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Analysis of aligned magnetic field, flow separation and stability in a porous medium saturated by hybrid nanofluids

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Abstract

This investigation is aimed to look at the significance of Joule heating and aligned magnetic effects in porous media saturated by a Ti-alloy/multi-wall carbon nanotube (MWCNT)-water based hybrid nanofluid in the presence of heat generation and velocity slip toward an exponentially shrinking surface. The Tiwari–Das approach is used to develop the mathematical modeling, which is then converted into system of nonlinear ODEs using appropriate similarity transformations. The dual solutions are noticed for the resultant boundary value problem using Newton–Raphson and Runge–Kutta based shooting scheme and then the comparative analysis is provided with existing results. Among these solutions, the first solution is found to be more stable and physically valid over time according to the smallest eigenvalue approach of linear temporal stability analysis. The important outcomes of this study, based on the stable solutions, are: (i) the hybrid nanofluid’s Nusselt number, skin friction, velocity and temperatures rise when the inclined magnetic parameter rises, (ii) the delay of boundary layer separation is noticed with enhancing values of the first-order slip, Darcy porosity and inclined magnetic parameters, (iii) the value of smallest eigenvalue is growing with growing values of the inclined magnetic parameter, and (iv) the thickness of momentum and thermal boundary layers is thinner for the first solution than the second solution. In addition, the delay in boundary layer separation occurs with increasing values of the first-order slip, porosity, and aligned magnetic parameters and decreasing values of the Joule heating parameter. Finally, the streamline patterns are provided in this study to have a better understanding of the fluid flow behavior.

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Abbreviations

\(a_1\) :

Velocity slip factor

b :

Velocity parameter of the shrinking surface

\(b_1\) :

Initial length of slip factor

B(x):

Variable magnetic field

\(B_{0}\) :

Strength of the magnetic field

\(C_\text {f}\) :

Skin friction coefficient

\(C_\text {p}\) :

Specific heat capacity with uniform pressure

f :

Dimensionless free stream function

J :

Joule parameter

k :

Thermal conductivity

\(k^*_\text {p}\) :

Permeability of the porous medium

\(k_\text {f}\) :

Thermal conductivity of the base fluid

\(k_\text {s}\) :

Thermal conductivity of the solid nanoparticles

\(k_\text {hnf}\) :

Thermal conductivity of the hybrid nanofluid

\(k_\text {nf}\) :

Thermal conductivity of the nanofluid

Q :

Heat generation parameter

L :

Characteristic length

M :

Magnetic parameter

\({\text {Nu}}_{\text {x}}\) :

Local Nusselt number

Pd:

Darcy porosity parameter

Pr:

Prandtl number

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

Surface heat flux

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

Local Reynolds number

S :

Dimensionless suction/injection parameter

t :

Time variable

T :

Fluid temperature

\(T_\text {o}\) :

Temperature parameter of the shrinking surface

\(T_\text {w}\) :

Wall temperature

\(T_{\infty }\) :

Ambient temperature

(u, v):

Velocity components in x and y directions

\(u_\text {w}(x)\) :

Velocity of the shrinking sheet

\(u_{\infty }(x)\) :

Free stream velocity

\(v_\text {w}(x)\) :

Suction or injection velocity

x, y :

Coordinates along and normal to the surface

\(\eta\) :

Dimensionless similarity variable

\(\epsilon\) :

Porosity

\(\alpha\) :

First-order slip parameter

\(\gamma\) :

Incline angle

\(\lambda\) :

Shrinking parameter

\(\mu _\text {f}\) :

Dynamic viscosity of the base fluid

\(\mu _\text {hnf}\) :

Dynamic viscosity of the hybrid nanofluid

\(\mu _\text {nf}\) :

Dynamic viscosity of the nanofluid

\(\nu _\text {f}\) :

Kinematic viscosity of the base fluid

\(\psi\) :

Stream function

\(\phi _{\text {M}}\) :

MWCNT nanoparticle’s volume fraction

\(\phi _{\text {T}}\) :

Ti-alloy nanoparticle’s volume fraction

\(\rho\) :

Density

\(\rho _\text {f}\) :

Density for the base fluid

\(\rho _\text {hnf}\) :

Density of the hybrid nanofluid

\(\rho _\text {nf}\) :

Density of the nanofluid

\(\rho _{\text{s}}\) :

Density of the solid nanoparticles

\((\rho C_\text {p})\) :

Heat capacity

\((\rho C_\text {p})_\text {f}\) :

Heat capacity of the base fluid

\((\rho C_\text {p})_\text {hnf}\) :

Heat capacity of the hybrid nanofluid

\((\rho C_\text {p})_\text {nf}\) :

Heat capacity of the nanofluid

\((\rho C_\text {p})_\text {s}\) :

Heat capacity of the solid nanoparticles

\(\sigma _{1}\) :

Unknown eigenvalue

\(\sigma _\text {f}\) :

Electric conductivity of the base fluid

\(\sigma _\text {hnf}\) :

Electric conductivity of the hybrid nanofluid

\(\sigma _\text {nf}\) :

Electric conductivity of the nanofluid

\(\sigma _{\text{s}}\) :

Electric conductivity of the solid nanoparticles

\(\tau\) :

Time variable

\(\tau _{\text {w}}\) :

Wall shear stress

\(\theta\) :

Dimensionless temperature

w:

Wall condition

\(\infty\) :

Ambient condition

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  1. Department of Mathematics, National Institute of Technology, Warangal, 506004, Telangana, India

    Har Lal Saran & Ch. RamReddy

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Saran, H.L., RamReddy, C. Analysis of aligned magnetic field, flow separation and stability in a porous medium saturated by hybrid nanofluids. J Therm Anal Calorim 148, 3765–3781 (2023). https://doi.org/10.1007/s10973-023-11946-3

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