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An improved heat conduction analysis in swirling viscoelastic fluid with homogeneous–heterogeneous reactions

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Abstract

This paper studies an unsteady magnetohydrodynamic (MHD) Maxwell fluid flow on a rotating as well as a vertically moving disk in the presence of homogeneous–heterogeneous reactions. An improved heat conduction theory, namely Cattaneo–Christov heat flux, is implemented instead of classical Fourier’s law to analyze the thermal features. The problem is basically an extension of the well-known von Karman viscous pump problem to the situation where a disk is rotating. The leading equations of motion are converted into a set of nonlinear differential equations by using von Karman transformations. A Matlab-based scheme, namely bvp4c, which uses finite difference method, is employed for numerical integration. It is noted that the wall motion of the rotating disk performs a similar effect to that of suction/injection. Further, it is observed that by elevating thermal relaxation time parameter, the temperature field diminishes. Moreover, stronger rates of homogeneous and heterogeneous reactions cause to reduce the concentration profile.

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Abbreviations

\(r,\varphi ,z\) :

Cylindrical coordinates

u, v, w :

Velocity components \(({\hbox {m s}}^{-1})\)

a(t):

Vertical distance \(\left( {\hbox {m}}\right)\)

\(\overset{\centerdot }{a}(t)\) :

Vertical velocity \(({\hbox {m s}}^{-1})\)

h :

Location of vertically moving disk \(\left( {\hbox {m}}\right)\)

T :

Temperature of fluid \(\left( {\hbox {K}}\right)\)

\(\varOmega (t)\) :

Angular velocity \(\left( {\hbox {rad s}}^{-1}\right)\)

S :

Wall motion parameter

\(\lambda _{1}\) :

Fluid relaxation time

\(\nu\) :

Kinematic viscosities \(({\hbox {m}}^{2}\,{\hbox {s}}^{-1})\)

M :

Magnetic field parameter

\(\alpha _{1}\) :

Wall temperature parameter \(\left( {\hbox {K}}\right)\)

\(\rho c_{p}\) :

Specific heat

G :

Dimensionless azimuthal velocity

\(B_{0}\) :

Magnetic field strength \(\left( {\hbox { N s C}}^{-1}\,\mathrm{m}^{-1}\right)\)

c :

Arbitrary constant

\(A^{*},B^{*}\) :

Chemical species

\(a^{*},b^{*}\) :

Concentration of chemical species

\(D_\mathrm{A},D_\mathrm{B}\) :

Diffusion coefficients

g :

Homogeneous concentration

h :

Heterogeneous concentration

\(k_\mathrm{c},k_{\rm s}\) :

Rate constants

\(\delta\) :

Ratio of mass diffusion coefficient

\(\alpha\) :

Thermal diffusivity \(\left( {\hbox {m}}^{2}\ {\hbox {s}}^{-1}\right)\)

Re:

Reynolds number

\(T_{\infty }\) :

Ambient fluid temperature

\(T_\mathrm{w}(t)\) :

Wall temperature \(\left( {\hbox {K}}\right)\)

\(\rho c_{p}\) :

Heat capacity of fluid \(\left( {\hbox {J m}}^{-2}\,\mathrm{K}^{-1}\right)\)

\(\eta\) :

Dimensionless variable

\(\theta\) :

Dimensionless temperature

\(\omega\) :

Rotation parameter

\(k_\mathrm{1}\) :

Strength of homogeneous reaction

H :

Dimensionless axial velocity

\(\beta _{1}\) :

Deborah number

\(k_{2}\) :

Strength of heterogeneous reaction

\({\mathbf {V}}\) :

Velocity field \(\left( {\hbox {m s}}^{-1}\right)\)

\({\mathbf {q}}\) :

Heat flux \(\left( {\hbox {W m}}^{-2}\right)\)

F :

Dimensionless radial velocity

\(\beta\) :

Wall permeability

\(\rho\) :

Fluid density \(\left( {\hbox {kg m}}^{-3}\right)\)

\(\lambda _{2}\) :

Thermal relaxation time parameter

k :

Thermal conductivity

\(\sigma\) :

Electric conductivity \(\left( {\hbox {S m}}^{-1}\right)\)

\(\beta _{2}\) :

Thermal relaxation parameter

Sc:

Schmidt number

\(\mu\) :

Dynamic viscosities \(({\hbox {kg m}}^{-1}\ \hbox{s}^{-1})\)

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Acknowledgements

This work has the financial supports from Higher Education Commission (HEC) of Pakistan under the project number: 6210.

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Authors and Affiliations

  1. Department of Mathematics, Quaid-i-Azam University, Islamabad, 44000, Pakistan

    Masood Khan, Jawad Ahmed & Wajid Ali

  2. Department of Basic Sciences, University of Engineering and Technology, Taxila, 47050, Pakistan

    Jawad Ahmed

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  1. Masood Khan
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Correspondence to Jawad Ahmed.

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Khan, M., Ahmed, J. & Ali, W. An improved heat conduction analysis in swirling viscoelastic fluid with homogeneous–heterogeneous reactions. J Therm Anal Calorim 143, 4095–4106 (2021). https://doi.org/10.1007/s10973-020-09342-2

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