several emerging dimensionless parameters, namely the Eyring–Powell parameter, Hartman number, Eckert number, local Reynolds number, and the angle between the two walls on velocity and temperature evolution in the boundary layer regime, is examined in detail.

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Data availability

The data that support the findings of this study are available within the article.

Abbreviations

Γ:

Extra-stress tensor (Nm−2)

μ:

Shear viscosity (kg m−1 s−1)

\(\beta\) and \(d\) :

Characteristics of the Eyring–Powell model

\({A}_{1}\) :

Kinematical tensor

\(V\) :

Velocity vector

\(J\) :

Joule current (A m−2)

\(E\) :

Electric field (V m−1)

\(B\) :

External magnetic field (kg s−2 A−1)

\(k\) :

Thermal conductivity (W m−1 K−1)

\(C_{{\text{P}}}\) :

Specific heat constant (J kg−1 K−1)

\(\sigma\) :

Electrical conductivity (S m−1)

\(Ep\) :

Eyring–Powell parameter

\(\lambda\) :

Local non-Newtonian parameter

\({\text{Re}}\) :

Local Reynolds number

\({\text{Ha}}\) :

Hartmann number

\({\text{Pr}}\) :

Prandtl number

\({\text{Ec}}\) :

Eckert number

\(q_{{\text{w}}}\) :

Surface heat flux (Wm−2)

\(C_{{\text{f}}}\) :

Wall shear stress

\(N_{{\text{u}}}\) :

Nusselt number

\(u, v\) :

Velocity components (ms−1)

\(T\) :

Fluid temperature (K)

\(T_{{\text{w}}}\) :

Wall temperature (K)

\(\rho\) :

Density (kg m−3)

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Acknowledgements

The authors wish to thank the reviewers profusely for their constructive comments which helped in improving the paper considerably.

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Authors and Affiliations

  1. Department of Mathematics, Bangalore University, Bengaluru, India

    S. Kumbinarasaiah & M. P. Preetham

  2. Department of Mathematics, Davangere University, Davangere, India

    K. R. Raghunatha

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  1. S. Kumbinarasaiah
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  2. K. R. Raghunatha
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  3. M. P. Preetham
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Appendix

Appendix

The coefficients and functions that appeared in the formulation of the problem are given below.

Derivation of kinematical tensor (\(A_{1}\))

To find \({A}_{1}=\nabla V+(\nabla {V)}^\text{T}\):

$$\nabla V = \nabla \otimes V = \left( {e_{{\text{r}}} \frac{\partial }{\partial r} + e_{{\uptheta }} \frac{1}{r}\frac{\partial }{\partial \theta }} \right) \otimes \left( {u\,e_{{\text{r}}} + v\,e_{{\uptheta }} } \right)$$
$$\nabla V = e_{{\text{r}}} \otimes \frac{{\partial \left( {u\,e_{{\text{r}}} } \right)}}{\partial r} + e_{{\text{r}}} \otimes \frac{{\partial \left( {v\,e_{{\uptheta }} } \right)}}{\partial r} + e_{{\uptheta }} \otimes \frac{1}{r}\frac{{\partial \left( {u\,e_{{\text{r}}} } \right)}}{\partial \theta } + e_{{\uptheta }} \otimes \frac{1}{r}\frac{{\partial \left( {v\,e_{{\uptheta }} } \right)}}{\partial \theta }$$
$$\nabla V = e_\text{r} \otimes u\frac{{\partial e_{{\text{r}}} }}{\partial r} + e_{{\text{r}}} \otimes \frac{\partial u}{{\partial r}}e_{{\text{r}}} + e_{{\text{r}}} \otimes v\frac{{\partial e_{{\uptheta }} }}{\partial r} + e_{{\text{r}}} \otimes e_{{\uptheta }} \frac{\partial v}{{\partial r}} + e_{{\uptheta }} \otimes \frac{u}{r}\frac{{\partial \,e_{{\text{r}}} }}{\partial \theta } + e_{{\uptheta }} \otimes \frac{1}{r}\,e_{{\text{r}}} \frac{\partial u}{{\partial \theta }} + e_{{\uptheta }} \otimes \frac{v}{r}\frac{{\partial e_{{\uptheta }} }}{\partial \theta } + e_{{\uptheta }} \otimes \frac{1}{r}e_{{\uptheta }} \frac{\partial v}{{\partial \theta }}$$

