September  2015, 20(7): 2107-2128. doi: 10.3934/dcdsb.2015.20.2107

Existence and uniqueness of steady flows of nonlinear bipolar viscous fluids in a cylinder

1. 

Allstate Insurance Company, 2775 Sanders Road, Suite D2W, Northbrook, IL 60062, United States

2. 

Northern Illinois University, Department of Mathematical Sciences, De Kalb, IL 60115

3. 

Northern Illinois University, Department of Mathematical Sciences, DeKalb, IL 60115-2888, United States

Received  April 2014 Revised  February 2015 Published  July 2015

The existence and uniqueness of solutions to the boundary-value problem for steady Poiseuille flow of an isothermal, incompressible, nonlinear bipolar viscous fluid in a cylinder of arbitrary cross-section is established. Continuous dependence of solutions, in an appropriate norm, is also established with respect to the constitutive parameters of the bipolar fluid model, as these parameters converge to zero, under the additional assumption that the cylinder has a circular cross-section.
Citation: Allen Montz, Hamid Bellout, Frederick Bloom. Existence and uniqueness of steady flows of nonlinear bipolar viscous fluids in a cylinder. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2107-2128. doi: 10.3934/dcdsb.2015.20.2107
References:
[1]

R. A. Adams, Sobolev Spaces, Academic Press, 1975.

[2]

G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, 1999.

[3]

H. Bellout and F. Bloom, Incompressible Bipolar and non-Newtonian Viscous Fluid Flow, Advances in Mathematical Fluid Mechanics, Birkhäuser/Springer, 2014. doi: 10.1007/978-3-319-00891-2.

[4]

H. Bellout, F. Bloom and J. Nečas, Phenomenological behavior of multipolar viscous fluids, Quarterly of Applied Mathematics, 50 (1992), 559-583.

[5]

H. Bellout and F. Bloom, Steady plane poiseuille flows of incompressible multipolar fluids, International Journal of Non-Linear Mechanics, 28 (1993), 503-518. doi: 10.1016/0020-7462(93)90043-K.

[6]

H. Bellout and F. Bloom, On the uniqueness of plane poiseuille solutions of the equations of incompressible dipolar viscous fluids, International Journal of Engineering Science, 31 (1993), 1535-1549. doi: 10.1016/0020-7225(93)90030-X.

[7]

H. Bellout and F. Bloom, Existence and asymptotic stability of time-dependent poiseuille flows of isothermal bipolar fluids, Applicable Analysis, 50 (1993), 115-130. doi: 10.1080/00036819308840188.

[8]

H. Bellout and F. Bloom, On the higher-order boundary conditions for incompressible nonlinear bipolar fluid flow, Quarterly of Applied Mathematics, 71 (2013), 773-785. doi: 10.1090/S0033-569X-2013-01330-9.

[9]

J. L. Bleustein and A. E. Green, Dipolar fluids, International Journal of Engineering Science, 5 (1967), 323-340. doi: 10.1016/0020-7225(67)90041-9.

[10]

F. Bloom and W. Hao, Steady flows of nonlinear bipolar viscous fluids between rotating cylinders, Quarterly of Applied Mathematics, LIII (1995), 143-171.

[11]

A. Chorin and J. Marsden, A Mathematical Introduction to Fluid Mechanics, Third Edition, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0883-9.

[12]

Q. Du and M. Gunzburger, Analysis of a Ladyzhenskaya model for incompressible viscous flow, Journal of Mathematical Analysis and Applications, 155 (1991), 21-45. doi: 10.1016/0022-247X(91)90024-T.

[13]

L. C. Evans, Partial Differential Equations, American Mathematical Society, RI, 1998. doi: 10.1090/gsm/019.

[14]

G. P. Galdi, Mathematical problems in classical and non-Newtonian fluid mechanics, in Hemodynamical Flows, Oberwolfach Semin., 37, Birkhäuser, Basel, 2008, 121-273. doi: 10.1007/978-3-7643-7806-6_3.

[15]

A. E. Green and R. S. Rivlin, Multipolar continuum mechanics, Archive for Rational Mechanics and Analysis, 17 (1964), 113-147.

[16]

A. E. Green and R. S. Rivlin, Simple force and stress multipoles, Archive for Rational Mechanics and Analysis, 16 (1964), 325-353.

[17]

O. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, 1969.

[18]

O. Ladyzhenskaya, New equations for the description of the viscous incompressible fluids and solvability in the large of the boundary value problems for them, in Boundary Value Problems of Mathematical Physics V, American Mathematical Society, Providence, RI, 1970.

[19]

J.-L. Lions, Quelques Mèthodes de Rèsolution des Problemes aux Limites Nonlineaires, Dunod; Gauthier-Villars, Paris, 1969.

[20]

A. Montz, Some Bipolar Viscous Fluid Flow Problems in Rigid and Compliant Domains, Ph.D thesis, Northern Illinois University, 2014.

