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Characteristics of melting heat transport of blood with time-dependent cross-nanofluid model using Keller–Box and BVP4C method

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Abstract

The blood flow with heat transportation has prominent clinical importance during the levels where the blood flow needs to be checked (surgery) and the heat transportation rate must be controlled (therapy). This work presents an analysis of the melting heat transport of blood, which consists of iron nanoparticles along free convection with cross-model and solution of the partial differential equation (PDEs) are emerged by the mathematical model. Being the importance of iron oxide nanoparticles in applications of the biomedical field due to their intrinsic properties such as colloidal stability, surface engineering capability and low toxicity, this study has been launched. Furthermore, PDEs of the problem are converted into a set of nonlinear ordinary differential equations (ODEs) by proper transformations. The solution of this system of ODEs is calculated through RK 4 method and Keller–Box scheme. Some leading points and numerical results of this study of both types of presence and absence of meting effects are tabulated.

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Data availability statement

No data are used to support this study.

Abbreviations

\(V,\tau\) :

Velocity, Cauchy tensor

\(p,I,A_{1}\) :

Pressure, identity tensor, Rivilin tensor

\(\mu_{\infty } ,\mu_{0}\) :

Lower shear rate and higher shear rate viscosity,

\(A = \frac{a}{c}\) :

Unsteady parameter

\(\lambda = \frac{{g\beta_{{\text{f}}} \left( {T_{2} - T_{{\text{m}}} } \right)}}{{u_{{\text{w}}}^{2} }}\) :

Convection parameter

\(M = \frac{{\left( {C_{{\text{p}}} } \right)_{{\text{f}}} \left( {T_{2} - T_{{\text{m}}} } \right)x}}{{\lambda^{*} + c_{{\text{s}}} \left( {T_{{\text{m}}} - T_{0} } \right)}}\) :

Melting parameter

\(\Pr = \frac{{\left( {\rho C_{{\text{p}}} } \right)_{{\text{f}}} }}{{\mu_{{{\text{nf}}}} }}\) :

Prandtl number

\({\text{Re}}_{x} = \tfrac{{\left( {u_{{\text{w}}} } \right)}}{{\upsilon_{{\text{f}}} }}\) :

Reynold number

\(\beta_{{{\text{nf}}}} \left( {\frac{1}{{\text{K}}}} \right)\) :

Coefficient of thermal expansion

\(k_{{{\text{nf}}}} \left( {\frac{{\text{W}}}{{{\text{Km}}}}} \right)\) :

Effective thermal conductivity

\(\rho_{{\text{f}}} \left( {\frac{{{\text{kg}}}}{{{\text{m}}^{3} }}} \right)\) :

Reference density of fluid

\(\rho_{{\text{s}}} \left( {\frac{{{\text{kg}}}}{{{\text{m}}^{3} }}} \right)\) :

Reference density of solid

\(\mu_{{\text{f}}} \left( {\frac{{{\text{Ns}}}}{{{\text{m}}^{2} }}} \right)\) :

Viscosity of fluid

\(\lambda^{*} \left( {\frac{{\text{J}}}{{{\text{kg}}}}} \right)\) :

Latent heat transfer of fluid

\(c_{{\text{s}}} \left( {\frac{{\text{J}}}{{\text{K}}}} \right)\) :

Heat capacity of solid surface

\({\text{We}}\) :

Weissenberg number

\(u\left( {\text{m/s}} \right)\) :

Velocity along x-axis

\(v\left( {\text{m/s}} \right)\) :

Velocity along y-axis

\(T_{1} \left( {\text{K}} \right)\) :

Temperature of nanofluid

\(T_{2} \left( {\text{K}} \right)\) :

Temperature of ambient fluid

\(\rho_{{{\text{nf}}}} \left( {\frac{{{\text{kg}}}}{{{\text{m}}^{3} }}} \right)\) :

Density of nanofluid

\(\mu_{{{\text{nf}}}} \left( {\frac{{{\text{Ns}}}}{{{\text{m}}^{2} }}} \right)\) :

Effective viscosity of nanofluid

\(g\left( {{\text{m/s}}^{2} } \right)\) :

Gravitational acceleration

\(\phi\) :

Visibility of concentration

\(k_{{\text{f}}} \left( {\text{W/Km}} \right)\) :

Thermal conductivity of fluid

\(k_{{\text{s}}} \left( {\text{W/Km}} \right)\) :

Thermal conductivity of solid

\(a,c\) :

Constant

\(n\) :

Cross-fluid index

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Hazara University, Mansehra, Pakistan

    Assad Ayub & Zulqurnain Sabir

  2. Department of General Studies, National Autonomous University of Chota, Cajamarca, Peru

    Gilder Cieza Altamirano

  3. Department of Mathematics, Zagazig Faculty of Engineering, Zagazig University, Zagazig, Egypt

    R. Sadat

  4. Department of Basic Science, Faculty of Engineering at Benha, Benha University, 13512, Egypt, Benha University, Banha, Egypt

    Mohamed R. Ali

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  1. Assad Ayub
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  2. Zulqurnain Sabir
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  4. R. Sadat
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Correspondence to Mohamed R. Ali.

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Ayub, A., Sabir, Z., Altamirano, G.C. et al. Characteristics of melting heat transport of blood with time-dependent cross-nanofluid model using Keller–Box and BVP4C method . Engineering with Computers 38, 3705–3719 (2022). https://doi.org/10.1007/s00366-021-01406-7

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