We know that \(\frac{{\partial e_{{\text{r}}} }}{\partial r} = \frac{{\partial e_{{\uptheta }} }}{\partial r} = 0;\,\,\frac{{\partial \,e_{{\text{r}}} }}{\partial \theta } = e_{{\uptheta }} ,\,\frac{{\partial e_{{\uptheta }} }}{\partial \theta } = - e_{{\text{r}}}\)

$$\nabla V = e_{{\text{r}}} \otimes u\left( 0 \right) + e_{{\text{r}}} \otimes \frac{\partial u}{{\partial r}}e_{{\text{r}}} + e_{{\text{r}}} \otimes v\left( 0 \right) + e_{{\text{r}}} \otimes e_{{\uptheta }} \frac{\partial v}{{\partial r}} + e_{{\uptheta }} \otimes \frac{u}{r}\left( {e_{{\uptheta }} } \right) + e_{{\uptheta }} \otimes \frac{1}{r}\,e_{{\text{r}}} \frac{\partial u}{{\partial \theta }} + e_{{\uptheta }} \otimes \frac{v}{r}\left( { - e_{{\text{r}}} } \right) + e_{{\uptheta }} \otimes \frac{1}{r}e_{{\uptheta }} \frac{\partial v}{{\partial \theta }}$$
$$\nabla V = e_{{\text{r}}} \otimes \frac{\partial u}{{\partial r}}e_{{\text{r}}} + e_{{\text{r}}} \otimes e_{{\uptheta }} \frac{\partial v}{{\partial r}} + e_{{\uptheta }} \otimes \frac{u}{r}\left( {e_{{\uptheta }} } \right) + e_{{\uptheta }} \otimes \frac{1}{r}\,e_{{\text{r}}} \frac{\partial u}{{\partial \theta }} + e_{{\uptheta }} \otimes \frac{v}{r}\left( { - e_{{\text{r}}} } \right) + e_{{\uptheta }} \otimes \frac{1}{r}e_{{\uptheta }} \frac{\partial v}{{\partial \theta }}$$
$$\nabla V = \frac{\partial u}{{\partial r}}e_{{\text{r}}} \otimes e_{{\text{r}}} + \frac{\partial v}{{\partial r}}e_{{\text{r}}} \otimes e_{{\uptheta }} + \frac{u}{r}e_{{\uptheta }} \otimes e_{{\uptheta }} + \frac{1}{r}\,\frac{\partial u}{{\partial \theta }}e_{{\uptheta }} \otimes e_{{\text{r}}} - \frac{v}{r}e_{{\uptheta }} \otimes e_{{\text{r}}} + \frac{1}{r}\frac{\partial v}{{\partial \theta }}e_{{\uptheta }} \otimes e_{{\uptheta }}$$
$$\nabla V = \frac{\partial u}{{\partial r}}e_{{\text{r}}} \otimes e_{{\text{r}}} + \frac{\partial v}{{\partial r}}e_{{\text{r}}} \otimes e_{{\uptheta }} + \left( {\frac{1}{r}\,\frac{\partial u}{{\partial \theta }} - \frac{v}{r}} \right)e_{{\uptheta }} \otimes e_{{\text{r}}} + \left( {\frac{1}{r}\frac{\partial v}{{\partial \theta }} + \frac{u}{r}} \right)e_{{\uptheta }} \otimes e_{{\uptheta }}$$

\(\nabla V = \left[ {\begin{array}{*{20}c} {\frac{\partial u}{{\partial r}}} & {\frac{\partial v}{{\partial r}}} \\ {\frac{1}{r}\,\frac{\partial u}{{\partial \theta }} - \frac{v}{r}} & {\frac{1}{r}\frac{\partial v}{{\partial \theta }} + \frac{u}{r}} \\ \end{array} } \right]\) and \(\left( {\nabla V} \right)^{T} = \left[ {\begin{array}{*{20}c} {\frac{\partial u}{{\partial r}}} & {\frac{1}{r}\,\frac{\partial u}{{\partial \theta }} - \frac{v}{r}} \\ {\frac{\partial v}{{\partial r}}} & {\frac{1}{r}\frac{\partial v}{{\partial \theta }} + \frac{u}{r}} \\ \end{array} } \right]\)

