In this article we analyze the boundary value problem governing stagnation-point flow of a fluid with a power law outer flow over a surface moving with a speed proportional to the outer flow. The flow is characterized by two physical parameters; epsilon, which measures the stretching (epsilon > 0) or shrinking (epsilon < 0) of the sheet relative to the outer flow, and n > 0, the power law exponent. In the case of aiding flow (epsilon > 0), where the (stretching) surface and the outer flow move in the same direction, we prove existence of a solution for all values of n. For opposing flow (epsilon < 0), where the (shrinking) surface and the outer flow move in opposite directions, the situation is much more complicated. For -1 < epsilon < 0 and all n we prove a solution exists. However, for epsilon <= -1, we prove there exists a value, epsilon(crit) (n) <= -1, such that no solutions exist for epsilon <= epsilon(crit). For n = 1/7 and n = 1/3 we prove that epsilon(crit) = -1. For other values of n, we derive bounds which illustrate the complicated nature of the existence/nonexistence boundary for opposing (epsilon < 0) flows.

In this article we analyze a recently proposed model for boundary layer flow of a nanofluid past a permeable stretching/shrinking sheet. The boundary value problem (BVP) resulting from this model is governed by two physical parameters; lambda, which controls the stretching (lambda > 0) or shrinking (lambda < 0) of the sheet, and S, which controls the suction (S > 0) or injection (S < 0) of fluid through the sheet. For lambda >= 0 and S is an element of R, we present a closed-form solution to the BVP and prove that this solution is unique. For lambda < 0 and S < 2 root-lambda we prove no solution exists. For lambda < 0 and S = 2 root-lambda we present a closed-form solution to the BVP and prove that it is unique. For lambda < 0 and S > 2 root-lambda we present two closed-form solutions to the BVP and prove the existence of an infinite number of solutions in this parameter range. The analytical results proved here differ from the numerical results reported in the literature. We discuss the mathematical aspects of the problem that lead to the difficulty in obtaining accurate numerical approximations to the solutions.

Several recent papers have investigated the two-dimensional stagnation point flow of an upper-convected Maxwell fluid by employing a similarity change of variable to reduce the governing PDEs to a nonlinear third order ODE boundary value problem (BVP). In these previous works, the BVP was studied numerically and several conjectures regarding the existence and behavior of the solutions were made. The purpose of this article is to mathematically verify these conjectures. We prove the existence of a solution to the BVP for all relevant values of the elasticity parameter. We also prove that this solution has monotonically increasing first derivative, thus verifying the conjecture that no "overshoot" of the boundary condition occurs. Uniqueness results are presented for a large range of parameter space and bounds on the skin friction coefficient are calculated.

Several recent papers investigate the boundary value problem ϕ′′(t)+λϕ′(t)+ϕ(t)2=0,t≥0subject to ϕ(0)=1,ϕ(∞)=0,which arises in certain situations of boundary layer flow. Previous work on the problem established the existence of a λmin∈[1,2∕3] such that solutions exist for λ≥λmin. It has been conjectured that for λ<λmin no solution exists. We improve existing results by proving that for λ<λ1≈.96105 no solution to the boundary value problem exists. The proof employs a novel application of Green’s Theorem and is applicable to other boundary value problems.

In this paper we investigate a boundary value problem (BVP) derived from a model of boundary layer flow past a suddenly heated vertical surface in a saturated porous medium. The surface is heated at a rate proportional to x k x^k where x x measures distance along the wall and k > − 1 k>-1 is constant. Previous results have established the existence of a continuum of solutions for − 1 > k > − 1 / 2 -1>k>-1/2 . Here we further analyze this continuum and determine that precisely one solution of this continuum approaches the boundary condition at infinity exponentially while all others approach algebraically. Previous results also showed that the solution to the BVP is unique for − 1 / 2 ≤ k > 0 -1/2 \leq k >0 . Here we extend the range of uniqueness to 0 ≤ k ≤ 1 0\leq k \leq 1 . Finally, the physical implications of the mathematical results are discussed and a comparison is made to the solutions for the related case of prescribed surface temperature on the surface.

This article considers two situations involving unsteady laminar boundary layer flow due to a stretching surface in a quiescent viscous incompressible fluid. In one configuration, the surface is impermeable with prescribed heat flux, in the other, the surface is permeable with prescribed temperature. The boundary value problems governing a similarity reduction for each of these situations are investigated and the existence of a solution is proved for all relevant values of physical parameters. The uniqueness of the solution is also proved for some (but not all) values of the parameters. Finally, a priori bounds are obtained for the skin friction coefficient and local Nusselt number.