[21]

J. Nečas, Direct Methods in the Theory of Elliptic Equations, Springer-Verlag, 2012. doi: 10.1007/978-3-642-10455-8.

[22]

J. Nečas and M. Šilhavý, Multipolar viscous fluids, Quarterly of Applied Mathematics, 49 (1991), 247-265.

[23]

Y. R. Ou and S. S. Sritharan, Analysis of regularized Navier-Stokes equations. I, II, Quarterly of Applied Mathematics, 49 (1991), 651-685, 687-728.

[24]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, North-Holland Pub. Co., 1977.

[25]

R. A. Toupin, Theories of elasticity with couple-stress, Archive for Rational Mechanics and Analysis, 17 (1964), 85-112.

show all references

References:
[1]

R. A. Adams, Sobolev Spaces, Academic Press, 1975.

[2]

G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, 1999.

[3]

H. Bellout and F. Bloom, Incompressible Bipolar and non-Newtonian Viscous Fluid Flow, Advances in Mathematical Fluid Mechanics, Birkhäuser/Springer, 2014. doi: 10.1007/978-3-319-00891-2.

[4]

H. Bellout, F. Bloom and J. Nečas, Phenomenological behavior of multipolar viscous fluids, Quarterly of Applied Mathematics, 50 (1992), 559-583.

[5]

H. Bellout and F. Bloom, Steady plane poiseuille flows of incompressible multipolar fluids, International Journal of Non-Linear Mechanics, 28 (1993), 503-518. doi: 10.1016/0020-7462(93)90043-K.

[6]

H. Bellout and F. Bloom, On the uniqueness of plane poiseuille solutions of the equations of incompressible dipolar viscous fluids, International Journal of Engineering Science, 31 (1993), 1535-1549. doi: 10.1016/0020-7225(93)90030-X.

[7]

H. Bellout and F. Bloom, Existence and asymptotic stability of time-dependent poiseuille flows of isothermal bipolar fluids, Applicable Analysis, 50 (1993), 115-130. doi: 10.1080/00036819308840188.

[8]

H. Bellout and F. Bloom, On the higher-order boundary conditions for incompressible nonlinear bipolar fluid flow, Quarterly of Applied Mathematics, 71 (2013), 773-785. doi: 10.1090/S0033-569X-2013-01330-9.

[9]

J. L. Bleustein and A. E. Green, Dipolar fluids, International Journal of Engineering Science, 5 (1967), 323-340. doi: 10.1016/0020-7225(67)90041-9.

[10]

F. Bloom and W. Hao, Steady flows of nonlinear bipolar viscous fluids between rotating cylinders, Quarterly of Applied Mathematics, LIII (1995), 143-171.

[11]

A. Chorin and J. Marsden, A Mathematical Introduction to Fluid Mechanics, Third Edition, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0883-9.

[12]

Q. Du and M. Gunzburger, Analysis of a Ladyzhenskaya model for incompressible viscous flow, Journal of Mathematical Analysis and Applications, 155 (1991), 21-45. doi: 10.1016/0022-247X(91)90024-T.

[13]

L. C. Evans, Partial Differential Equations, American Mathematical Society, RI, 1998. doi: 10.1090/gsm/019.

[14]

G. P. Galdi, Mathematical problems in classical and non-Newtonian fluid mechanics, in Hemodynamical Flows, Oberwolfach Semin., 37, Birkhäuser, Basel, 2008, 121-273. doi: 10.1007/978-3-7643-7806-6_3.

[15]

A. E. Green and R. S. Rivlin, Multipolar continuum mechanics, Archive for Rational Mechanics and Analysis, 17 (1964), 113-147.

[16]

A. E. Green and R. S. Rivlin, Simple force and stress multipoles, Archive for Rational Mechanics and Analysis, 16 (1964), 325-353.

[17]

O. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, 1969.

[18]

O. Ladyzhenskaya, New equations for the description of the viscous incompressible fluids and solvability in the large of the boundary value problems for them, in Boundary Value Problems of Mathematical Physics V, American Mathematical Society, Providence, RI, 1970.

[19]

J.-L. Lions, Quelques Mèthodes de Rèsolution des Problemes aux Limites Nonlineaires, Dunod; Gauthier-Villars, Paris, 1969.

[20]

A. Montz, Some Bipolar Viscous Fluid Flow Problems in Rigid and Compliant Domains, Ph.D thesis, Northern Illinois University, 2014.

[21]

J. Nečas, Direct Methods in the Theory of Elliptic Equations, Springer-Verlag, 2012. doi: 10.1007/978-3-642-10455-8.

[22]

J. Nečas and M. Šilhavý, Multipolar viscous fluids, Quarterly of Applied Mathematics, 49 (1991), 247-265.

[23]

Y. R. Ou and S. S. Sritharan, Analysis of regularized Navier-Stokes equations. I, II, Quarterly of Applied Mathematics, 49 (1991), 651-685, 687-728.

[24]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, North-Holland Pub. Co., 1977.

[25]

R. A. Toupin, Theories of elasticity with couple-stress, Archive for Rational Mechanics and Analysis, 17 (1964), 85-112.

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