Abstract
The paper investigates analytically the instability of Casson binary nanofluids and further numerically by taking blood as the base fluid. The blood being a non-Newtonian fluid is characterized by Casson model and is assumed to undergo nano-scale, solutal and buoyancy effects which induce convection currents in the flow. Nanoparticle volume fraction is taken to be so small in the fluid that initially it is assumed to be constant. Small disturbances are added to the solution and normal mode technique is used to convert partial differential equations into ordinary. Further, these equations are solved using one term Galerkin method for free–free, rigid–free and rigid–rigid boundaries. The complex expressions of Rayleigh number are simplified with valid approximations to get analytical results. Thermal, solutal and nano-scale effects are found to contribute equally and independent of each other while their overall impact is inversely proportional to Casson parameter. To get a complete insight of the problem, complex analytical expressions are explored numerically and the effects of various parameters on the instability of blood are depicted graphically using the software Mathematica. The alumina nanoparticles have more destabilizing impact on blood than copper and could play a significant role on human health, more specifically in the cardiovascular system.
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Abbreviations
- a :
-
Wave number
- c :
-
Nanofluid specific heat
- C:
-
Solute concentration
- \(C_{0} \) :
-
Solute concentration at upper layer
- \(C_{1} \) :
-
Solute concentration at lower layer
- d :
-
Depth of the layer
- \(d_\mathrm{p}\) :
-
Diameter of nanoparticle
- \(D_\mathrm{b} \) :
-
Brownian diffusion coefficient
- \(D_\mathrm{{ct}}\) :
-
Diffusivity of soret type
- \(D_\mathrm{s} \) :
-
Solutal diffusivity
- \(D_\mathrm{t} \) :
-
Thermophoresis diffusion coefficient
- \(D_\mathrm{{tc}} \) :
-
Diffusivity of Dufour type
- \(e_{ij} \) :
-
Deformation rate
- g:
-
Acceleration due to gravity
- \(H_\mathrm{y}\) :
-
Yield stress for Casson fluid
- k :
-
Thermal conductivity
- \(k_\mathrm{b}\) :
-
Boltzmann’s constant
- \(m_{x} \) :
-
Wave number in x-pivot
- \(n_{y}\) :
-
Wave number in y-pivot
- p :
-
Pressure
- s :
-
Growth rate
- t:
-
Time
- T :
-
Temperature
- \(T_{0}\) :
-
Temperature at upper layer
- \(T_{1} \) :
-
Temperature at lower layer
- v :
-
Velocity of fluid (\(v=(u,v,w))\)
- \(\alpha _\mathrm{f}\) :
-
Thermal diffusivity of fluid
- \(\beta \) :
-
Casson parameter
- \(\beta _\mathrm{c}\) :
-
Solutal volumetric coefficient
- \(\beta _\mathrm{t} \) :
-
Thermal volumetric coefficient
- \(\mu \) :
-
Viscosity of the nanofluid
- \(\mu _\mathrm{d} \) :
-
Dynamic viscosity
- \(\tau \) :
-
Stress tensor
- \(\rho _\mathrm{p} \) :
-
Nanoparticle density
- \(\rho _{0}\) :
-
Fluid density at temperature \(T_{0} \)
- \(\rho c\) :
-
Heat capacity
- \(\phi _\mathrm{b}\) :
-
Initial nanoparticle volume fraction
- \(L_\mathrm{e}\) :
-
Solute Lewis number
- \(L_\mathrm{n} \) :
-
Nanofluid Lewis number
- \(N_\mathrm{a}\) :
-
Diffusivity ratio
- \(N_\mathrm{b} \) :
-
Particle density increment
- \(N_\mathrm{{ct}}\) :
-
Soret parameter
- \(N_\mathrm{{tc}}\) :
-
Dufour parameter
- \(p_\mathrm{r}\) :
-
Prandtl number
- \(R_\mathrm{a} \) :
-
Thermal Rayleigh number
- \(R_\mathrm{m}\) :
-
Basic-density Rayleigh number
- \(R_\mathrm{n} \) :
-
Nanoparticle Rayleigh number
- \(R_\mathrm{s}\) :
-
Solute Rayleigh number
- \(\sim \) :
-
Basic solution
- \(\hat{\ }\) :
-
Perturbed quantities
- \(*\) :
-
Non-dimensional variable
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Acknowledgements
One of the authors, Mamta Devi is thankful to Council of Scientific and Industrial Research, New Delhi-110012, India for the financial assistance in the form of JRF [Ref. No.: 09/135 (0895)/2019-EMR-1]. The authors express their gratefulness to the Reviewers for their valuable comments which have gone a long way in improving the quality of the paper.
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Gupta, U., Sharma, J. & Devi, M. Double-diffusive instability of Casson nanofluids with numerical investigations for blood-based fluid. Eur. Phys. J. Spec. Top. 230, 1435–1445 (2021). https://doi.org/10.1140/epjs/s11734-021-00053-9
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DOI: https://doi.org/10.1140/epjs/s11734-021-00053-9