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Alexei I Zhurov - One of the best experts on this subject based on the ideXlab platform.
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one dimensional reductions and functional separable solutions to unsteady plane and axisymmetric Boundary Layer Equations for non newtonian fluids
International Journal of Non-linear Mechanics, 2016Co-Authors: Andrei D. Polyanin, Alexei I ZhurovAbstract:Abstract The paper deals with Equations describing the unsteady axisymmetric Boundary Layer of power-law non-Newtonian fluids on a body of revolution. The axisymmetric Boundary-Layer equation for the stream function is shown to reduce to a single third-order PDE of the form w tz + w z w xz − w x w zz = κ r n + 1 ( x ) w zz n − 1 w zzz + F ( t , x ) , where n is a rheological parameter of the fluid; the function r(x), determining the shape of the body, is assumed arbitrary. For this non-linear PDE, we describe one-dimensional reductions as well as a number of new generalized and functional separable solutions, which depend on two to five arbitrary functions. The solutions are obtained with the direct method of functional separation of variables by using particular solutions to an auxiliary ODE and systems of first-order PDEs. Many of the solutions are expressed in terms of elementary functions, which, together with significant arbitrariness, makes them especially useful for solving certain model problems and testing numerical and approximate analytical methods in fluid dynamics. Apart from power-law fluids, the paper looks at three-parameter polynomial and generalized Sisko models of non-Newtonian fluids. The unsteady plane Boundary-Layer Equations for the general non-Newtonian fluid model are shown to admit a reduction to an ODE. The paper presents new exact solutions to the plane Boundary-Layer Equations for power-law fluids as well as some other rheologically complex fluids.
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Unsteady axisymmetric Boundary-Layer Equations: Transformations, properties, exact solutions, order reduction and solution method
International Journal of Non-Linear Mechanics, 2015Co-Authors: Andrei D. Polyanin, Alexei I ZhurovAbstract:The paper deals with Equations describing the unsteady axisymmetric Boundary Layer on a body of revolution. The shape of the body is assumed arbitrary. The axisymmetric Boundary-Layer equation for the stream function is shown to reduce to a plane Boundary-Layer equation with a streamwise-coordinate-dependent viscosity of the form wtz + wzwxz − wxwzz = νr 2 (x)wzzz + F(t;x): We describe a number of new generalized and functional separable solutions to this non-linear equation, which depend on two to five arbitrary functions. The solutions are obtained with a new method based on using particular solutions to an auxiliary ODE. Many of the solutions are expressed in terms of elementary functions, provided that the arbitrary functions are also elementary. Two theorems are stated that enable one to generalize exact solutions of unsteady axisymmetric Boundary-Layer Equations by including additional arbitrary functions. Furthermore, we specify a von Mises-type transformation that reduces the unsteady axisymmetric Boundary-Layer equation to a non-linear second-order PDE. We also
Andrei D. Polyanin - One of the best experts on this subject based on the ideXlab platform.
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Transformations, properties, and exact solutions of unsteady axisymmetric Boundary Layer Equations for non-Newtonian fluids
Theoretical Foundations of Chemical Engineering, 2017Co-Authors: Andrei D. Polyanin, V. F. ZaitsevAbstract:Unsteady axisymmetric Boundary Layer Equations for power-law non-Newtonian fluids are analyzed. A number of new exact solutions containing arbitrary functions and free parameters are constructed using generalized or functional separation of variables. The solutions are obtained using a Crocco-type transformation reducing the order of the Equations examined and simpler point transformations. Along with the exact solutions to axisymmetric Boundary Layer Equations, some new exact solutions to planar Boundary Layer Equations for non-Newtonian fluids are constructed. Several properties have been discovered that allow the exact solutions of the unsteady axisymmetric Boundary Layer Equations to be generalized by including additional arbitrary functions therein. All results refer to an arbitrarily shaped streamlined solid of revolution.
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one dimensional reductions and functional separable solutions to unsteady plane and axisymmetric Boundary Layer Equations for non newtonian fluids
International Journal of Non-linear Mechanics, 2016Co-Authors: Andrei D. Polyanin, Alexei I ZhurovAbstract:Abstract The paper deals with Equations describing the unsteady axisymmetric Boundary Layer of power-law non-Newtonian fluids on a body of revolution. The axisymmetric Boundary-Layer equation for the stream function is shown to reduce to a single third-order PDE of the form w tz + w z w xz − w x w zz = κ r n + 1 ( x ) w zz n − 1 w zzz + F ( t , x ) , where n is a rheological parameter of the fluid; the function r(x), determining the shape of the body, is assumed arbitrary. For this non-linear PDE, we describe one-dimensional reductions as well as a number of new generalized and functional separable solutions, which depend on two to five arbitrary functions. The solutions are obtained with the direct method of functional separation of variables by using particular solutions to an auxiliary ODE and systems of first-order PDEs. Many of the solutions are expressed in terms of elementary functions, which, together with significant arbitrariness, makes them especially useful for solving certain model problems and testing numerical and approximate analytical methods in fluid dynamics. Apart from power-law fluids, the paper looks at three-parameter polynomial and generalized Sisko models of non-Newtonian fluids. The unsteady plane Boundary-Layer Equations for the general non-Newtonian fluid model are shown to admit a reduction to an ODE. The paper presents new exact solutions to the plane Boundary-Layer Equations for power-law fluids as well as some other rheologically complex fluids.
