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Laminar mixed convection of permeable fluid overlaying immiscible nanofluid

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Abstract

Immiscible flow has been extensively emerged in science and technology. Researchers and architects were delighted by the concept of multiple fluid transport by the means of shear pressure. The reliance of drag impact of the two immiscible liquids is very much aspired but yet challenging. A mathematical examination has been conveyed to understand the free convection inside a vertical vessel. There are two immiscible liquids filled in the enclosure which are synthesized as two discrete regions encompassing a nanofluid and permeable fluid. The Tiwari–Das model and Dupuit–Forchheimer is utilized to define the nanofluid and permeable fluid, respectively. Southwell over-relaxation technique subject to suitable interface and boundary conditions is bestowed to solve the conservation equations. Essential criteria defining the fluid flow and energy transfer are studied deliberately. The outcomes demonstrate that the Grashof, Brinkman and Darcy numbers augment the velocity, whereas inertial, solid volume fraction, viscosity and thermal conductivity ratios depletes the momentum. The temperature distributions are not much modulated with any of the controlling parameters. By sagging nanoparticles, the flow is not much reformed but reckoning copper nanoparticle as ethylene glycol–mineral oil base fluid regulates the supreme flow. Diamond nanoparticle dropped in water catalyzes the highest rate of heat transfer.

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Abbreviations

\(A_{i} \) :

Ratio of length to width \(\left( {\frac{a_{i} }{2b}} \right) \)(-)

\(a_{i} \) :

Length of the conduit (m)

b :

Breadth of conduit (m)

Br :

Brinkman number \(\left( {\frac{\mu _{f}^{3}}{K_{f} \,\,\rho _{f} ^{2}\,\,b^{2}\,\,\left( {T_{w2} \,-\,T_{w1} } \right) \,\,}} \right) \)

Da :

Darcy number \(\left( {\frac{\kappa }{b^{2}}} \right) \)

Gr :

Grashof number \(\left( {\frac{g\,\,\rho _{f}^{2} \,\beta _{f} b^{3}\left( {T_{w2} \,-\,T_{w1} } \right) }{\mu _{f}^{2} }} \right) \)

\(K_{nf} \) :

Thermal conductivity (W/mK)

\(Nx_{i} ,\,Ny\) :

Number of grid points in x and y directions (-)

n :

Ratio of densities \(\left( {\frac{\rho _{2} }{\rho _{f} }} \right) \) (-)

P :

Pressure (Pa)

p :

Pressure gradient in no dimension \(\left( {\frac{\rho _{f} \,\,b^{3}}{\mu _{f}^{2} \,\,}\,\frac{\partial P}{\partial Z}} \right) \)(-)

\(T_{i} \) :

Temperature (K)

\(T_{wi} \) :

Temperature on the walls (K)

\(W_{i} \) :

Velocity (m/s)

\(w_{i} \) :

Velocity with no dimension (-)

x, y, z :

Coordinates with zero dimension (-)

\(\Delta x_{i} \) and \(\Delta y\) :

Grid size (m)

\(X_{i} \,,\,\,Y,Z\) :

Dimensional ordinates (-)

\(\lambda \) :

Proportion of the viscosity \(\left( {\frac{\mu _{2} }{\mu _{f} }} \right) \)(-)

\(\beta _{i} \) :

Thermal expansion coefficient (K)

\(\beta \) :

Proportion of heat expansion coefficients \(\left( {\frac{\beta _{2} }{\beta _{f} }} \right) \) (-)

K :

Proportion of the thermal conductivity \(\left( {\frac{K_{2} }{K_{f} }} \right) \) (-)

\(\mu \) :

Dynamic viscosity (-)

\(\theta \) :

Dimensionless temperature (-)

\(\rho \) :

Density of the fluid (kg/m\(^{3})\)

\(i\,=\,1,2\) :

Quantities for region-1 and region-2, respectively.

nf :

Nanofluid

f :

Base fluid

s :

Solid nanoparticles

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  1. Department of Mathematics, Gulbarga University, Gulbarga, Karnataka, 585 106, India

    J. C. Umavathi

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Umavathi, J.C. Laminar mixed convection of permeable fluid overlaying immiscible nanofluid. Eur. Phys. J. Spec. Top. 231, 2583–2603 (2022). https://doi.org/10.1140/epjs/s11734-022-00585-8

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