Communications in Nonlinear Science and Numerical Simulation
Nano boundary layers over stretching surfaces
Introduction
In classical boundary layer theory, the condition of no-slip near solid walls is usually applied. The fluid velocity component is assumed to be zero relative to the solid boundary. This is not true for fluid flows at the micro and nano scale. Investigations show that the condition of no-slip is no longer valid. Instead, a certain degree of tangential slip must be allowed. To describe the phenomenon of slip, Navier [1] introduced a boundary condition which states the component of the fluid velocity tangential to the boundary walls is proportional to the tangential stress. Later, several researchers [2], [3], [4] extended the Navier boundary condition.
The development of new micro- and nano-technologies is fast-paced, but there are still many issues to resolve in the thermofluids engineering of small-scale devices; from the fundamental simulation of fluid flow and heat transfer, to the optimization of design for fabrication. This attempt is hopefully a modest one to understand the effects of nano boundary layers over a stretching sheet.
We consider the model proposed by Wang [5] describing the viscous flow due to a stretching surface with both surface slip and suction (or injection). As in Wang, we consider two geometries situations: (i) the two-dimensional stretching surface and (ii) the axisymmetric stretching surface. A similarity transform is applied to convert the Navier–Stokes equations into a nonlinear ordinary differential equation. The existence and uniqueness results for each of the two problems were presented in Wang [5] along with some numerical results.
Motivated by this interesting and important study of Wang [5], in the present paper, we apply the Homotopy Analysis Method to obtain analytical solutions. Further, such analytical solutions can be used as asymptotic solutions for large values of the independent variable η. These analytical solutions can be obtained only with a very few iterations by choosing an appropriate initial approximation and a convergence control parameter . Furthermore, it has frequently been demonstrated that the Homotopy Analysis Method (HAM) gives one the ability to adjust and control the convergence region of obtained solutions, by use of the convergence control parameter ℏ and an appropriate initial guess to the solution. Moreover, the HAM includes the other perturbation methods as a special case and HAM can be easily applied to find some new solutions (see for details [7], [8], [9]) which are not discovered by the other solution processes.
It may be noted that by using a boundary value problem solver, we also obtain numerical solutions, which display a number of qualitative properties of the similarity solutions. Our method of obtaining numerical solutions differs from that of Wang [5] in that we employ a boundary value problem solver, while Wang [6] converts the boundary value problem into an initial value problem first and then obtains a solution via the Runge–Kutta method. Our results agree with those in Wang [5] up to the number of decimal places provided. For instance, the numerical solutions for the shear stress at the surface f″(0) are given to four decimal places in Wang [5] and to three decimal places in Wang [6]. We consider up to 10 decimal places, and the first few digits of our results agree with those of Wang [5], [6].
Section snippets
Formulation of the problem
Let (u, v, w) be the velocity components in the (x, y, z) directions, respectively, and let p be the pressure. Then the Navier–Stokes equations for the steady viscous fluid flow can be written aswhere ν is the kinematic viscosity and ρ is the density of the fluid. The continuity equation can be written asAs in [5], we take the velocity on the stretching surface (on the plane z = 0) as
Analytical solutions
In Crane [10], when m = 1 and s = K = 0, the exact solution to (10) is given asWhen m = 1, Wang [5] gives a solution of the formwhere C is the maximal root of the equationBy Descartes’ rule of signs and from the fact that K > 0, there is only one positive root; this will be the root we consider. For m ≠ 1, solutions are much harder to obtain. However, as shown in [5], solutions for the interesting cases of m = 1, 2 do exist. We now consider analytical
Numerical solutions
We obtain numerical solutions to the boundary value problem (10) using the boundary value problem solver in Maple 11 (see, for instance, [17], [18]). Thus, our procedure differs from that of [5] where Eq. (10) was solved as an initial value problem via the Runge–Kutta method. In particular, an initial “guess” for f′′(0) is not required in the boundary value problem solver. The presented results are obtained with a relative error less than 10−11 in all cases. The numerical results are obtained
Discussion of the results
From Fig. 2, we see that in the case of suction s > 0, the normal velocity f(η) is always in the direction of the stretching surface. But for injection s < 0, the normal velocity is away from the stretching surface, and vanishes at some finite distance. For fixed values of m = 1, 2 and when K ⩾ 0, we find that f(η) increases over the entire domain uniformly with an increase in the suction/injection parameter s.
Further, from the profiles of f″(η) (presented in Fig. 2), we see that the similarity
Acknowledgments
Authors’ thanks are due to the anonymous referees for their constructive comments which led to definite improvement in the paper. One of the authors (R.A. Van Gorder) was supported in part by NSF Grant DMS-0505566.
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