Geodesic
On the Unique Solvability of the Problem of the Flow of an Aqueous Solution of Polymers near a Critical Point
Matematičeskie zametki, Tome 106 (2019) no. 5, pp. 723-735.

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider the boundary-value problem in a semibounded interval for a fourth-order equation with “double degeneracy”: the small parameter in the equation multiplies the product of the unknown function vanishing on the boundary and its highest derivative. Such a problem arises in the description of the motion of weak solutions of polymers near a critical point. For the zero value of the parameter, the solution is the classical Hiemenz solution. We prove the unique solvability of the problem for nonnegative values of the parameter not exceeding $1$.
Keywords: flow of an aqueous solution of polymers, boundary-value problem, unique solvability.
Mots-clés : Hiemenz solution 
@article{MZM_2019_106_5_a6,
     author = {A. G. Petrova},
     title = {On the {Unique} {Solvability} of the {Problem} of the {Flow} of an {Aqueous} {Solution} of {Polymers} near a {Critical} {Point}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {723--735},
     publisher = {mathdoc},
     volume = {106},
     number = {5},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2019_106_5_a6/}
}
TY  - JOUR
AU  - A. G. Petrova
TI  - On the Unique Solvability of the Problem of the Flow of an Aqueous Solution of Polymers near a Critical Point
JO  - Matematičeskie zametki
PY  - 2019
SP  - 723
EP  - 735
VL  - 106
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2019_106_5_a6/
LA  - ru
ID  - MZM_2019_106_5_a6
ER  - 
%0 Journal Article
%A A. G. Petrova
%T On the Unique Solvability of the Problem of the Flow of an Aqueous Solution of Polymers near a Critical Point
%J Matematičeskie zametki
%D 2019
%P 723-735
%V 106
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2019_106_5_a6/
%G ru
%F MZM_2019_106_5_a6
A. G. Petrova. On the Unique Solvability of the Problem of the Flow of an Aqueous Solution of Polymers near a Critical Point. Matematičeskie zametki, Tome 106 (2019) no. 5, pp. 723-735. http://geodesic.mathdoc.fr/item/MZM_2019_106_5_a6/

[1] G. Shlikhting, Teoriya pogranichnogo sloya, Nauka, M., 1974 | MR

[2] K. Hiemenz, “Die Grenzschicht an einem in den gleichförmigen Flüssigkeitsstrom eingetauchten geraden Kreiszylinder”, Dingler's Politech. J., 326 (1911), 321–324

[3] L. Howarth, “On the calculation of steady flow in the boundary layer near the surface of a cylinder in a stream”, Reports and Memoranda. Aeronautical Research Committee, 1632 (1935)

[4] L. Howarth, “The boundary layer in three dimensional flow. Part II. The flow near a stagnation point”, Philos. Mag. (7), 42 (1951), 1433–1440 | MR | Zbl

[5] S. V. Meleshko, B. B. Pukhnachev, “Ob odnom klasse chastichno invariantnykh reshenii uravnenii Nave–Stoksa”, Prikl. mekh. i tekhn. fiz., 40:2 (1999), 24–33 | MR | Zbl

[6] S. N. Aristov, D. V. Knyazev, A. D. Polyanin, “Tochnye resheniya uravnenii Nave–Stoksa s lineinoi zavisimostyu komponent skorosti ot dvukh prostranstvennykh peremennykh”, Teoret. osnovy khim. tekhnol., 43:5 (2009), 547–566

[7] G. K. Rajeswari, S. L. Rathna, “Flow of a particular class of non-Newtonian visko-elastic and visko-inelastic fluid near a stagnation point”, Z. Angew. Math. Phys., 13 (1962), 43–57 | DOI | MR | Zbl

[8] K. Sadeghy, H. Hajibeygi, S.-M. Taghavi, “Stagnation-point flow of upper-convected Maxwell fluid”, Int. J. Non-Linear Mech., 41:10 (2006), 1242–1247 | DOI

[9] J. E. Paullet, “Analysis of stagnation point flow of an upper-convected Maxwell fluid”, Electron. J. Differential Equations, 2017, Paper No. 302 | MR | Zbl

[10] J. B. McLeod, K. R. Rajagopal, “On the uniqueness of flow of a Navier–Stokes fuid due to a stretching boundary”, Arch. Rational Mech. Anal., 98:4 (1987), 385–393 | DOI | MR | Zbl

[11] D. Riabouchinsky, “Quelques considérations sur les mouvements plans rotationnels d'un liquide”, C. R. Hebdomadaires Acad. Sci., 179 (1924), 1133–1136 | Zbl

[12] O. A. Frolovskaya, “Avtomodelnoe nestatsionarnoe techenie vyazkoi zhidkosti vblizi kriticheskoi tochki”, Prikl. mekh. i tekhn. fiz., 57 (2016), 3–8 | MR | Zbl

[13] V. A. Galaktionov, J. L. Vazquez, “Blow-up of a class of solutions with free boundaries for the Navier–Stokes equations”, Adv. Differential Equations, 4:3 (1999), 297–321 | MR | Zbl

[14] A. G. Petrova, V. V. Pukhnachev, O. A. Frolovskaya, “Analiticheskoe i chislennoe issledovanie zadachi o nestatsionarnom techeniii vblizi kriticheskoi tochki”, Prikl. matem. i mekh., 80 (2016), 304–316 | MR

[15] Ya. I. Voitkunskii, V. B. Amfilokhiev, V. A. Pavlovskii, “Uravneniya dvizheniya zhidkosti s uchetom ee relaksatsionnykh svoistv”, Tr. Leningr. korablestroit. in-ta, 69 (1970), 9–26

[16] V. A. Pavlovskii, “K voprosu o teoreticheskom opisanii slabykh vodnykh rastvorov polimerov”, Dokl. AN SSSR, 200:4 (1971), 809–812

[17] O. A Frolovskaya, V. V. Pukhnachev, “Analysis of the models of motion of aqueous solutions of polymers on the basis of their exact solutions”, Polymers, 10:6 (2018), 684–696 | DOI

[18] Yu. D. Bozhkov, V. V. Pukhnachev, “Gruppovoi analiz uravnenii dvizheniya vodnykh rastvorov polimerov”, Dokl. AN, 460:5 (2015), 536–539 | MR

[19] A. Fridman, Uravneniya s chastnymi proizvodnymi parabolicheskogo tipa, M., Nauka, 1968 | MR | Zbl

[20] E. Kamke, Spravochnik po obyknovennym differentsialnym uravneniyam, Nauka, M., 1961 | MR | Zbl

[21] T. P. Pukhnacheva, “Zadacha ob osesimmetrichnom techenii vodnogo rastvora polimerov vblizi kriticheskoi tochki”, Trudy seminara po geometrii i matematicheskomu modelirovaniyu, 2, Altaiskii gos. un-t, Barnaul, 2016, 75–80