Exact analytical solutions for the Poiseuille and Couette-Poiseuille flow of third grade fluid between parallel plates. (English) Zbl 1245.35093
Summary: Exact analytical solutions for the velocity profiles and flow rates have been obtained in explicit forms for the Poiseuille and Couette-Poiseuille flow of a third grade fluid between two parallel plates. These exact solutions match well with their numerical counter parts and are better than the recently developed approximate analytical solutions. Besides, effects of various parameters on the velocity profile and flow rate have been studied.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 76D07 | Stokes and related (Oseen, etc.) flows |
| 74K20 | Plates |
References:
| [1] | Oldroyd, J. G., On the formulation of rheological equations of state, Proc R Soc Lond A, 200, 523-541 (1950) · Zbl 1157.76305 |
| [2] | Rivlin, R. S.; Ericksen, J. L., Stress-deformation relations for isotropic materials, J Ration Mech Anal, 3, 323-425 (1955) · Zbl 0064.42004 |
| [3] | Bird, R. B.; Armstrong, R. C.; Hassager, O., Dynamics of polymeric liquids, (Fluid mechanics (1987), Wiley-Interscience: Wiley-Interscience New York), vol. 1 |
| [4] | Bird, R. B.; Stewart, W. E.; Lightfoot, E. N., Transport phenomena (2002), John Wiley & Sons: John Wiley & Sons New York |
| [5] | Fosdick, R. L.; Rajagopal, K. R., Thermodynamics and stability of fluids of third grade, Proc R Soc Lond A, 339, 351-377 (1980) · Zbl 0441.76002 |
| [6] | Rajagopal, K. R., On the stability of third grade fluids, Arch Ration Mech Anal, 32, 867-875 (1980) · Zbl 0462.76003 |
| [7] | Patria, M. C., Stability questions for a third-grade fluid in exterior domains, Int J Nonlinear Mech, 24, 451-457 (1989) · Zbl 0695.76006 |
| [8] | Bellout, H.; Nečas, J.; Rajagopal, K. R., On the existence and uniqueness of flows of multipolar fluids of grade 3 and their stability, Int J Eng Sci, 37, 75-96 (1999) · Zbl 1210.76011 |
| [9] | Passerni, A.; Patria, M. C., Existence, uniqueness and stability of steady flows of second and third grade fluids in an unbounded “pipe-like” domain, Int J Nonlinear Mech, 35, 1081-1103 (2000) · Zbl 1006.76003 |
| [10] | Vajravelu, K.; Cannon, J. R.; Rollins, D.; Leto, J., On solutions of some nonlinear differential equations arising in third grade fluid flows, Int J Eng Sci, 40, 1791-1805 (2002) · Zbl 1211.76015 |
| [11] | Rajagopal, K. R.; Sciubba, E., Pulsating Poiseuille flow of a non-Newtonian fluid, Math Comput Simul, 26, 276-288 (1984) · Zbl 0534.76005 |
| [12] | Majhi, S. N.; Nair, V. R., Pulsatile flow of third grade fluids under body acceleration-modelling blood flow, Int J Eng Sci, 32, 839-846 (1994) · Zbl 0925.76975 |
| [13] | Akyildiz, F. T.; Bellout, H.; Vajravelu, K., Exact solutions of nonlinear differential equations arising in third grade fluid flows, Int J Nonlinear Mech, 39, 1571-1578 (2004) · Zbl 1348.76011 |
| [14] | Hayat, T.; Ellahi, R.; Mahomed, F. M., Exact solutions for thin film flow of a third grade fluid down an inclined plane, Chaos Solitons Fract, 38, 1336-1341 (2008) · Zbl 1154.76319 |
| [15] | Sajid, M.; Hayat, T., The application of homotopy analysis method to thin film flows of a third grade fluid, Chaos Solitons Fract, 38, 506-515 (2008) · Zbl 1146.76588 |
| [16] | Siddiqui, A. M.; Mahmood, R.; Ghori, Q. K., Homotopy perturbation method for thin film flow of a third grade fluid down an inclined plane, Chaos Solitons Fract, 35, 140-147 (2008) · Zbl 1135.76006 |
| [17] | Ayub, M.; Rasheed, A.; Hayat, T., Exact flow of a third grade fluid past a porous plate using homotopy analysis method, Int J Eng Sci, 41, 2091-2103 (2003) · Zbl 1211.76076 |
| [18] | Siddiqui, A. M.; Zeb, A.; Ghori, Q. K.; Benharbit, A. M., Homotopy perturbation method for heat transfer flow of a third grade fluid between parallel plates, Chaos Solitons Fract, 36, 182-192 (2008) · Zbl 1152.80327 |
| [19] | Siddiqui, A. M.; Hameed, M.; Siddiqui, B. M.; Ghori, Q. K., Use of Adomian decomposition method in the study of parallel plate flow of a third grade fluid, Commun Nonlinear Sci Numer Simul, 15, 2388-2399 (2010) · Zbl 1222.76079 |
| [20] | Rice, R. G.; Do, D. D., Applied mathematics and modeling for chemical engineers (1994), John Wiley & Sons Inc.: John Wiley & Sons Inc. New York |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