$$A_{1} = \nabla V + \left( {\nabla V} \right)^\text{T} = \left[ {\begin{array}{*{20}c} {\frac{\partial u}{{\partial r}}} & {\frac{\partial v}{{\partial r}}} \\ {\frac{1}{r}\,\frac{\partial u}{{\partial \theta }} - \frac{v}{r}} & {\frac{1}{r}\frac{\partial v}{{\partial \theta }} + \frac{u}{r}} \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {\frac{\partial u}{{\partial r}}} & {\frac{1}{r}\,\frac{\partial u}{{\partial \theta }} - \frac{v}{r}} \\ {\frac{\partial v}{{\partial r}}} & {\frac{1}{r}\frac{\partial v}{{\partial \theta }} + \frac{u}{r}} \\ \end{array} } \right]$$
$$A_{1} = \nabla V + \left( {\nabla V} \right)^\text{T} = \left[ {\begin{array}{*{20}c} {2\frac{\partial u}{{\partial r}}} & {\frac{1}{r}\,\frac{\partial u}{{\partial \theta }} + \frac{\partial v}{{\partial r}} - \frac{v}{r}} \\ {\frac{1}{r}\,\frac{\partial u}{{\partial \theta }} + \frac{\partial v}{{\partial r}} - \frac{v}{r}} & {2\left( {\frac{1}{r}\frac{\partial v}{{\partial \theta }} + \frac{u}{r}} \right)} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {A_\text{rr} } & {A_{\text{r}\uptheta }} \\ {A_{\uptheta\text{r}} } & {A_{\uptheta \uptheta } } \\ \end{array} } \right]$$
$$A_{1}^{2} = \left[ {\begin{array}{*{20}c} {2\frac{\partial u}{{\partial r}}} & {\frac{1}{r}\,\frac{\partial u}{{\partial \theta }} + \frac{\partial v}{{\partial r}} - \frac{v}{r}} \\ {\frac{1}{r}\,\frac{\partial u}{{\partial \theta }} + \frac{\partial v}{{\partial r}} - \frac{v}{r}} & {2\left( {\frac{1}{r}\frac{\partial v}{{\partial \theta }} + \frac{u}{r}} \right)} \\ \end{array} } \right]\,\,\,\left[ {\begin{array}{*{20}c} {2\frac{\partial u}{{\partial r}}} & {\frac{1}{r}\,\frac{\partial u}{{\partial \theta }} + \frac{\partial v}{{\partial r}} - \frac{v}{r}} \\ {\frac{1}{r}\,\frac{\partial u}{{\partial \theta }} + \frac{\partial v}{{\partial r}} - \frac{v}{r}} & {2\left( {\frac{1}{r}\frac{\partial v}{{\partial \theta }} + \frac{u}{r}} \right)} \\ \end{array} } \right]$$
$$A_{1}^{2} = \left[ {\begin{array}{*{20}l} {4\left( {\frac{\partial u}{{\partial r}}} \right)^{2} + \left( {\frac{1}{r}\,\frac{\partial u}{{\partial \theta }} + \frac{\partial v}{{\partial r}} - \frac{v}{r}} \right)^{2} } \hfill & W \hfill \\ W \hfill & {\left( {\frac{1}{r}\,\frac{\partial u}{{\partial \theta }} + \frac{\partial v}{{\partial r}} - \frac{v}{r}} \right)^{2} + 4\left( {\frac{1}{r}\frac{\partial v}{{\partial \theta }} + \frac{u}{r}} \right)^{2} } \hfill \\ \end{array} } \right]$$

where \(W = 2\frac{\partial u}{{\partial r}}\left( {\frac{1}{r}\,\frac{\partial u}{{\partial \theta }} + \frac{\partial v}{{\partial r}} - \frac{v}{r}} \right) + 2\left( {\frac{1}{r}\frac{\partial v}{{\partial \theta }} + \frac{u}{r}} \right)\left( {\frac{1}{r}\,\frac{\partial u}{{\partial \theta }} + \frac{\partial v}{{\partial r}} - \frac{v}{r}} \right)\) and

$$\dot{\gamma }^{2} = \frac{1}{2}Tr\left( {A_{1}^{2} } \right) = 2\left( {\frac{\partial u}{{\partial r}}} \right)^{2} + \left( {\frac{1}{r}\,\frac{\partial u}{{\partial \theta }} + \frac{\partial v}{{\partial r}} - \frac{v}{r}} \right)^{2} + 2\left( {\frac{1}{r}\frac{\partial v}{{\partial \theta }} + \frac{u}{r}} \right)^{2} .$$

Now, \(\Gamma = \left( {\mu + \frac{1}{\beta d} - \frac{{\dot{\gamma }^{2} }}{{6\beta d^{3} }}} \right)A_{1}\).

We know that \(A_{1} = \left[ {\begin{array}{*{20}c} {A_\text{rr} } & {A_{\text{r}\uptheta } } \\ {A_{\uptheta \text{r}} } & {A_{\uptheta \uptheta } } \\ \end{array} } \right]\) then

$$\Gamma_{{{\text{rr}}}} = \left( {\mu + \frac{1}{\beta d} - \frac{{\dot{\gamma }^{2} }}{{6\beta d^{3} }}} \right)A_{{{\text{rr}}}}$$
$$\Gamma _{{\text{r}\uptheta }} = \left( {\mu + \frac{1}{{\beta d}} - \frac{{\dot{\gamma }^{2} }}{{6\beta d^{3} }}} \right)A_{{\text{r}\uptheta }}$$
$$\Gamma _{{\theta \theta }} = \left( {\mu + \frac{1}{{\beta d}} - \frac{{\dot{\gamma }^{2} }}{{6\beta d^{3} }}} \right)A_{{\theta \theta }}$$

Derivation of general momentum equation:

$$\rho \left[ {\frac{{\partial \vec{q}}}{\partial t} + \left( {\vec{q} \cdot \nabla } \right)\vec{q}} \right] = - \nabla p + \nabla \cdot \Gamma$$

