Application of Recursive Theory of Slow Viscoelastic Flow to the Hydrodynamics of Second-Order Fluid Flowing through a Uniformly Porous Circular Tube
Abstract
:1. Introduction
2. Governing Equations
3. Langlois Recursive Approach
4. Problem Description
5. Problem Solution
5.1. First Order System and the Solution
5.2. Second Order System and the Solution
5.3. Third Order System and the Solution
5.4. Expression for Stream Function Correct to Third Order
6. Velocity Components
6.1. First Order Velocity Terms
6.2. Second Order Velocity Terms
6.3. Third Order Velocity Terms
6.4. Expressions for Velocity Components Correct to Third Order
7. Pressure Distribution
7.1. First Order Characteristic Pressure Terms and Pressure Drop
7.2. Second Order Characteristic Pressure Terms
7.3. Third Order Characteristic Pressure Terms
7.4. Characteristic Pressure Correct to Third Order
8. Various Important Expressions in Dimensionless Form
8.1. Velocity Components
8.2. Volume Flow Rate
8.3. Wall Shear Stress
8.4. Fractional Rabsorption
8.5. Leakage Flux
8.6. Mean Pressure Drop
8.7. Stream Function
9. Graphical Results and Discussion
10. Conclusions
- If the elastic parameter , the results obtained by Narasimhan [16] are achieved.
- If , the results obtained by Macey [1] are achieved.
- Elastic parameter does not bring any significant change in any of the flow variables in case of slow flow with small amount of cross flow.
- The magnitude of axial velocity component decreases if the magnitude of suction increases. Reverse flow is observed when wall porosity parameter assumes value threshold value of .
- Volume flow rate is found to be independent of both the elastic parameter and the cross-viscosity parameter .
- Volume flow rate and wall shear stress decrease in major flow direction if increases.
- Mean pressure drop in major flow direction increases with the increase of the value of wall Reynolds number . It is also observed that elastic parameter does not bring significant change in pressure drop, however pressure drop increases with increase of cross-viscosity parameter .
- Fractional reabsorption (FR) also increases with increase of , this relationship is given in Table 2.
- The radial velocity component is independent of and attains maximum at . The magnitude of radial velocity component increases with increase of .
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Abbreviations
Elastic parameter | |
Cross-viscosity parameter | |
Ratio of radius to length | |
Dimensionless coordinate of tube’s transversal axis, | |
Dimensionless elastic parameter | |
Dimensionless cross viscosity parameter | |
Dimensionless stream function | |
Stream function | |
Wall shear stress | |
Dimensionless wall shear stress | |
Coordinate of tube’s azimuthal axis | |
Dimensionless coordinate of tube’s longitudinal axis, | |
Mean pressure drop in major flow direction in the tube at point | |
Characteristic pressure in the tube at point | |
Fractional reabsorption | |
L | Length of the tube |
Inlet flow Reynolds number | |
P | Dimensionless pressure in the tube at point |
p | Pressure in the tube at point |
Q | Volume flow rate at any point z |
Dimensionless volume flow rate at any point | |
Inlet volume flow rate | |
q | leakage flux |
Dimensionless leakage flux | |
R | Radius of the tube |
r | Coordinate of tube’s transversal axis |
Dimensionless fluid velocity along r-direction | |
Dimensionless fluid velocity along r-direction | |
Dimensionless fluid velocity along z-direction | |
Fluid velocity along z-direction | |
Cross flow radial velocity at wall of the pipe | |
Wall Reynolds number | |
z | Coordinate of tube’s longitudinal axis |
Appendix A. Expression of Characteristic Pressure Correct to Third Order
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Quantity | Symbol | Value |
---|---|---|
Length of the Tube | L | cm |
Radius of the Tube | R | cm |
Inlet Volume Flow Rate | cm/s | |
Dynamic Viscosity | dyn·s/cm |
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Bhatti, K.; Siddiqui, A.M.; Bano, Z. Application of Recursive Theory of Slow Viscoelastic Flow to the Hydrodynamics of Second-Order Fluid Flowing through a Uniformly Porous Circular Tube. Mathematics 2020, 8, 1170. https://doi.org/10.3390/math8071170
Bhatti K, Siddiqui AM, Bano Z. Application of Recursive Theory of Slow Viscoelastic Flow to the Hydrodynamics of Second-Order Fluid Flowing through a Uniformly Porous Circular Tube. Mathematics. 2020; 8(7):1170. https://doi.org/10.3390/math8071170
Chicago/Turabian StyleBhatti, Kaleemullah, Abdul Majeed Siddiqui, and Zarqa Bano. 2020. "Application of Recursive Theory of Slow Viscoelastic Flow to the Hydrodynamics of Second-Order Fluid Flowing through a Uniformly Porous Circular Tube" Mathematics 8, no. 7: 1170. https://doi.org/10.3390/math8071170
APA StyleBhatti, K., Siddiqui, A. M., & Bano, Z. (2020). Application of Recursive Theory of Slow Viscoelastic Flow to the Hydrodynamics of Second-Order Fluid Flowing through a Uniformly Porous Circular Tube. Mathematics, 8(7), 1170. https://doi.org/10.3390/math8071170