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Mathematical modelling of modified hybrid nanofluid in a peristaltic diverging tube with MHD and convective boundary conditions

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Abstract

Numerous applications of magneto-hydrodynamics are applied in daily life medics and particularly where phenomena of peristaltic are used for delivery of drugs to the affected area in order to cure the disease. The delivery method of zinc, titanium and copper modified hybrid nanoparticles to target area through oesophagus necessitated elaboration in order to achieve best desired results. The significance of the current problem specifies the model of a magneto-hydrodynamic peristaltically driven flow when the cross section of the tube is diverging. The manuscript focuses to consider zinc, titanium and copper modified hybrid nanoparticles with water as carrier fluid. The impact of wall properties, slip and convective boundary conditions is intricate. The Cauchy Euler’s method is accomplished to obtain the exact solution of consequential system of nonlinear ODE’s raised from governing PDE’s and the boundary conditions for computation are evolved by using appropriate transformations. The computational software packages “Mathematica” used as a main tool to show pumping phenomenon against velocity and temperature profile using emerging parameters and bolus formation. The results have supported the improvement in delivery system of above discussed particles especially in the field of photodynamic therapy.

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Abbreviations

\({r}^{\breve{\,}}\),\({z}^{\breve{\,}}\) :

Radial and axial direction in wave frame

\({l}^{\breve{\,}}\) :

Radius of inner tube

\({U}^{\breve{\,}},{W}^{\breve{\,}}\) :

Velocity components in radial and axial direction

\({d}^{\breve{\,}}\) :

Wave speed

λ :

Wave length

\(K\) :

Non-uniform parameter

ε :

Amplitude ratio

\(Re\) :

Reynolds’s number

\(L\) :

Velocity slip parameter

\({\mathcalligra{c}}_{\mathcalligra{p}}\) :

Specific heat

\({{\mathcal{K}}^{\breve{\,}}}_{bf}\) :

Thermal conductivity of base fluid

\({{\mathcal{K}}^{\breve{\,}}}_{{\mathcalligra{nf}}}\) :

Thermal conductivity of nanofluid

\({{\mathcal{K}}^{\breve{\,}}}_{h\mathcalligra{nf}}\) :

Thermal conductivity of hybrid nanofluid

\({{\mathcal{K}}^{\breve{\,}}}_{mh\mathcalligra{nf}}\) :

Thermal conductivity of Modified hybrid nanofluid

\({{\rho }^{\breve{\,}}}_{nf}\) :

Density of nanofluid

\({{\left(\rho {c}_{p}\right)}^{\breve{\,}}}_{nf}\) :

Heat capacity of nanofluid

\({{\left(\rho {c}_{p}\right)}^{\breve{\,}}}_{hnf}\) :

Heat capacity of hybrid nanofluid

\({{\left(\rho {c}_{p}\right)}^{\breve{\,}}}_{mhnf}\) :

Heat capacity of Modified hybrid nanofluid

\({{\mu }^{\breve{\,}}}_{nf}\) :

Dynamic Viscosity of nanofluid

\({{\mu }^{\breve{\,}}}_{hnf}\) :

Dynamic viscosity of Hybrid nanofluid

\({{\mu }^{\breve{\,}}}_{mhnf}\) :

Dynamic viscosity of Modified hybrid nanofluid

\({{\beta }^{\breve{\,}}}_{\mathcalligra{nf}}\) :

Thermal expansion of nanofluid

\({{\beta }^{\breve{\,}}}_{h\mathcalligra{nf}}\) :

Thermal expansion of Hybrid nanofluid

\({{\beta }^{\breve{\,}}}_{mh\mathcalligra{nf}}\) :

Thermal expansion of Modified Hybrid nanofluid

\({{\sigma }^{\breve{\,}}}_{nf}\) :

Electrical conductivity of the nanofluid

\({{\sigma }^{\breve{\,}}}_{hnf}\) :

Electrical conductivity of Hybrid nanofluid

\({{\sigma }^{\breve{\,}}}_{mhnf}\) :

Electrical conductivity of Modified hybrid nanofluid

\({{\alpha }^{\breve{\,}}}_{\mathcalligra{nf}}\) :

Thermal diffusivity of nanofluid

\({{\alpha }^{\breve{\,}}}_{h\mathcalligra{nf}}\) :

Thermal diffusivity of hybrid nanofluid

\({{\alpha }^{\breve{\,}}}_{mh\mathcalligra{nf}}\) :

Thermal diffusivity of Modified hybrid nanofluid

\({m}^{\breve{\,}}\) :

Mass per unit area

\({\sigma }^{\breve{\,}}\) :

Elastic tension in the membrane

\({\mathcal{C}}^{\breve{\,}}\) :

Coefficient of viscous damping force

\(\phi \) :

Nano particle volume fraction

\(m\) :

Different shape models

E1 :

Rigidity parameter

E2 :

Stiffness parameter

E3 :

Viscous damping force parameter

\(M\) :

Hartmann number

\({{G}_{r}}^{\breve{\,}}\) :

Grashof number

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Authors and Affiliations

  1. Department of Mathematics, BUITEMS, Quetta, Pakistan

    Naheeda Iftikhar

  2. DBS&H, CEME, National University of Sciences and Technology, Islamabad, Pakistan

    Hina Sadaf

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  1. Naheeda Iftikhar
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Correspondence to Naheeda Iftikhar.

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Iftikhar, N., Sadaf, H. Mathematical modelling of modified hybrid nanofluid in a peristaltic diverging tube with MHD and convective boundary conditions. Comp. Part. Mech. 10, 1477–1491 (2023). https://doi.org/10.1007/s40571-023-00553-6

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