(i) Expression of \(\left( {\vec{q} \cdot \nabla } \right)\vec{q}\):

$$\vec{q} \cdot \nabla = \left( {u\,e_{{\text{r}}} + v\,e_{{\uptheta }} } \right) \cdot \left( {e_{{\text{r}}} \frac{\partial }{\partial r} + e_{{\uptheta }} \frac{1}{r}\frac{\partial }{\partial \theta }} \right) = u\frac{\partial }{\partial r} + v\frac{1}{r}\frac{\partial }{\partial \theta }$$
$$\left( {\vec{q} \cdot \nabla } \right)\vec{q} = \left( {u\frac{\partial }{\partial r} + v\frac{1}{r}\frac{\partial }{\partial \theta }} \right)\left( {u\,e_{{\text{r}}} + v\,e_{{\uptheta }} } \right)$$
$$\left( {\vec{q} \cdot \nabla } \right)\vec{q} = u\frac{{\partial \left( {u\,e_{{\text{r}}} } \right)}}{\partial r} + u\frac{{\partial \left( {v\,e_{{\uptheta }} } \right)}}{\partial r} + v\frac{1}{r}\frac{{\partial \left( {u\,e_{{\text{r}}} } \right)}}{\partial \theta } + v\frac{1}{r}\frac{{\partial \left( {v\,e_{{\uptheta }} } \right)}}{\partial \theta }$$
$$\left( {\vec{q} \cdot \nabla } \right)\vec{q} = u\,\,u\frac{{\partial \left( {\,e_{{\text{r}}} } \right)}}{\partial r} + u\,\frac{\partial u}{{\partial r}}e_{{\text{r}}} + u\,v\frac{{\partial \left( {e_{{\uptheta }} } \right)}}{\partial r} + u\,\,\frac{\,\partial v}{{\partial r}}e_{{\uptheta }} + u\,v\frac{1}{r}\frac{{\partial \left( {e_{{\text{r}}} } \right)}}{\partial \theta } + v\frac{1}{r}\frac{\partial u}{{\partial \theta }}e_{{\text{r}}} + v\,v\frac{1}{r}\frac{{\partial \left( {\,e_{{\uptheta }} } \right)}}{\partial \theta } + v\frac{1}{r}\frac{\partial v}{{\partial \theta }}e_{{\uptheta }}$$

We know that \(\frac{{\partial \left( {\,e_{{\text{r}}} } \right)}}{\partial r} = \frac{{\partial \left( {e_{{\uptheta }} } \right)}}{\partial r} = 0;\frac{{\partial \left( {e_{{\text{r}}} } \right)}}{\partial \theta } = e_{{\uptheta }} ,\,\,\frac{{\partial \left( {\,e_{{\uptheta }} } \right)}}{\partial \theta } = - e_{{\text{r}}}\)

$$\left( {\vec{q} \cdot \nabla } \right)\vec{q} = u\,\,u\left( 0 \right) + u\,\frac{\partial u}{{\partial r}}e_{{\text{r}}} + u\,v\left( 0 \right) + u\,\,\frac{\,\partial v}{{\partial r}}e_{{\uptheta }} + u\,v\frac{1}{r}\left( {e_{{\uptheta }} } \right) + v\frac{1}{r}\frac{\partial u}{{\partial \theta }}e_{{\text{r}}} + v\,v\frac{1}{r}\left( { - e_{{\text{r}}} } \right) + v\frac{1}{r}\frac{\partial v}{{\partial \theta }}e_{{\uptheta }}$$
$$\left( {\vec{q} \cdot \nabla } \right)\vec{q} = \left( {u\,\frac{\partial u}{{\partial r}} + \frac{v}{r}\frac{\partial u}{{\partial \theta }} - \frac{{v^{2} }}{r}} \right)e_{{\text{r}}} + \left( {u\,\,\frac{\,\partial v}{{\partial r}} + \frac{v}{r}\frac{\partial v}{{\partial \theta }} + \frac{u\,v}{r}} \right)e_{{\uptheta }}$$

(ii) Expressions of the shears of Eyring–Powell fluid in polar coordinates (\(\Gamma_\text{rr} ,\,\Gamma_{\text{r}\uptheta } ,\,\Gamma_{\uptheta \uptheta }\)):

