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Steady flow and heat transfer of the power-law fluid between two stretchable rotating disks with non-uniform heat source/sink

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Abstract

This study aims to analyze the heat transfer phenomenon of power-law fluid with the occurrence of non-uniform heat source/sink within two stretchable disks which are parted with the constant distance and are co-axially rotating. The thermal conductivity is obeying the similar properties of power-law as that of viscosity. Von Karman’s generalized similarity transformation has been used firstly to reduce the physically modeled partial differential equations to nonlinear coupled ordinary differential equations and then tackled numerically with shooting method by finding missing initial conditions with the help of Newton–Raphson method and then system of equations are handled by means of RK-method. The influence of physical parameters for instance rotation as well as stretching, power-law index, Prandtl number, heat sink/source parameters upon non-dimensional velocity and temperature profiles are studied profoundly, later on, comprehensive analysis is expressed in discussion and results segment. The results which are computed numerically illustrate that the emerging parameters have substantial influences on velocity and temperature fields. In addition, rotation enhances the velocity components but temperature is predicting two diverse behaviors for shear-thinning and shear-thickening fluids, whenever upper and lower disk stretching it leads to an upsurge in radial and axial velocities but causes a decline in tangential velocity and temperature. Moreover, velocity and temperature distributions are in increasing trend except for the tangential component of the velocity which is decreasing by boosting the index of power-law. Furthermore, temperature decreases along with the similarity variable with the increasing Prandtl number but enhances with the enhancement in heat source/sink parameters. Finally, the skin friction in radial direction and local Nusselt number are escalating along the stretching parameters and Prandtl number but skin friction in tangential direction plummeting.

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Abbreviations

\(u\) :

Radial velocity \(\left( {\text{m s}}^{-1} \right)\)

\(v\) :

Tangential velocity \(\left( {\text{m s}}^{-1} \right)\)

\(w\) :

Axial velocity \(\left( {\text{m s}}^{-1} \right)\)

\(r\) :

Cylindrical coordinate (m)

\(\varphi\) :

Cylindrical coordinate (m)

\(z\) :

Cylindrical coordinates (m)

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

Local Nusselt number

\(c_{\rm p}\) :

Specific heat \(\left( {\text{J}}\, {\text{kg}}^{{-1}} \,{\text{K}}^{-1} \right)\)

\(k_{0}\) :

Positive constant

\(k\) :

Thermal conductivity \(\left( {\text{W }}\,{\text{m}}^{-1} \,{\text{K}}^{-1} \right)\)

\(s_{1}\) :

Lower disk stretching rate \(\left( {\text{rad }}\,{\text{s}}^{-1} \right)\)

\(s_{2}\) :

Upper disk stretching rate \(\left( {\text{rad }}\,{\text{s}}^{-1} \right)\)

\(T\) :

Temperature of the fluid (K)

\(T_{1}\) :

Temperature at lower wall (K)

\(T_{2}\) :

Temperature at upper wall (K)

\(S_{1}\) :

Lower disk stretching parameter

\(S_{2}\) :

Upper disk stretching parameter

\(F\) :

Dimensionless radial velocity

\(G\) :

Dimensionless tangential velocity

\(H\) :

Dimensionless axial velocity

\({ \Pr }\) :

Prandtl number

\(C_{\text{Fr}}\) :

Skin friction in radial directions

\(C_{{\rm F}\theta }\) :

Skin friction in tangential direction

\(q_{\rm w}\) :

Constant heat flux \(\left( {{\text{W}}\, {\text{m}}^{-2}} \right)\)

\(q^{\prime\prime\prime}\) :

Non-uniform heat source/sink \(\left( {\text{k }}\,{\text{s}}^{-1} \right)\)

\(B^{*}\) :

Temperature-dependent heat source/sink parameter

\(A^{*}\) :

Temperature-dependent heat source/sink parameter

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

Local Reynolds number

\(\theta\) :

Dimensionless temperature

\(\rho\) :

Effective density \(\left( {{\text{kg }}\,{\text{m}}^{-3}} \right)\)

\(\nu\) :

Kinematic viscosity \(\left( {\text{m}}^{2} \,{\text{s}}^{-1} \right)\)

\(\varOmega\) :

Rotational parameter

\(\mu\) :

Effective dynamic viscosity \(\left( {\text{kg}}\,{\text{m}}^{-1} {\text{s}}^{-1} \right)\)

\(\mu_{0}\) :

Consistency coefficient

\(\varOmega_{1}\) :

Lower disk angular velocity \(\left( {\text{rad}}\, {\text{s}}^{-1} \right)\)

\(\varOmega_{2}\) :

Upper disk angular velocity \(\left( {\text{rad}}\, {\text{s}}^{-1} \right)\)

\(\xi\) :

Dimensionless similarity variable

\(\tau_{\rm rz}\) :

Shear stress in radial direction \(\left( {\text{Pa}} \right)\)

\(\tau_{\uptheta {\rm{z}}}\) :

Shear stress in tangential direction \(\left( {\text{Pa}} \right)\)

\('\) :

Derivative w. r. t \(\xi\)

\(n\) :

Power-law index

\(\varvec{*}\) :

Dimensionless variables

p:

Pressure \(\left( {\text{Pa}} \right)\)

N:

Effective variable

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Authors and Affiliations

  1. Beijing Key Laboratory for Magneto-Photoelectrical Composite and Interface Science, Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing, 100083, China

    Usman

  2. Division of Mathematics, University of Dundee, Dundee, DD1 4HN, Scotland, UK

    Ping Lin

  3. Department of Mathematics, University of Education, Lahore (Attock Campus), Attock, 43600, Pakistan

    Abuzar Ghaffari

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  1. Usman
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Usman, Lin, P. & Ghaffari, A. Steady flow and heat transfer of the power-law fluid between two stretchable rotating disks with non-uniform heat source/sink. J Therm Anal Calorim 146, 1735–1749 (2021). https://doi.org/10.1007/s10973-020-10142-x

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