Peristaltic transportation of hybrid nano-blood through a ciliated micro-vessel subject to heat source and Lorentz force

Abstract

The center of interest of this research study is to unfold the phenomena in the electric double layer (EDL) adjacent to the indicted peristaltic wall and its impact on a peristaltic transport of ionized non-Newtonian blood (Jeffrey liquid model) infused with hybridized copper and gold nanoparticles through a ciliated micro-vessel under the buoyancy and Lorentz forces’ action. The energy equation is found with consideration of viscous dissipation and internal heat source impacts. The complicated normalized flow equations are abridged by adopting lubrication and Debye–Hückel linearization postulates. The homotopy perturbation approach is devoted to yield the optimal series solutions of the resulting equations. The amendment in the pertinent hemodynamical characteristics against the significant flow parameters is canvassed via plentiful graphical designs. Outcomes confess that a higher assisting the electric body force and thin EDL significantly opposes the blood flow nearby the ciliated micro-vessel wall. The heat exchange rate for hybrid nano-blood (26% for Cu-Au/blood) is greatly evaluated to nano-blood (20% for Au-blood and 11.4% for Cu-blood). The trapped bolus is expanded due to thinner EDL or longer cilia length. This simulation could help to design electro-osmotic blood pumps, diagnostic devices, pharmacological systems, etc.

Appendices

Appendix A:

$$ X = - \frac{2\pi \alpha \beta \delta \cos 2\pi z}{{1 - 2\pi \alpha \beta \delta \cos 2\pi z}} - 1,{\kern 1pt} $$

$$ A_{1} = X + \frac{{a_{0} }}{{k^{4} }}(a_{2} + k^{2} )I_{0} (hk),{\kern 1pt} $$

$$ \begin{aligned} A_{2} = \frac{1}{256}\left[ {\frac{64}{{k^{2} }}a_{0} a_{2} I_{0} (hk) - a_{1} \left\{ { - 16h^{2} (a_{2} - a_{5} - 1) + a_{4} \left( {\frac{\text{d}p}{{\text{d}z}}} \right)^{2} h^{4} (1 + \lambda_{1} )^{2} + 64} \right\}} \right. \\ + \;\left. {a_{2}^{2} h^{2} \left\{ {3\frac{{\text{d}}p}{\text{d}z}h^{2} (1 + \lambda_{1} ) - 16(1 + X)} \right\} - 16a_{2} (a_{3} h^{2} - 4(1 + X)) + 64a_{3} } \right],{\kern 1pt} \\ \end{aligned} $$

$$ A_{3} = \frac{1}{256}\left[ {4a_{1} ( - a_{2} + a_{5} + 1) + a_{2} \left\{ { - a_{2} \frac{dp}{{dz}}h^{2} (1 + \lambda_{1} ) + 4a_{2} (X + 1) + 4a_{3} } \right\}} \right], $$

$$ A_{4} = \frac{1}{2304}(1 + \lambda_{1} )\frac{dp}{{dz}}\left\{ {a_{1} a_{4} \frac{dp}{{dz}}(1 + \lambda_{1} ) + a_{2}^{2} } \right\}, $$

$$ A_{5} = - \frac{{a_{0} }}{{k^{4} }}(a_{2} + k^{2} ),{\kern 1pt} $$

$$ A_{6} = - \frac{{a_{5} }}{4}, $$

$$ \begin{gathered} A_{7} = \frac{1}{{128}}a_{4} \frac{{\text{d}p}}{{\text{d}z}}(1 + \lambda _{1} )\left[ {4a_{1} + a_{2} \frac{{dp}}{{dz}}h^{2} (1 + \lambda _{1} ) - 4a_{2} (X + 1)} \right. \hfill \\ \left. {\quad \;\; + 2\left( { - 2a_{3} + \frac{{dp}}{{dz}}(1 + \lambda _{1} )} \right)} \right] \hfill \\ \end{gathered} $$

$$ A_{8} = - \frac{1}{576}a_{2} a_{4} \left( {\frac{\text{d}p}{\text{dz}}} \right)^{2} (1 + \lambda_{1} )^{2} , $$

$$ A_{9} = \frac{{a_{0} a_{4} }}{{k^{4} }}\frac{\text{d}p}{\text{dz}}(1 + \lambda_{1} ),{\kern 1pt} $$

$$ B_{1} = \frac{5}{3072}\pi a_{1} a_{4} h^{8} (1 + \lambda_{1} )^{2} , $$

$$ B_{2} = \frac{1}{{3072x_{1} }}\pi h^{4} (1 + \lambda_{1} )[a_{2} h^{2} (11a_{2} h^{2} x_{1} - 64) + 384], $$

$$ \begin{aligned} B_{3} = & \frac{1}{{8k^{5} }}\pi a_{0} h\;\left[ {hk\{ k^{2} (8 - a_{2} h^{2} ) + 8a_{2} \} I_{0} (hk)} \right. \\ & \left. { - 16(a_{2} + k^{2} )I_{1} (hk)} \right] \\ & \frac{1}{{48}}\pi h^{4} \{ a_{1} (h^{2} ( - a_{2} + a_{5} + 1) + 6) \\ & + a_{2} (X + 1)(a_{2} h^{2} - 6)\} + \pi h^{2} X - Q \\ \end{aligned} $$