$$\nabla \cdot \Gamma = \left( {e_\text{r} \frac{\partial }{{\partial r}} + e_{\uptheta } \frac{1}{r}\frac{\partial }{{\partial \theta }}} \right) \cdot \left[ {\Gamma _\text{{rr}} e_\text{r} \otimes e_\text{r} + \Gamma _{{\text{r}\uptheta }} e_\text{r} \otimes e_{\uptheta } + \Gamma _{{\uptheta \text{r}}} e_{\uptheta } \otimes e_\text{r} + \Gamma _{{\uptheta \uptheta }} e_{\uptheta } \otimes e_{\uptheta } } \right]$$
$$\begin{gathered} \nabla \cdot \Gamma = \left( {e_\text{r} \frac{\partial }{{\partial r}}} \right) \cdot \left[ {\Gamma _\text{{rr}} e_\text{r} \otimes e_\text{r} + \Gamma _{{\text{r}\uptheta }} e_\text{r} \otimes e_{\uptheta } + \Gamma _{{\uptheta \text{ r}}} e_{\uptheta } \otimes e_\text{r} + \Gamma _{{\uptheta \uptheta }} e_{\uptheta } \otimes e_{\uptheta } } \right] + \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\left( {e_{\theta } \frac{1}{r}\frac{\partial }{{\partial \theta }}} \right) \cdot \left[ {\Gamma _\text{{rr}} e_\text{r} \otimes e_\text{r} + \Gamma _{{\text{r}\uptheta }} e_\text{r} \otimes e_{\uptheta } + \Gamma _{{\uptheta \text{r}}} e_{\uptheta } \otimes e_\text{r} + \Gamma _{{\uptheta \uptheta }} e_{\uptheta } \otimes e_{\uptheta } } \right] \hfill \\ \end{gathered}$$
$$\begin{gathered} \nabla \cdot \Gamma = \left( {e_\text{r} \frac{\partial }{{\partial r}}} \right) \cdot \left( {\Gamma _\text{{rr}} e_\text{r} \otimes e_\text{r} } \right) + \left( {e_\text{r} \frac{\partial }{{\partial r}}} \right) \cdot \left( {\Gamma _{{\text{r}\uptheta }} e_\text{r} \otimes e_{\uptheta } } \right) + \left( {e_\text{r} \frac{\partial }{{\partial r}}} \right) \cdot \left( {\Gamma _{{\uptheta\text{r}}} e_{\uptheta } \otimes e_\text{r} } \right) + \left( {e_\text{r} \frac{\partial }{{\partial r}}} \right) \cdot \left( {\Gamma _{{\uptheta \uptheta }} e_{\uptheta } \otimes e_{\uptheta } } \right) + \hfill \\ \left( {\frac{{e_{\uptheta } }}{\text{r}}\frac{\partial }{{\partial \theta }}} \right) \cdot \left( {\Gamma _\text{{rr}} e_\text{r} \otimes e_\text{r} } \right) + \left( {\frac{{e_{\uptheta } }}{\text{r}}\frac{\partial }{{\partial \theta }}} \right) \cdot \left( {\Gamma _{{\text{r}\uptheta }} e_\text{r} \otimes e_{\uptheta } } \right) + \left( {\frac{{e_{\uptheta } }}{\text{r}}\frac{\partial }{{\partial \theta }}} \right) \cdot \left( {\Gamma _{{\uptheta\text{r}}} e_{\uptheta } \otimes e_\text{r} } \right) + \left( {\frac{{e_{\uptheta } }}{r}\frac{\partial }{{\partial \theta }}} \right) \cdot \left( {\Gamma _{{\uptheta \uptheta }} e_{\uptheta } \otimes e_{\uptheta } } \right) \hfill \\ = a_{1} + a_{2} + a_{3} + a_{4} + a_{5} + a_{6} + a_{7} + a_{8} \hfill \\ \end{gathered}$$
$$\begin{aligned} a_{1} = & \left( {e_{{\text{r}}} \frac{\partial }{\partial r}} \right) \cdot \left( {\Gamma_{{{\text{rr}}}} e_{{\text{r}}} \otimes e_{{\text{r}}} } \right) = e_{{\text{r}}} \cdot \frac{\partial }{\partial r}\left( {\Gamma_{{{\text{rr}}}} e_{{\text{r}}} \otimes e_{{\text{r}}} } \right) = e_{{\text{r}}} \cdot \left[ {\frac{{\partial \Gamma_{{{\text{rr}}}} }}{\partial r}\left( {e_{{\text{r}}} \otimes e_{{\text{r}}} } \right) + \Gamma_{{{\text{rr}}}} \left( {\frac{{\partial e_{{\text{r}}} }}{\partial r} \otimes e_{{\text{r}}} } \right) + \Gamma_{{{\text{rr}}}} \left( {e_{{\text{r}}} \otimes \frac{{\partial e_{{\text{r}}} }}{\partial r}} \right)} \right] \\ = & \frac{{\partial \Gamma_{{{\text{rr}}}} }}{\partial r}e_{{\text{r}}} \,\,\,\,\,\,\,\,\left( {\because \frac{{\partial e_{{\text{r}}} }}{\partial r} = 0} \right) \\ \end{aligned}$$
