| Citation: |
Haroon Tahira, Abdul Majeed Siddiqui, Hamee Ullah, Dianche Lu. FLOW OF MAXWELL FLUID IN A CHANNEL WITH UNIFORM POROUS WALLS[J]. Journal of Applied Analysis & Computation, 2021, 11(3): 1322-1347. doi: 10.11948/20200158
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FLOW OF MAXWELL FLUID IN A CHANNEL WITH UNIFORM POROUS WALLS
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Abstract
In this paper a theoretical study of incompressible Maxwell fluid in a channel with uniform porous walls is presented. Along with the viscoelasticity, inertial effects are also considered. Six nonlinear partial differential equations (PDEs) with non-homogeneous boundary conditions in two dimensions are solved using recursive approach. Expressions for stream function, velocity components, volumetric flow rate, pressure distribution, shear and normal stresses in general and on the walls of the channel, fractional absorption and leakage flux are obtained. The volumetric flow rate and mean flow rate are found to be very useful to understand the flow phenomena through the channel and while defining non-dimensional parameters. Points of maximum velocity components are also identified. A graphical study is carried out to show the effect of absorption, Reynolds number, material parameter on above mentioned resulting expressions. It is observed that velocity of the fluid decreases with the increase in absorption parameter, Reynolds number and also with Maxwell parameter. These results enforce the presence of inertia terms. As all three parameters, mentioned above play very important role in the stability of fluid flow. The limited cases are in full agreement with the available literature. Above mentioned solution technique proved itself a best and easy to handle technique for the solutions of highly nonlinear PDEs with non-homogeneous boundary conditions, a great help to mathematical community. This theoretical study has significant importance in industry and also in biosciences.
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Keywords:
- Maxwell fluid /
- laminar flow /
- porous walls /
- constant absorption /
- recursive approach
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References
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Haroon Tahira, Abdul Majeed Siddiqui, Hamee Ullah, Dianche Lu. FLOW OF MAXWELL FLUID IN A CHANNEL WITH UNIFORM POROUS WALLS[J]. Journal of Applied Analysis & Computation, 2021, 11(3): 1322-1347. doi: 10.11948/20200158
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- Figure 1. Geometry of the Problem
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Figure 2. Variation in axial velocity due to absorption
$ S $ when$ R_e = 0, \ \delta = 0 $ , at (a)$ x = 0.1, $ (b)$ x = 0.5, $ (c)$ x = 0.9 $ -
Figure 3. Variation in axial velocity due to absorption when
$ R_e = 15, \ \delta = 0 $ , at (a)$ x = 0.1, $ (b)$ x = 0.5, $ (c)$ x = 0.9 $ -
Figure 4. Variation in axial velocity due to
$ \delta $ when$ R_e = 15, \ S = 0.4 $ , at (a)$ x = 0.1, $ (b)$ x = 0.5, $ (c)$ x = 0.9 $ -
Figure 5. Variation in axial velocity due to
$ S $ when$ R_e = 5, \ \delta = 0.4 $ , at (a)$ x = 0.1, $ (b)$ x = 0.5, $ (c)$ x = 0.9 $ -
Figure 6. Variation in axial velocity due to
$ S $ when$ R_e = 15, \ \delta = 0.4 $ , at (a)$ x = 0.1, $ (b)$ x = 0.5, $ (c)$ x = 0.9 $ . -
Figure 7. Variation in axial velocity due to
$ R_e $ when$ S = 0.4, \ \delta = 0.0 $ , at (a)$ x = 0.1, $ (b)$ x = 0.5, $ (c)$ x = 0.9 $ -
Figure 8. Variation in axial velocity due to
$ R_e $ when$ S = 0.4, \ \delta = 0.0 $ , at (a)$ x = 0.1, $ (b)$ x = 0.5, $ (c)$ x = 0.9 $ -
Figure 9. Variation in axial velocity due to
$ R_e $ when$ S=0.4,\ \delta=0.4 $ , at (a)$ x=0.1, $ (b)$ x=0.5, $ (c)$ x=0.9 $ -
Figure 10. Variation in axial velocity due to
$ R_e $ when$ S=0.4,\ \delta=0.2 $ , at (a)$ x=0.1, $ (b)$ x=0.5, $ (c)$ x=0.9 $ -
Figure 11. Variation in radial velocity due to absorption for
$ R_e = 45, $ and (a)$ \delta = 0.0 $ (b)$ \delta = 0.4 $ (c)$ \delta = 0.8 $ -
Figure 12. Radial velocity due to variation in
$ S $ for$ \delta = 0.4 $ and (a)$ R_e = 5, $ (b)$ R_e = 25, $ (c)$ R_e = 45. $ -
Figure 13. Stream lines for
$ R_e = 15, \ \delta = 0.0 $ (a)$ S = 0.2, $ (b)$ S = 0.4, $ (c)$ S = 0.8 $ . -
Figure 14. Stream lines for
$ R_e = 45, \ S = 0.4 $ (a)$ \delta = 0.4, $ (b)$ \delta = 0.6, $ (c)$ \delta = 0.8 $ . -
Figure 15. Stream lines for
$ \delta = 0.4, \ S = 0.4 $ (a)$ R_e = 60, $ (b)$ R_e = 80, $ (c)$ R_e = 100 $ . -
Figure 16. Variation in wall shear stress with
$ \delta $ when (a)$ S = 0.0, R_e = 0 $ , (b)$ S = 0.4, R_e = 0, $ (c)$ S = 0.4, R_e = 15. $ -
Figure 17. Variation in wall shear stress due to
$ S $ when (a)$ \delta = 0.0, R_e = 0 $ , (b)$ \delta = 0.0, R_e = 15, $ (c)$ \delta = 0.4, R_e = 0. $ -
Figure 18. Variation in wall shear stress due to (a)
$ S $ when$ \delta = 0.2, R_e = 15.0 $ , (b)$ R_e $ when$ \delta = 0.2, S = 0.4, $ (c)$ R_e $ when$ \delta = 0.6, S = 0.4, $ -
Figure 19. Effect on normal stresses difference of
$ \delta $ when (a)$ S = 0.2, R_e = 15 $ , (b)$ S = 0.4, R_e = 15 $ , and of$ S $ when (c)$ \delta = 0.2, R_e = 15, $ -
Figure 20. Effect on normal stresses difference of
$ S $ when (a)$ \delta = 0.6, R_e = 15, $ and of$ R_e $ when (b)$ \delta = 0.2, S \to 0, $ (c)$ \delta = 0.6, S \to 0. $ -
Figure 21. Pressure difference
$ \Delta p $ due to$ S $ when$ \delta = 0, $ (a)$ R_e = 0.5, $ (b)$ R_e = 5 $ , (c)$ R_e = 15. $ -
Figure 22. Pressure difference
$ \Delta p $ due to$ \delta $ when$ S = 0.4 $ (a)$ R_e = 0.5, $ (b)$ R_e = 5, $ and (c)$ R_e = 15 $