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Mixed initial-boundary value problems describing motions of Maxwell fluids with linear dependence of viscosity on the pressure

  • Constantin Fetecau ORCID logo , Dumitru Vieru ORCID logo EMAIL logo , Abdul Rauf ORCID logo and Tahir Mushtaq Qureshi
Published/Copyright: October 13, 2021
Zeitschrift für Naturforschung A
From the journal Zeitschrift für Naturforschung A Volume 76 Issue 12

Abstract

Some mixed initial-boundary value problems are analytically studied. They correspond to unsteady motions of the incompressible upper-convected Maxwell (IUCM) fluids with linear dependence of viscosity on the pressure between infinite horizontal parallel plates. The fluid motion is generated by the upper plate that applies time-dependent shear stresses to the fluid. Exact solutions are established for the dimensionless velocity and nontrivial shear stress fields using a suitable change of the spatial variable and the Laplace transform technique. They are presented as sum of the steady-state and transient components and are used to determine the required time to reach the permanent state. Comparisons between exact and numerical solutions indicate an excellent agreement. Analytical solutions for the unsteady motion of the same fluids induced by an exponential shear stress on the boundary are obtained as limiting cases of the general solutions. Moreover, the steady-state solutions corresponding to the ordinary IUCM fluids performing the initial motions are provided by means of asymptotic approximations of standard Bessel functions. Finally, spatial variation of starting solutions and the influence of physical parameters on the fluid motion are graphically underlined and discussed.

Keywords: exact solutions; parallel plates; pressure-dependent viscosity; upper-convected Maxwell fluids
Corresponding author: Dumitru Vieru, Department of Theoretical Mechanics, Technical University of Iasi, Iasi 700050, Romania, E-mail:

Acknowledgments

The authors would like to express their sincere gratitude to the anonymous reviewers for the careful assessment, fruitful remarks, valuable suggestions and kind appreciations regarding the initial version of the manuscript.

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: None declared.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Appendix

(A1) J 0 ( − z ) = J 0 ( z ) , J 1 ( − z ) = − J 1 ( z ) , Y 0 ( − z ) = Y 0 ( z ) + 2 i J 0 ( z ) , Y 1 ( − z ) = − Y 1 ( z ) − 2 i J 1 ( z ) ,

(A2) d d s J 1 [ ξ ( s ) ] Y 0 [ ξ ( s ) β + 1 ] − Y 1 [ ξ ( s ) ] J 0 [ ξ ( s ) β + 1 ] = 2 s W e + 1 π s ( s W e + 1 ) J 1 2 [ ξ ( s ) ] − J 0 2 [ ξ ( s ) β + 1 ] J 1 [ ξ ( s ) ] J 0 [ ξ ( s ) β + 1 ] ,

(A3) R e z u ̄ c ( y , s ) e s t s = s 1 n + R e z u ̄ c ( y , s ) e s t s = s 2 n = β π q n 2 s 1 n ( s 2 n 2 + ω 2 ) ( 2 s 2 n W e + 1 ) e s 1 n t + s 2 n ( s 1 n 2 + ω 2 ) × ( 2 s 1 n W e + 1 ) e s 2 n t ( s 1 n 2 + ω 2 ) s 2 n 2 + ω 2 × ( 2 s 1 n W e + 1 ) ( 2 s 2 n W e + 1 )   × J 0 ( q n β + 1 ) J 1 ( q n ) J 0 2 ( q n β + 1 ) − J 1 2 ( q n ) B 0 ( β ( 1 − y ) + 1 , q n ) ,

where s 1n and s 2n are roots of the algebraic equation 4 W e s 2 + 4 s − β q n 2 = 0 .

(A4) ( s 1 n 2 + ω 2 ) ( s 2 n 2 + ω 2 ) = ( β 2 q n 2 − 4 W e ω 2 ) 2 + 16 ω 2 16 W e 2 ,

(A5) ( 2 s 1 n W e + 1 ) ( 2 s 2 n W e + 1 ) = − 4 a n 2 W e 2 = W e β 2 q n 2 − 1 ,

(A6) s 1 n ( s 2 n 2 + ω 2 ) ( 2 s 2 n W e + 1 ) = ( β 2 q n 2 − 4 W e ω 2 ) ( 1 − W e β 2 q n 2 ) 8 W e 2 + ( β 2 q n 2 + 4 W e ω 2 ) 1 − W e β 2 q n 2 8 W e 2 ,

(A7) s 2 n ( s 1 n 2 + ω 2 ) ( 2 s 1 n W e + 1 ) = ( β 2 q n 2 − 4 W e ω 2 ) ( 1 − W e β 2 q n 2 ) 8 W e 2 − ( β 2 q n 2 + 4 W e ω 2 ) 1 − W e β 2 q n 2 8 W e 2 ,

(A8) J 0 ( z ) ≈ 1 , J 1 ( z ) ≈ z 2 , Y 0 ( z ) ≈ 2 π ln z 2 + α , Y 1 ( z ) ≈ − 2 π z   f o r z ≪ 1 ,

where α = 0.5772 is the Euler–Mascheroni constant.