$$\begin{aligned} a_{2} = & \left( {e_\text{r} \frac{\partial }{{\partial r}}} \right) \cdot \left( {\Gamma _{{\text{r}\uptheta }} e_\text{r} \otimes e_{\uptheta } } \right) = e_\text{r} \cdot \frac{\partial }{{\partial r}}\left( {\Gamma _{{\text{r}\uptheta }} e_\text{r} \otimes e_{\uptheta } } \right) = e_\text{r} \cdot \left[ {\frac{{\partial \Gamma _{{\text{r}\uptheta }} }}{{\partial r}}\left( {e_\text{r} \otimes e_{\uptheta } } \right) + \Gamma _{{\text{r}\uptheta }} \left( {\frac{{\partial e_{r} }}{{\partial r}} \otimes e_{\uptheta } } \right) + \Gamma _{{\text{r}\uptheta }} \left( {e_\text{r} \otimes \frac{{\partial e_{\uptheta } }}{{\partial r}}} \right)} \right] \\ = & \frac{{\partial \Gamma _{{\text{r}\uptheta }} }}{{\partial r}}e_{\uptheta } \,\left( {\because \frac{{\partial e_\text{r} }}{{\partial r}} = \frac{{\partial e_{\uptheta } }}{{\partial r}} = 0} \right) \\ \end{aligned}$$
$$\begin{aligned} a_{3} = & \left( {e_{{\text{r}}} \frac{\partial }{\partial r}} \right) \cdot \left( {\Gamma_{{{\uptheta \text{r}}}} e_{{\uptheta }} \otimes e_{{\text{r}}} } \right) = e_{{\text{r}}} \cdot \frac{\partial }{\partial r}\left( {\Gamma_{{{\uptheta \text{r}}}} e_{{\uptheta }} \otimes e_{{\text{r}}} } \right) = e_{{\text{r}}} \cdot \left[ {\frac{{\partial \Gamma_{{{\uptheta \text{r}}}} }}{\partial r}\left( {e_{{\uptheta }} \otimes e_{{\text{r}}} } \right) + \Gamma_{{{\uptheta \text{r}}}} \left( {\frac{{\partial e_{{\uptheta }} }}{\partial r} \otimes e_{{\text{r}}} } \right) + \Gamma_{{{\uptheta \text{r}}}} \left( {e_{{\uptheta }} \otimes \frac{{\partial e_{{\text{r}}} }}{\partial r}} \right)} \right] = 0 \\ & \left( {\because e_{{\text{r}}} \cdot \left( {e_{{\uptheta }} \otimes e_{{\text{r}}} } \right) = \frac{{\partial e_{{\uptheta }} }}{\partial r} = \frac{{\partial e_{{\text{r}}} }}{\partial r} = 0} \right) \\ \end{aligned}$$
$$\begin{gathered} a_{4} = \left( {e_\text{r} \frac{\partial }{{\partial r}}} \right) \cdot \left( {\Gamma _{{\uptheta \uptheta }} e_{\uptheta } \otimes e_{\uptheta } } \right) = e_\text{r} \cdot \frac{\partial }{{\partial r}}\left( {\Gamma _{{\uptheta \uptheta }} e_{\uptheta } \otimes e_{\uptheta } } \right) = e_\text{r} \cdot \left[ {\frac{{\partial \Gamma _{{\uptheta \uptheta }} }}{{\partial r}}\left( {e_{\uptheta } \otimes e_{\uptheta } } \right) + \Gamma _{{\uptheta \uptheta }} \left( {\frac{{\partial e_{\uptheta } }}{{\partial r}} \otimes e_{\uptheta } } \right) + \Gamma _{{\uptheta \uptheta }} \left( {e_{\theta } \otimes \frac{{\partial e_{\uptheta } }}{{\partial r}}} \right)} \right] = 0 \hfill \\ \left( {\because e_\text{r} \cdot \left( {e_{\uptheta } \otimes e_{\uptheta } } \right) = \frac{{\partial e_{\uptheta } }}{{\partial r}} = \frac{{\partial e_\text{r} }}{{\partial r}} = 0} \right) \hfill \\ \end{gathered}$$
$$\begin{aligned} a_{5} = & \left( {\frac{{e_{{\uptheta }} }}{r}\frac{\partial }{\partial \theta }} \right) \cdot \left( {\Gamma_{{{\text{rr}}}} e_{{\text{r}}} \otimes e_{{\text{r}}} } \right) = \frac{{e_{{\uptheta }} }}{r} \cdot \frac{\partial }{\partial \theta }\left( {\Gamma_{{{\text{rr}}}} e_{{\text{r}}} \otimes e_{{\text{r}}} } \right) = \frac{{e_{{\uptheta }} }}{r} \cdot \left[ {\frac{{\partial \Gamma_{{{\text{rr}}}} }}{\partial \theta }\left( {e_{{\text{r}}} \otimes e_{{\text{r}}} } \right) + \Gamma_{{{\text{rr}}}} \left( {\frac{{\partial e_{{\text{r}}} }}{\partial \theta } \otimes e_{{\text{r}}} } \right) + \Gamma_{{{\text{rr}}}} \left( {e_{{\text{r}}} \otimes \frac{{\partial e_{{\text{r}}} }}{\partial \theta }} \right)} \right] \\ = & \frac{{e_{\theta } }}{r} \cdot \left[ {\frac{{\partial \Gamma_{{{\text{rr}}}} }}{\partial \theta }\left( {e_{{\text{r}}} \otimes e_{{\text{r}}} } \right) + \Gamma_{{{\text{rr}}}} \left( {e_{{\uptheta }} \otimes e_{{\text{r}}} } \right) + \Gamma_{{{\text{rr}}}} \left( {e_{{\text{r}}} \otimes e_{{\uptheta }} } \right)} \right] \\ = & \frac{{\Gamma_{{{\text{rr}}}} }}{r}e_{r} \,\left( {\because \frac{{\partial e_{{\text{r}}} }}{\partial \theta } = e_{{\uptheta }} ;\,\,\,\,e_{{\uptheta }} \cdot \left( {e_{{\text{r}}} \otimes e_{{\text{r}}} } \right) = e_{{\uptheta }} \cdot \left( {e_{{\text{r}}} \otimes e_{{\uptheta }} } \right) = 0} \right) \\ \end{aligned}$$