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Unsteady axisymmetric Boundary-Layer Equations: Transformations, properties, exact solutions, order reduction and solution method
International Journal of Non-Linear Mechanics, 2015Co-Authors: Andrei D. Polyanin, Alexei I ZhurovAbstract:The paper deals with Equations describing the unsteady axisymmetric Boundary Layer on a body of revolution. The shape of the body is assumed arbitrary. The axisymmetric Boundary-Layer equation for the stream function is shown to reduce to a plane Boundary-Layer equation with a streamwise-coordinate-dependent viscosity of the form wtz + wzwxz − wxwzz = νr 2 (x)wzzz + F(t;x): We describe a number of new generalized and functional separable solutions to this non-linear equation, which depend on two to five arbitrary functions. The solutions are obtained with a new method based on using particular solutions to an auxiliary ODE. Many of the solutions are expressed in terms of elementary functions, provided that the arbitrary functions are also elementary. Two theorems are stated that enable one to generalize exact solutions of unsteady axisymmetric Boundary-Layer Equations by including additional arbitrary functions. Furthermore, we specify a von Mises-type transformation that reduces the unsteady axisymmetric Boundary-Layer equation to a non-linear second-order PDE. We also
Mehmet Pakdemirli - One of the best experts on this subject based on the ideXlab platform.
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Similarity Solutions for Boundary Layer Equations of a Powel-Eyring Fluid
Mathematical and Computational Applications, 2013Co-Authors: Tasawar Hayat, Mehmet Pakdemirli, Yiğit AksoyAbstract:Boundary Layer Equations are derived for the first time for the Powel-Eyring fluid model, a non-Newtonian model proposed for pseudoplastic behavior. Using a scaling symmetry of the Equations, partial differential system is transferred to an ordinary differential system. Resulting Equations are numerically solved using a finite difference algorithm. Effects of non-Newtonian parameters on the solutions are discussed.
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Boundary Layer Equations and Lie Group Analysis of a Sisko Fluid
Journal of Applied Mathematics, 2012Co-Authors: Gözde Sarı, Mehmet Pakdemirli, Tasawar Hayat, Yiğit AksoyAbstract:Boundary Layer Equations are derived for the Sisko fluid. Using Lie group theory, a symmetry analysis of the Equations is performed. A partial differential system is transferred to an ordinary differential system via symmetries. Resulting Equations are numerically solved. Effects of non-Newtonian parameters on the solutions are discussed.
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Symmetries of Boundary Layer Equations of power-law fluids of second grade
Acta Mechanica Sinica, 2008Co-Authors: Mehmet Pakdemirli, Muhammet Yürüsoy, Yiğit Aksoy, Chaudry Masood KhaliqueAbstract:A modified power-law fluid of second grade is considered. The model is a combination of power-law and second grade fluid in which the fluid may exhibit normal stresses, shear thinning or shear thickening behaviors. The Equations of motion are derived for two dimensional incompressible flows, and from which the Boundary Layer Equations are derived. Symmetries of the Boundary Layer Equations are found by using Lie group theory, and then group classification with respect to power-law index is performed. By using one of the symmetries, namely the scaling symmetry, the partial differential system is transformed into an ordinary differential system, which is numerically integrated under the classical Boundary Layer conditions. Effects of power-law index and second grade coefficient on the Boundary Layers are shown and solutions are contrasted with the usual second grade fluid solutions.
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Boundary Layer Equations and stretching sheet solutions for the modified second grade fluid
International Journal of Engineering Science, 2007Co-Authors: Yigˇit Aksoy, Mehmet Pakdemirli, Chaudry Masood KhaliqueAbstract:Abstract A modified second grade non-Newtonian fluid model is considered. The model is a combination of power-law and second grade fluids in which the fluid may exhibit normal stresses, shear thinning or shear thickening behaviors. The Equations of motion are derived for two dimensional incompressible flows. The Boundary Layer Equations are derived from the Equations. Symmetries of the Boundary Layer Equations are calculated using Lie Group theory. For a special power law index of m = −1, the principal Lie algebra extends. Using one of the symmetries, the partial differential system is transferred to an ordinary differential system. The ordinary differential Equations are numerically integrated for the stretching sheet Boundary conditions. Effects of power-law index and second grade coefficient on the Boundary Layers are shown and solutions are contrasted with the usual second grade fluid solutions. The shear stress on the Boundary is also calculated.