Appendix B: Summary of Some important formulas:

(1) The constitutive formulas for the extra-stress tensor $\tilde{S}$ in Jeffrey fluid model in component form are as:

$$ \tilde{S}_{{\tilde{\text{R}}\tilde{\text{R}}}} = \frac{{2\mu_{{{\text{hnf}}}} }}{{1 + \lambda_{1}^{ * } }}\left[ {1 + \lambda_{2} \left( {\tilde{U}\frac{\partial }{{\partial \tilde{R}}} + \tilde{W}\frac{\partial }{{\partial \tilde{Z}}}} \right)} \right]\frac{{\partial \tilde{U}}}{{\partial \tilde{R}}}, $$

$$ \tilde{S}_{{\tilde{\text{R}}\tilde{\text{Z}}}} = \tilde{S}_{{\tilde{\text{Z}}\tilde{\text{R}}}} = \frac{{\mu_{{{\text{hnf}}}} }}{{1 + \lambda_{1}^{ * } }}\left[ {1 + \lambda_{2} \left( {\tilde{U}\frac{\partial }{{\partial \tilde{R}}} + \tilde{W}\frac{\partial }{{\partial \tilde{Z}}}} \right)} \right]\left( {\frac{{\partial \tilde{U}}}{{\partial \tilde{Z}}} + \frac{{\partial \tilde{W}}}{{\partial \tilde{R}}}} \right), $$

$$ \tilde{S}_{{\tilde{\theta }\tilde{\theta }}} = \frac{{2\mu_{{{\text{hnf}}}} }}{{1 + \lambda_{1}^{ * } }}\left[ {1 + \lambda_{2} \left( {\tilde{U}\frac{\partial }{{\partial \tilde{R}}} + \tilde{W}\frac{\partial }{{\partial \tilde{Z}}}} \right)} \right]\left( {\frac{{\tilde{U}}}{{\tilde{R}}}} \right), $$

$$ \tilde{S}_{{\tilde{\text{Z}}\tilde{\text{Z}}}} = \frac{{2\mu_{{{\text{hnf}}}} }}{{1 + \lambda_{1}^{ * } }}\left[ {1 + \lambda_{2} \left( {\tilde{U}\frac{\partial }{{\partial \tilde{R}}} + \tilde{W}\frac{\partial }{{\partial \tilde{Z}}}} \right)} \right]\frac{{\partial \tilde{W}}}{{\partial \tilde{Z}}}, $$

(2) The electric potential $\tilde{\Phi }$ across the EDL is expressed as follows [15, 70]:

$$ \nabla^{2} \tilde{\Phi } = - \frac{{\rho_{\text{e}} }}{{\varepsilon_{0} }}, $$

(3) In a unit volume of the ionic blood, the electric charge density is rewritten as:

$$ \rho_{{\text{e}}} = - 2n_{0} e\overline{z}\sinh \left( {\frac{{e\overline{z}\tilde{\Phi }}}{{K_{{\text{B}}} T_{{\text{a}}} }}} \right). $$

(4) The simplified Poisson–Boltzmann equation is as:

$$ \nabla^{2} \tilde{\Phi } = \frac{{2n_{0} e\overline{z}}}{{\varepsilon_{0} }}\sinh \left( {\frac{{e\overline{z}\tilde{\Phi }}}{{K_{{\text{B}}} T_{{\text{a}}} }}} \right), $$

(5) Debye–Hückel linearization approximation: When the thermal energy of the ions is greater than the electric potential energy, i.e.,$\left| {e\overline{z}\tilde{\Phi }} \right| \ll \left| {K_{\text{B}} T_{\text{a}} } \right|$, then $\left| {\frac{{e\overline{z}\tilde{\Phi }}}{{K_{{\text{B}}} T_{{\text{a}}} }}} \right| \ll 1$; accordingly $\sinh \left( {\frac{{e\overline{z}\tilde{\Phi }}}{{K_{{\text{B}}} T_{{\text{a}}} }}} \right) \approx \frac{{e\overline{z}\tilde{\Phi }}}{{K_{{\text{B}}} T_{{\text{a}}} }}$.

(6) Adopting Debye–Hückel linearization approximation, Poisson–Boltzmann equation is:

$$ \frac{1}{{\tilde{R}}}\frac{\partial }{\partial }\left( {\tilde{R}\frac{{\partial \tilde{\Phi }}}{{\partial \tilde{R}}}} \right) + \frac{{\partial^{2} \tilde{\Phi }}}{{\partial \tilde{Z}^{2} }} = \frac{1}{{\lambda_{{\text{D}}}^{2} }}\tilde{\Phi }. $$