$$\begin{aligned} a_{6} = & \left( {\frac{{e_{\uptheta } }}{r}\frac{\partial }{{\partial \theta }}} \right) \cdot \left( {\Gamma _{{\text{r}\uptheta }} e_\text{r} \otimes e_{\uptheta } } \right) = \frac{{e_{\uptheta } }}{\text{r}} \cdot \frac{\partial }{{\partial \theta }}\left( {\Gamma _{{\text{r}\uptheta }} e_\text{r} \otimes e_{\uptheta } } \right) = \frac{{e_{\uptheta } }}{\text{r}} \cdot \left[ {\frac{{\partial \Gamma _{{\text{r}\uptheta }} }}{{\partial \theta }}\left( {e_\text{r} \otimes e_{\uptheta } } \right) + \Gamma _{{\text{r}\uptheta }} \left( {\frac{{\partial e_\text{r} }}{{\partial \theta }} \otimes e_{\uptheta } } \right) + \Gamma _{{\text{r}\uptheta }} \left( {e_\text{r} \otimes \frac{{\partial e_{\uptheta } }}{{\partial \theta }}} \right)} \right] \\ = & \frac{{e_{\uptheta } }}{\text{r}} \cdot \left[ {\frac{{\partial \Gamma _{{\text{r}\uptheta }} }}{{\partial \theta }}\left( {e_\text{r} \otimes e_{\uptheta } } \right) + \Gamma _{{\text{r}\uptheta }} \left( {e_{\uptheta } \otimes e_{\uptheta } } \right) + \Gamma _{{\text{r}\uptheta }} \left( {e_\text{r} \otimes \left( { - e_\text{r} } \right)} \right)} \right] \\ = & \frac{{\Gamma _{{\text{r}\uptheta }} }}{\text{r}}e_{\uptheta } \\ \end{aligned}$$
$$\begin{aligned} a_{7} = & \left( {\frac{{e_{{\uptheta }} }}{r}\frac{\partial }{\partial \theta }} \right) \cdot \left( {\Gamma_{{{\uptheta \text{r}}}} e_{{\uptheta }} \otimes e_{{\text{r}}} } \right) = \frac{{e_{{\uptheta }} }}{r} \cdot \frac{\partial }{\partial \theta }\left( {\Gamma_{{{\uptheta \text{r}}}} e_{{\uptheta }} \otimes e_{{\text{r}}} } \right) = \frac{{e_{{\uptheta }} }}{r} \cdot \left[ {\frac{{\partial \Gamma_{{{\uptheta \text{r}}}} }}{\partial \theta }\left( {e_{{\uptheta }} \otimes e_{{\text{r}}} } \right) + \Gamma_{{{\uptheta \text{r}}}} \left( {\frac{{\partial e_{{\uptheta }} }}{\partial \theta } \otimes e_{{\text{r}}} } \right) + \Gamma_{{{\uptheta \text{r}}}} \left( {e_{{\uptheta }} \otimes \frac{{\partial e_{{\text{r}}} }}{\partial \theta }} \right)} \right] \\ = & \frac{{e_{{\uptheta }} }}{r} \cdot \left[ {\frac{{\partial \Gamma_{{{\uptheta \text{r}}}} }}{\partial \theta }\left( {e_{{\uptheta }} \otimes e_{{\text{r}}} } \right) + \Gamma_{{{\uptheta \text{r}}}} \left( { - e_{{\text{r}}} \otimes e_{{\text{r}}} } \right) + \Gamma_{{{\uptheta \text{r}}}} \left( {e_{{\uptheta }} \otimes e_{{\uptheta }} } \right)} \right] \\ = & \frac{1}{r}\frac{{\partial \Gamma_{{{\uptheta \text{r}}}} }}{\partial \theta }e_{{\text{r}}} + \frac{{\Gamma_{{{\uptheta \text{r}}}} }}{r}e_{{\uptheta }} \\ \end{aligned}$$
$$\begin{gathered} a_{8} = \left( {\frac{{e_{\uptheta } }}{r}\frac{\partial }{\partial \theta }} \right) \cdot \left( {\Gamma_{\uptheta \uptheta } e_{\uptheta } \otimes e_{\uptheta } } \right) = \frac{{e_{\uptheta } }}{\text{r}} \cdot \frac{\partial }{\partial \theta }\left( {\Gamma_{\uptheta \uptheta } e_{\uptheta } \otimes e_{\uptheta } } \right) = \frac{{e_{\uptheta } }}{\text{r}} \cdot \left[ {\frac{{\partial \Gamma_{\uptheta \uptheta } }}{\partial \theta }\left( {e_{\uptheta } \otimes e_{\uptheta } } \right) + \Gamma_{\uptheta \uptheta } \left( {\frac{{\partial e_{\uptheta } }}{\partial \theta } \otimes e_{\uptheta } } \right) + \Gamma_{\uptheta \uptheta } \left( {e_{\uptheta } \otimes \frac{{\partial e_{\uptheta } }}{\partial \theta }} \right)} \right] \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{e_{\uptheta } }}{\text{r}} \cdot \left[ {\frac{{\partial \Gamma_{\uptheta \uptheta } }}{\partial \theta }\left( {e_{\uptheta } \otimes e_{\uptheta } } \right) + \Gamma_{\uptheta \uptheta } \left( { - e_\text{r} \otimes e_{\uptheta } } \right) + \Gamma_{\uptheta \uptheta } \left( {e_{\uptheta } \otimes \left( { - e_\text{r} } \right)} \right)} \right] \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{1}{r}\frac{{\partial \Gamma_{\uptheta \uptheta } }}{\partial \theta }e_{\uptheta } - \frac{{\Gamma_{\uptheta \uptheta } }}{\text{r}}e_\text{r} \hfill \\ \end{gathered}$$