The article named above appeared recently in Applied Mathematics Letters and investigated a boundary value problem governing viscous flow over a nonlinearly stretching sheet. The authors of the work assert existence and (under certain restrictions) uniqueness of a solution to the problem for all relevant values of the parameter governing the stretching rate of the sheet. Unfortunately, several proofs presented in the article are incorrect. We will prove that for a range of parameter space the solution to the BVP is not unique. For these parameter values there are infinitely many solutions to the problem. The same incorrect analysis is reproduced in several other papers (see the references). Some of the claims of these papers are contradicted by established results on, for example, the Falkner–Skan problem.

This paper investigates a boundary value problem (BVP) governing Marangoni convection over a flat surface. The BVP involves a temperature gradient parameter k>−1. Previous numerical and analytical studies report the existence of one solution for each value of k. Here we show that the nature of the solutions varies greatly depending on the value of k. For each −10.

We investigate the nonlinear boundary value problem (BVP) that is derived from a similarity transformation of the Navier–Stokes equations governing fluid flow toward a stretching permeable cylinder. Existence of a solution is proven for all values of the Reynolds number and for both suction and injection, and uniqueness results are obtained in the case of a monotonic solution. A priori bounds on the skin friction coefficient are also obtained. These bounds achieve any desired order of accuracy as the injection parameter tends to negative infinity.

In this note we consider axisymmetric stagnation point flow of one fluid impinging on a disk covered with a second fluid. A similarity reduction is employed to reduce the governing PDEs to a nonlinear ODE boundary value problem. Previous numerical investigations of the problem in the literature indicate the existence of one solution. Here we prove the existence of at least two solutions to the BVP. We also obtain results concerning the possibility of further solutions and present numerical approximations to the solutions. AMS (MOS) Subject Classification. 35B15, 76D03, 76D10

We investigate a generalization of the Lotka-Volterra equations to a three species food chain. This simplified model yields a more complicated dynamical system than models involving logistic-type equations and lends itself to a study or discussion in a modeling of dierential equations course.

There has been much recent interest in the stagnation point flow of a fluid toward a stretching sheet. Investigations that may include oblique stagnation flow and heat transfer to a horizontal plate all involve the same boundary value problem (BVP):f‴+ff″-(f′)2+b2=0,f(0)=0,f′(0)=1,f′(∞)=b.Here b is the ratio of the stagnation flow strain rate to the stretch rate of the sheet. Through numerical analysis of the problem, several authors have conjectured the existence of a solution for all values of b>0. In this note we present numerical evidence that a second solution exists for 00 there exists a monotonic solution to the BVP and if b>1, this solution is unique. If b<1 it can be shown that any further solutions cannot be monotonic and the second solution found here numerically is non-monotonic. The asymptotic behavior of solutions near b=0 and 1 is also presented. Finally, a stability analysis is performed to show that solutions on the upper branch of the dual solutions are linearly stable, while those on the lower branch are linearly unstable.

This paper investigates the large time (final state flow) solutions for unsteady mixed convection boundary layer flow near a stagnation point on a vertical surface embedded in a Darcian fluid-saturated porous medium. Through numerical computations Nazar et al. [R. Nazar, N. Amin, I. Pop, Unsteady mixed convection boundary layer flow near the stagnation point on a vertical surface in a porous medium, Int. J. Heat Mass Transfer 47 (2004) 2681–2688] concluded that for values of the mixed convection parameter λ>−1, the governing boundary value problem (BVP) had a unique solution. If λc≈−1.4175<λ⩽−1 two solutions were reported, and if λ<λc then no solutions were found. The purpose of this note is to provide further mathematical and numerical analysis of this problem. We prove existence of a solution to the governing BVP for all λ>−1. We also present numerical evidence that a second solution exists for λ>−1, thus giving dual solutions for all λ>λc. It is also proven that if λ<−2.9136 no solution to the BVP exists. Finally, a stability analysis is performed to show that solutions on the upper branch are linearly stable while those on the lower branch are linearly unstable.

In this note a proof for nonexistence of a solution for reverse radial stagnation flow on a cylinder is given. The results are valid for either suction or blowing on the cylinder wall.

(2002). A Lotka-Volterra Three-species Food Chain. Mathematics Magazine: Vol. 75, No. 4, pp. 243-255.

We investigate the boundary value problem (BVP) resulting from an exact reduction of the Navier–Stokes equations for radial stagnation flow impinging on a porous cylinder with uniform transpiration. Existence of a solution is proven for all values of the Reynolds number and for both suction and blowing. The results are obtained using a topological shooting argument which varies a parameter related to the axial shear stress. This formulation permits a derivation of useful bounds on the axial shear stress in a straightforward manner. For a certain parameter range, uniqueness results are also obtained.