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Finite Element Analysis of Viscoelastic Nanofluid Flow with Energy Dissipation and Internal Heat Source/Sink Effects

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Abstract

A numerical study is conducted of laminar viscoelastic nanofluid polymeric boundary layer stretching sheet flow. Viscous dissipation, surface transpiration (suction/injection), internal heat generation/absorption and work done due to deformation are incorporated using a second grade viscoelastic non-Newtonian nanofluid with non-isothermal associated boundary conditions. The nonlinear boundary value problem is solved using a higher order finite element method. The influence of viscoelasticity parameter, Brownian motion parameter, thermophoresis parameter, Eckert number, Lewis number, Prandtl number, internal heat generation and also wall suction on thermofluid characteristics is evaluated in detail. Validation with earlier non-dissipative studies is also included. The hp-finite element method achieves the desired accuracy at \(p=8\) with comparatively less CPU cost per iteration (with less degrees of freedom, DOF) as compared to lower order finite element methods. The simulations have shown that greater polymer fluid viscoelasticity \(({k}_{1})\) accelerates the flow. A rise in Brownian motion parameter (Nb) and thermophoresis parameter (Nt) elevates temperatures and reduce the heat transfer rates (local Nusselt number function). Increasing Eckert number increases temperatures whereas increasing Prandtl number (Pr) strongly lowers temperatures. Increasing internal heat generation \(({Q} > 0)\) elevates temperatures and reduces the heat transfer rate (local Nusselt number function) whereas heat absorption \(({Q} < 0)\) generates the converse effect. Increasing suction \(({f}_{w}>0)\) reduces velocities and temperatures but elevates enhances mass transfer rates (local Sherwood number function), whereas increasing injection \(({f}_{w}<0)\) accelerate the flow, increases temperatures and depresses wall mass transfer rates. The study finds applications in rheological nano-bio-polymer manufacturing.

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Abbreviations

\(u_w\) :

Sheet velocity (m/s)

A, B, E :

Constants

Q :

Internal heat source/sink

C :

Nanoparticle volume fraction

\(C_w\) :

Nanoparticle volume fraction

\(C_\infty \) :

Ambient nanoparticle volume fraction

Nt :

Thermophoresis parameter

(x, y):

Cartesian coordinates (m)

\(T_w\) :

Temperature at the sheet (K)

\(T_\infty \) :

Ambient temperature attained (K)

T :

Temperature on the sheet (K)

Pr :

Prandtl number

\(q_m\) :

Wall mass flux (kg/s)

\(q_w\) :

Wall heat flux \((\hbox {W}/\hbox {m}^{2})\)

\(D_B\) :

Brownian diffusion coefficient \((\hbox {m}^{2}/\hbox {s})\)

\(D_T\) :

Thermophoretic diffusion coefficient \((\hbox {m}^{2}/\hbox {s})\)

\(u_{w}\) :

Velocity of stretching sheet (m/s)

\(f(\eta )\) :

Dimensionless stream function

\(g(\eta )\) :

Gravitational acceleration \((\hbox {m}/\hbox {s}^2)\)

Nb :

Brownian motion parameter

Le :

Lewis number

\(k_{1}\) :

Viscoelastic parameter

\(Nu_{x}\) :

Nusselt number

\(A_{1}, A_{2}\) :

Rivlin–Ericksen tensors in the constitutive relation \((\hbox {N}/\hbox {m}^2)\)

Ec :

Eckert number

\(Sh_{x}\) :

Sherwood number

\(C_{f}\) :

Skin friction

\(u,\upsilon \) :

Velocity components along x–y axes (m/s)

\(f_w\) :

Suction/injection parameter

m :

Power-law parameter

\(\Gamma \) :

Stress tensor \((\hbox {N}/\hbox {m}^{2})\)

\(\tau \) :

Parameter defined by \(\varepsilon \left( {\rho c} \right) _p /\left( {\rho c} \right) _f\)

\(\left( {\rho c} \right) _f\) :

Heat capacity of the fluid \((\hbox {J}/\hbox {kg}^{3}\,\hbox {K})\)

\(\phi \left( \eta \right) \) :

Rescaled nanoparticle volume fraction

\(\eta \) :

Similarity variable

\(\theta \left( \eta \right) \) :

Dimensionless temperature

\(\left( {\rho c} \right) _p\) :

Effective heat capacity of the nanoparticle material \((\hbox {J}/\hbox {kg}^{3}\,\hbox {K})\)

\(\rho _{f}\) :

Fluid density \((\hbox {kg}/\hbox {m}^{3})\)

\(\beta \) :

Volumetric expansion coefficient of the fluid (l/K)

\(\rho _{p}\) :

Nanoparticle mass density \((\hbox {kg}/\hbox {m}^3)\)

\(\psi \) :

Stream function

\(\nu \) :

Fluid kinematic viscosity \((\hbox {m}^2/\hbox {s})\)

\(\alpha _m\) :

Thermal diffusivity \((\hbox {m}^2/\hbox {s})\)

\(\alpha _1 , \alpha _2\) :

Material moduli \((\hbox {N}/\hbox {m}^2)\)

\(\beta _1 , \beta _2 , \beta _3\) :

Higher order viscosities \((\hbox {m}^2/\hbox {s})\)

w :

Condition on the sheet (wall)

\(\infty \) :

Condition far away from the sheet (free stream)

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Acknowledgments

Dr. O. Anwar Bég is grateful to the late Professor Howard Brenner (1929-2014) of Chemical Engineering, MIT, USA, for some excellent discussions regarding viscoelastic characteristics of biopolymers.

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Authors and Affiliations

  1. Department of Mathematics, Jaypee Institute of Information Technology, Noida, India

    P. Rana

  2. Department of Mathematics, Indian Institute of Technology, Roorkee, 247667, India

    R. Bhargava

  3. Spray Research Group, School of Computing, Science and Engineering, Newton Bldg, The Crescent, University of Salford, Manchester, M54WT, England, UK

    O. Anwar Bég & A. Kadir

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Rana, P., Bhargava, R., Bég, O.A. et al. Finite Element Analysis of Viscoelastic Nanofluid Flow with Energy Dissipation and Internal Heat Source/Sink Effects. Int. J. Appl. Comput. Math 3, 1421–1447 (2017). https://doi.org/10.1007/s40819-016-0184-5

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