Thus,

$$\begin{gathered} \nabla \cdot \Gamma = a_{1} + a_{2} + a_{3} + a_{4} + a_{5} + a_{6} + a_{7} + a_{8} \hfill \\ \nabla \cdot \Gamma = \frac{{\partial \Gamma _\text{{rr}} }}{{\partial r}}e_\text{r} \, + \frac{{\partial \Gamma _{{\text{r}\uptheta }} }}{{\partial r}}e_{\uptheta } + 0 + 0 + \frac{{\Gamma _\text{{rr}} }}{r}e_\text{r} + \frac{{\Gamma _{{\text{r}\uptheta }} }}{\text{r}}e_{\uptheta } + \frac{1}{r}\frac{{\partial \Gamma _{{\uptheta\text{r}}} }}{{\partial \theta }}e_\text{r} + \frac{{\Gamma _{{\uptheta \text{r}}} }}{\text{r}}e_{\uptheta } + \frac{1}{r}\frac{{\partial \Gamma _{{\uptheta \uptheta }} }}{{\partial \theta }}e_{\uptheta } - \frac{{\Gamma _{{\uptheta \uptheta }} }}{r}e_\text{r} \hfill \\ \end{gathered}$$
$$\nabla \cdot \Gamma = \left( {\frac{{\partial \Gamma _\text{{rr}} }}{{\partial r}} + \frac{{\Gamma _\text{{rr}} }}{\text{r}} + \frac{1}{r}\frac{{\partial \Gamma _{{\uptheta \text{r}}} }}{{\partial \theta }} - \frac{{\Gamma _{{\uptheta \uptheta }} }}{\text{r}}} \right)e_\text{r} \, + \left( {\frac{{\partial \Gamma _{{\text{r}\uptheta }} }}{{\partial r}} + \frac{{\Gamma _{{\text{r}\uptheta }} }}{r} + \frac{{\Gamma _{{\uptheta \text{r}}} }}{\text{r}} + \frac{1}{r}\frac{{\partial \Gamma _{{\uptheta \uptheta }} }}{{\partial \theta }}} \right)e_{\uptheta }$$
$$\nabla \cdot \Gamma = \left( {\frac{1}{r}\frac{{\partial \left( {r\Gamma _\text{{rr}} } \right)}}{{\partial r}} + \frac{1}{r}\frac{{\partial \Gamma _{{\text{r}\uptheta }} }}{{\partial \theta }} - \frac{{\Gamma _{{\uptheta \uptheta }} }}{\text{r}}} \right)e_\text{r} \, + \left( {\frac{{\partial \Gamma _{{\text{r}\uptheta }} }}{{\partial r}} + \frac{1}{r}\frac{{\partial \Gamma _{{\uptheta \uptheta }} }}{{\partial \theta }} + 2\frac{{\Gamma _{{\text{r}\uptheta }} }}{\text{r}}} \right)e_{\uptheta } .$$

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Kumbinarasaiah, S., Raghunatha, K.R. & Preetham, M.P. Applications of Bernoulli wavelet collocation method in the analysis of Jeffery–Hamel flow and heat transfer in Eyring–Powell fluid. J Therm Anal Calorim 148, 1173–1189 (2023). https://doi.org/10.1007/s10973-022-11706-9

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