(A9) ( s 1 n 2 + ω 2 ) ( 2 s 1 n W e + 1 ) = β 2 q n 2 + 4 W e ω 2 + 4 s 1 n − 2 s 1 n W e ( β 2 q n 2 − 4 W e ω 2 ) 4 W e ,

(A10) ( s 2 n 2 + ω 2 ) ( 2 s 2 n W e + 1 ) = β 2 q n 2 + 4 W e ω 2 + 4 s 2 n − 2 s 2 n W e ( β 2 q n 2 − 4 W e ω 2 ) 4 W e ,

(A11) lim s → − 1 / W e Y 0 [ ξ ( s ) β + 1 ] J 1 [ ξ ( s ) β ( 1 − y ) + 1 ] − J 0 [ ξ ( s ) β + 1 ] Y 1 [ ξ ( s ) β ( 1 − y ) + 1 ] J 1 [ ξ ( s ) ] Y 0 [ ξ ( s ) β + 1 ] − Y 1 [ ξ ( s ) ] J 0 [ ξ ( s ) β + 1 ] = 1 β ( 1 − y ) + 1 ,

(A12) s 1 n 2 ( s 2 n 2 + ω 2 ) ( 2 s 2 n W e + 1 ) = 8 ω 2 ( 1 − W e β 2 q n 2 − 1 ) − β 2 q n 2 ( β 2 q n 2 − 4 W e ω 2 ) 16 W e 2 × 1 − W e β 2 q n 2 ,

(A13) s 2 n 2 ( s 1 n 2 + ω 2 ) ( 2 s 1 n W e + 1 ) = 8 ω 2 ( 1 − W e β 2 q n 2 + 1 ) + β 2 q n 2 ( β 2 q n 2 − 4 W e ω 2 ) 16 W e 2   × 1 − W e β 2 q n 2 ,

(A14) s 1 n ( s 2 n 2 + ω 2 ) ( 2 s 2 n W e + 1 ) = β 2 q n 2 + 4 W e ω 2 + ( β 2 q n 2 − 4 W e ω 2 ) 1 − W e β 2 q n 2 8 W e 2   × 1 − W e β 2 q n 2 ,

(A15) s 2 n ( s 1 n 2 + ω 2 ) ( 2 s 1 n W e + 1 ) = − ( β 2 q n 2 + 4 W e ω 2 ) + ( β 2 q n 2 − 4 W e ω 2 ) 1 − W e β 2 q n 2 8 W e 2   × 1 − W e β 2 q n 2 ,

(A16) J ν ( z ) ≈ 2 π z cos z − ( 2 ν + 1 ) π 4 , Y ν ( z ) ≈ 2 π z sin z − ( 2 ν + 1 ) π 4   f o r z ≫ 1 .

In order to prove the equality (A2), for instance, we used the next known results

J ′ 0 ( z ) = − J 1 ( z ) , Y ′ 0 ( z ) = − Y 1 ( z ) , J ′ 1 ( z ) = 1 z J 0 ( z ) − J 1 ( z ) , Y ′ 1 ( z ) = 1 z Y 0 ( z ) − Y 1 ( z ) , J ν ( z ) Y ν + 1 ( z ) − J ν + 1 ( z ) Y ν ( z ) = − 2 π z

and the fact that

J 1 [ ξ ( s ) ] Y 0 [ ξ ( s ) β + 1 ] − Y 1 [ ξ ( s ) ] J 0 [ ξ ( s ) β + 1 ] = 0 .

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Received: 2021-07-29
Revised: 2021-09-12
Accepted: 2021-09-15
Published Online: 2021-10-13
Published in Print: 2021-12-20

© 2021 Walter de Gruyter GmbH, Berlin/Boston

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https://doi.org/10.1515/zna-2021-0212
Keywords for this article
exact solutions; parallel plates; pressure-dependent viscosity; upper-convected Maxwell fluids

Articles in the same Issue

  1. Frontmatter
  2. Dynamical Systems & Nonlinear Phenomena
  3. Study of shocks in a nonideal dusty gas using Maslov, Guderley, and CCW methods for shock exponents
  4. Nonlinear excitations and dynamic features of dust ion-acoustic waves in a magnetized electron–positron–ion plasma
  5. Bifurcation analysis in a predator–prey model with strong Allee effect
  6. Hydrodynamics
  7. Mixed initial-boundary value problems describing motions of Maxwell fluids with linear dependence of viscosity on the pressure
  8. Quantum Theory
  9. Spin coherent states, Bell states, spin Hamilton operators, entanglement, Husimi distribution, uncertainty relation and Bell inequality
  10. Solid State Physics & Materials Science
  11. Tailoring the optical properties of tin oxide thin films via gamma irradiation
  12. Thermodynamics & Statistical Physics
  13. Impact of partially thermal electrons on the propagation characteristics of extraordinary mode in relativistic regime
  14. The first-order phase transition of melting for molecular crystals by Frost–Kalkwarf vapor- and sublimation-pressure equations
  15. Mathieu-state reordering in periodic thermodynamics
  16. Corrigendum
  17. Corrigendum to: Study of ferrofluid flow and heat transfer between cone and disk
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