F. Talay Akyildiz

A generalization of surfaces family with common spatial geodesic

Abstract We analyzed the problem of constructing a surfaces family from a given spatial geodesic curve as in the work of Wang et al. [G.-J. Wang, K. Tang, C.-L. Tai, Parametric representation of a surface pencil with a common spatial geodesic, Comp. Aided Des. 36 (5) (2004) 447–459], who derived the sufficient condition on the marching-scale functions for which the curve C is an isogeodesic curve on a given surface. They assumed that these functions have a factor decomposition. In this work, we generalized their assumption to more general marching-scale functions and derived the sufficient conditions on them for which the curve C is an isogeodesic curve on a given surface. Finally using generalized marching-scale functions, we demonstrated some surfaces about subject.

A note on the flow of a non-Newtonian fluid film

Abstract Film flow of liquids simulated by the third-grade model down a vertical longitudinallyoscillating wall is investigated. The non-linear partial differential equation resulting from themomentum equation is solved by the method of Galerkin and flow enhancement is predicted fordifferent material constants. It is also found that an increase in either the amplitude or frequency ofthe vibration always leads to an increase in the magnitude of the flow enhancement.

Existence and uniqueness results for a nonlinear differential equation arising in viscous flow over a nonlinearly stretching sheet

Abstract We establish the existence and uniqueness results for a class of nonlinear third order ordinary differential equations arising in the viscous flow over a nonlinearly stretching sheet. In particular, we consider solutions over the semi-infinite interval [ 0 , ∞ ) . These results generalize the results of Vajravelu and Cannon [K. Vajravelu, J.R. Cannon, Applied Mathematics and Computation 181 (2006) 609], where they considered the finite interval [ 0 , R ] . Also in this paper, we answer their open question of finding the existence and uniqueness results for the problem over the semi-infinite domain and discuss the properties of the solution.

A numerical study for computation of geodesic curves

Consideration is given to the numerical computation of geodesic on surfaces. The method of finite-difference is used for governing non-linear system of differential equations. Then the resulting non-linear algebraic equations are solved by both iterative and Newtons method. It is shown that iterative method (IM) gives better result than Newtons method. Finally, we demonstrated our finding on several surfaces.

Surfaces with common geodesic in Minkowski 3-space

In this paper, we analyzed the problem of constructing a family of surfaces from a given spacelike (or timelike) geodesic curve. Using the Frenet trihedron frame of the curve in Minkowski space, we express the family of surfaces as a linear combination of the components of this frame, and derive the necessary and sufficient conditions for the coefficients to satisfy both the geodesic and the isoparametric requirements. Finally, examples are given to show the family of surfaces with common geodesic.

Analytical and numerical results for the Swift-Hohenberg equation

The problem of the Swift-Hohenberg equation is considered in this paper. Using homotopy analysis method (HAM) the series solution is developed and its convergence is discussed and documented here for the first time. In particular, we focus on the roles of the eigenvalue parameter @a and the length parameter l on the large time behaviour of the solution. For a given time t, we obtain analytical expressions for eigenvalue parameter @a and length l which show how different values of these parameters may lead to qualitatively different large time profiles. Also, the results are presented graphically. The results obtained reveal many interesting behaviors that warrant further study of the equations related to non-Newtonian fluid phenomena, especially the shear-thinning phenomena. Shear thinning reduces the wall shear stress.

Viscoelastic Lubrication With Phan-Thein-Tanner Fluid (PTT)

We analyze the lubrication flow of a viscoelastic fluid to account for the time dependent nature of the lubricant. The material obeys the constitutive equation for Phan-Thein-Tanner fluid (PTT). An explicit expression of the velocity field is obtained. This expression shows the effect of the Deborah number (De =λU/L, λ is the relaxation time). Using this velocity field, we derive the generalized Reynolds equation for PTT fluids. This equation reduces to the Newtonian case as De→0. Finally, the effect of the Deborah number on the pressure field is explored numerically in detail and the results are documented graphically.

Existence results and numerical simulation of magnetohydrodynamic viscous flow over a shrinking sheet with suction

This paper presents an analysis and a semi-analytical study of the boundary value problem resulting from a self-similar transformation of the partial differential equations governing the magnetohydrodynamic (MHD) viscous flow due to a shrinking sheet with suction. Existence of a solution is shown using the fixed point argument and the boundary value problem is solved. Galerkin-Legendre spectral method with domain truncation is used to obtain approximate analytical solutions. The approximate solutions computed confirm the theoretical analysis.

A note on the flow of a third-grade fluid between heated parallel plates

Abstract Flow of an incompressible third-grade fluid between heated parallel plates is investigated analytically. An exact solution of the energy equation is given for a constant heat flux at the walls. Results are discussed and compared with the numerical work of Szeri and Rajagopal (Int. J. Non-Linear Mech. 20 (1985) 91–101).

Orthogonal cubic spline collocation method for the nonlinear parabolic equation arising in non-Newtonian fluid flow

Using the orthogonal cubic spline collocation method, solution for the nonlinear parabolic equation arising in magneto-hydrodynamic unsteady Poiseuille flow of the generalized Newtonian fluid (Carreau rheological model) is obtained. Also, using the Lyapunov functional, a bound for the maximum norm of the semi-discrete solution is derived. Moreover, optimal error estimates are established for the semi-discrete solution. Numerical results thus obtained are presented graphically and the salient features of the solution are discussed, for various values of the parameters. The results obtained reveal many interesting behaviors that warrant further study on the parabolic equations related to non-Newtonian fluid phenomena. Furthermore the analysis can be used to study the mathematical models that involve the flow of viscous fluids with shear rate-dependent properties: For example, models dealing with polymer processing, tribology and lubrication, and food processing.