Nano boundary layer equation with nonlinear Navier boundary condition

Abstract

At the micro and nano scale the standard no slip boundary condition of classical fluid mechanics does not apply and must be replaced by a boundary condition that allows some degree of tangential slip. In this study the classical laminar boundary layer equations are studied using Lie symmetries with the no-slip boundary condition replaced by a nonlinear Navier boundary condition. This boundary condition contains an arbitrary index parameter, denoted by n > 0, which appears in the coefficients of the ordinary differential equation to be solved. The case of a boundary layer formed in a convergent channel with a sink, which corresponds to n = 1/2, is solved analytically. Another analytical but non-unique solution is found corresponding to the value n = 1/3, while other values of n for n > 1/2 correspond to the boundary layer formed in the flow past a wedge and are solved numerically. It is found that for fixed slip length the velocity components are reduced in magnitude as n increases, while for fixed n the velocity components are increased in magnitude as the slip length is increased.

Analytical Solutions of Nano Boundary Layer Flows by Using He's Homotopy Perturbation Method

“…By using similarity solution the coupled differential equations, (1) and (2), are reduced to (8). Equation (8) with the boundary conditions, (9), is solved by using HPM.…”

“…The linear Navier condition was first used for nano boundary layer by M.T.Matthews and J.M. Hill [2]. They used similarity solutions to produce one ordinary differential equation and then solved it with numerical methods.…”

Double diffusive magnetohydrodynamic squeezing flow of nanofluid between two parallel disks with slip and temperature jump boundary conditions

“…Therefore, additional boundary conditions are required to adequately predict the low flow pressure or low system characteristics size. Such slip boundary condition was first initiated by Navier [26] upon which other researchers have built [2,24]. Most of the above reviews have been limited to the analysis of squeezing flow under no slip and no temperature jump boundary conditions.…”

Thermo-Magneto-Solutal Squeezing Flow of Nanofluid between Two Parallel Disks Embedded in a Porous Medium: Effects of Nanoparticle Geometry, Slip and Temperature Jump Conditions

“…Moreover, it has been established that in many cases of uid and ow problems, such as polymeric liquids, thin lm problems, nano uids, rare ed uid problems, uids containing concentrated suspensions, and ow on multiple interfaces, slip condition prevails at the boundary of the ow [10][11][12][13][14][15][16][17][18][19][20][21]. Such slip boundary condition was rst initiated by Navier [22] upon which other researchers have built their analysis [10,11]. erefore, in recent years, the e ects of slip e ect on uid ow have been considered by many researchers [12][13][14][15][16][17][18][19][20][21] due to its signi cance to most practical uid ow situations.…”