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similarity analysis of Boundary Layer Equations of a class of non newtonian fluids
International Journal of Non-linear Mechanics, 1994Co-Authors: Mehmet PakdemirliAbstract:Abstract A similarity analysis of three-dimensional Boundary Layer Equations of a class of non-Newtonian fluids in which the stress is an arbitrary function of rates of strain is made. It is shown that under scaling transformation, for an arbitrary stress function, only 90° of wedge flow leads to similarity solutions, whereas for a specific more restricted form, similarity solutions exist for arbitrary wedge angles. In the case of spiral group transformation, no similarity solutions exist if we force the stress function to remain arbitrary after the transformation, whereas for a specific more restricted form, similarity solutions exist for arbitrary wedge angles. For both transformations, similarity Equations for power-law and Newtonian fluids are presented as special cases of the analysis. Finally the conditions for invariance and the form of the stress function for a two-dimensional case are also presented.
Klaus Gersten - One of the best experts on this subject based on the ideXlab platform.
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Numerical Integration of the Boundary–Layer Equations
Boundary-Layer Theory, 2016Co-Authors: Herrmann Schlichting, Klaus GerstenAbstract:Numerical solutions of the Boundary–Layer Equations are based on the assumption that the differential expressions in the partial differential Equations can be approximated by difference expressions. This approximation, called discretisation can be obtained from a series expansion for the velocity components in the coordinate directions. These series expansions do not necessarily have to consist of Taylor series. Since only a certain number of terms in any expansion can be taken, there is a discretisation or truncation error, and this is dependent on the number and size of the terms neglected.
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General Properties and Exact Solutions of the Boundary–Layer Equations for Plane Flows
Boundary-Layer Theory, 2016Co-Authors: Herrmann Schlichting, Klaus GerstenAbstract:Before further examples of the calculation of Boundary Layers are treated in the next chapter, some general properties of Boundary-Layer Equations will be discussed. We will confine ourselves to steady, two-dimensional, incompressible Boundary Layers.
Tong Yang - One of the best experts on this subject based on the ideXlab platform.
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magnetic effects on the solvability of 2d mhd Boundary Layer Equations without resistivity in sobolev spaces
Journal of Functional Analysis, 2020Co-Authors: Chengjie Liu, Tong Yang, Feng Xie, Dehua WangAbstract:Abstract In this paper, we are concerned with the magnetic effect on the Sobolev solvability of Boundary Layer Equations for the 2D incompressible MHD system without resistivity. The MHD Boundary Layer is described by the Prandtl type Equations derived from the incompressible viscous MHD system without resistivity under the no-slip Boundary condition on the velocity. Assuming that the initial tangential magnetic field does not degenerate, a local-in-time well-posedness in Sobolev spaces is proved without the monotonicity condition on the velocity field. Moreover, we show that if the tangential magnetic field of shear Layer is degenerate at one point, then the linearized MHD Boundary Layer system around the shear Layer profile is ill-posed in the Gevrey function space provided that the initial velocity shear flow is non-degenerately critical at the same point.
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magnetic effects on the solvability of 2d mhd Boundary Layer Equations without resistivity in sobolev spaces
arXiv: Analysis of PDEs, 2020Co-Authors: Chengjie Liu, Tong Yang, Feng Xie, Dehua WangAbstract:In this paper, we are concerned with the magnetic effect on the Sobolev solvability of Boundary Layer Equations for the 2D incompressible MHD system without resistivity. The MHD Boundary Layer is described by the Prandtl type Equations derived from the incompressible viscous MHD system without resistivity under the no-slip Boundary condition on the velocity. Assuming that the initial tangential magnetic field does not degenerate, a local-in-time well-posedness in Sobolev spaces is proved without the monotonicity condition on the velocity field. Moreover, we show that if the tangential magnetic field shear Layer is degenerate at one point, then the linearized MHD Boundary Layer system around the shear Layer profile is ill-posed in the Sobolev settings provided that the initial velocity shear flow is non-degenerately critical at the same point.
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Long Time Existence of Solutions to MHD Boundary Layer Equations with Small Perturbed Analytic Initial Data
arXiv: Analysis of PDEs, 2018Co-Authors: Feng Xie, Tong YangAbstract:In this paper, we consider the long time existence and uniqueness of solutions to the MHD Boundary Layer Equations with small perturbed analytic initial data around some special solutions in two dimensional spatial variables. Here, the initial data is required to be real-analytic with respect to tangential variable $x$ and belongs to a weighted $L^2$ Sobolev space with respect to normal variable $y$. It is proven that if the initial data is a small perturbation around the given special solutions with the size of $\varepsilon$, then the MHD Boundary Layer Equations admit a unique solution with the lifespan being larger than $\varepsilon^{-2+\delta_0}$, where $\delta_0$ can be an arbitrary small positive constant. It is noted that the strength of the special background solutions are not necessarily small. And the proof depends on the uniform energy estimates in analytic settings.