. Convex liquid bridge between two balls has studied by Vogel [ 13 ], [ 14 ]. Recently, Gagneux et al. [ 15 ] has researched the capillary bridge properties with the use of analytical calculation of bridge profile and the experimental image data. In this article, on the basis of the Young-Laplace equation, exact solution of capillary force between two separate unequal spheres is investigated. In Section 2, we analyze the liquid bridge geometry between two particles by shooting method. Based on the gorge method, the capillary force is calculated. In Section 3, the effects

. \end{align}$$ 3 Numerical solution The above system of equations is solved numerically by using fourth order Runge-Kutta method along with shooting technique. For this purpose the Eqs. (20-22 ) and ( 24 - 26 ) are transformed in to first order due to the higher order equation at r = δ (boundary-layer thickness) being unavailable. Then, the boundary value problem is solved by shooting method. 3.1 Numerical solution of Reynolds model Eqs. (20) and (21) are solved numerically by Ruge-Kutta method corresponding to the boundary conditions (22) by

‡ DIM & CMM (UMR CNRS no. 2071), FCFM Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile e-mail: manasevi@dim.uchile.cl Communicated by Jon Jacobsen Abstract We consider radial solutions of an elliptic equation involving the p-Laplace operator and prove by a shooting method the existence of compactly supported solutions with any pre- scribed number of nodes. The method is based on a change of variables in the phase plane corresponding to an asymptotic Hamiltonian system and provides qualitative properties of the solutions. ∗Partially supported by ECOS

stretching sheet were studied by Cobble [ 18 ], Helmy [ 19 ], Chiam [ 20 ], Anjali Devi and Thiyagarajan [ 21 ] and very recently by Ishak et al. [ 22 , 23 ]. Although many authors have studied non-Newtonian fluid over stretching cylinder up to some extent the Williamson fluid for steady, hydromagnetic flow over a stretching cylinder in the presence of uniform transverse magnetic field has not yet been studied. Our focus will be on its numerical solution by shooting method with Runge–Kutta–Fehlberg method. The effects of variation in stretching ratio parameter and

developed by homotopy analysis method (HAM). The Numerical so- lutions are obtained by using shooting method with fourth-order Runge–Kutta integration technique. The fields are influence appreciably with the variation of embedding parameters. We noticed that the velocity ratio has a dual behaviour on the momentum boundary layer. On the other hand the thermal boundary layer thins when the velocity ratio is increased. The results indicate a significant increase in the velocity and a decrease in thermal boundary layer thickness with an intensification in the viscoelastic

, velocity profile, temperature profile, pressure gradient, and stream- lines have been discussed graphically at the end of the article. Key words: Variable Viscosity; Shooting Method; Comparison. 1. Introduction The peristaltic flow has paramount importance in physiology and engineering science. Occurrence of such flows are quite prevalent in nature. Particularly, these flows are encountered in the urinary tract, chyme movement in the gastrointestinal tract, swallowing of food through esophagus, and many others. The pi- oneering work on this topic has been initiated by

as the shooting method was used to find their solution. Introducing the stream function defined as (31) u = ∂ ψ ∂ y , v = − ∂ ψ ∂ x . $$u = {{\partial \psi } \over {\partial y}},\;v = - {{\partial \psi } \over {\partial x}}.$$ With the help of equation (31), equations (29)–(30) can be written as (32) ∂ ∂ y [ ( 1 + 2 α 1 ( ∂ 2 ψ ∂ y 2 ) 2 1 + 2 α 2 ( ∂ 2 ψ ∂ y 2 ) 2 ) ∂ 2 ψ ∂ y 2 ] = d p d x . $${\partial \over {\partial y}}\left[ {\left( {{{1 + 2{\alpha _1}{{\left( {{{{\partial ^2}\psi } \over {\partial {y^2}}}} \right)}^2}} \over {1 + 2{\alpha _2

(19) C f = μ ρ V 1 2 ∂ u ∂ y S h = − a ( C 2 − C 1 ) ∂ C ∂ y N u = − a ( T 2 − T 1 ) ∂ T ∂ y $$\begin{align} & {C_f} = \frac{\mu }{{\rho V_1^2}}\frac{{\partial u}}{{\partial y}}\,\,\,\,\,\,\,\,\,\,\,Sh = \frac{{ - a}}{{({C_2} - {C_1})}}\frac{{\partial C}}{{\partial y}}\nonumber\\ & Nu = \frac{{ - a}}{{({T_2} - {T_1})}}\frac{{\partial T}}{{\partial y}} \end{align}$$ 3 Solution procedure A nonlinear system of first order equations (21) is formulated from equations (13)-(17). Shooting method is implemented by reducing the BVP to IVP and an algebraic nonlinear

1.70 3.249 × 10 −5 1.48 1.019 × 10 −4 1.33 Table 3 Example 5.2 with several values of α . Comparison with the exact solution at t = 0.5 (the value in the boundary condition) and t = 1 with several values of the stepsize h (shooting method with Method 2 to solve the IVP). α = 1/4 α = 1/2 α = 2/3 h y (0) e ϵ h $ e^h_{\epsilon} $ (0.5) e ϵ h $ e^h_{\epsilon} $ (1) y (0) e ϵ h $ e^h_{\epsilon} $ (0.5) e ϵ h $ e^h_{\epsilon} $ (1) y (0) e ϵ h $ e^h_{\epsilon} $ (0.5) e ϵ h $ e^h_{\epsilon} $ (1) 1/20 9.313 × 10 −11 3.014 × 10 −3 5.280 × 10 −3 9.313 × 10

## Abstract

In this paper, the steady two-dimensional stagnation-point flow of a viscoelastic Walters’ B’ fluid over a stretching surface is examined. It is assumed that the fluid impinges on the wall obliquely. Using similarity variables, the governing partial differential equations are transformed into a set of two non-dimensional ordinary differential equations. These equations are then solved numerically using the shooting method with a finite-difference technique.