Some similarity solutions for three-dimensional boundary layers

Abstract

A similarity solution of a three-dimensional boundary layer is investigated. The outer flow is given by U = ( − xz , − yz , z 2 ), corresponding to an axisymmetric poloidal circulation with constant potential vorticity. This flow is an exact solution of the Navier–Stokes. A wall is introduced at y = 0 along which a boundary layer develops towards x = 0. We show that a similarity reduction to a system of ODEs is possible. Two distinct solutions are found, one of them through numerical path-continuation, and their stability is investigated. A second three-dimensional solution is also identified for two-dimensional outer flow. The solutions are generalized for outer flows scaling with different powers of z and similar results are found. This behaviour is related to the non-uniqueness of the Falkner–Skan flows in a three-dimensional sense, with a transverse wall-jet.

A new similarity solution with stability analysis for the three-dimensional boundary layer of hybrid nanofluids

“…We consider the steady, three-dimensional boundary layer flow of a hybrid nanofluid past a permeable surface, as shown in Figure 1, where (x, y, z) are the Cartesian coordinates, with the surface in the plane at y = 0 along which a boundary layer develops toward x = 0. Following Vaz and Mestel (2019), we assume that the outer flow (inviscid flow) is given by:…”

“…The axisymmetric is broken by the introduction of a wall at y = 0, where there is a slip velocity and so a boundary layer develops there. Under these assumptions, the basic boundary layer equations of this problem from the steady, incompressible Navier-Stokes equations can be written as (Vaz and Mestel, 2019;Devi and Devi, 2016b)…”

“…where @p @x ¼ 0 and @p @z ¼ À2C 2 Z 3 . The corresponding boundary condition of these equations is (Vaz and Mestel, 2019):…”

“…Other numerical works on threedimensional hybrid nanofluids also have been scrutinized by Yousefi et al (2018), Dinarvand (2019, Dinarvand and Rostami (2020), Chahregh and Dinarvand (2020) and Waini et al (2019c). Vaz and Mestel (2019) investigated the three-dimensional boundary layer flow where the outer flow corresponding to an axisymmetric poloidal circulation with constant potential vorticity. They presented several cases of three-dimensional flow using the proposed similarity transformations.…”

“…They presented several cases of three-dimensional flow using the proposed similarity transformations. Thus, inspired by the work of Vaz and Mestel (2019) and Devi and Devi (2016b), the novelty of the present work is to present the solution profiles of the three-dimensional hybrid nanofluid flow using the new kind of similarity transformation. The dual solutions are observed for both permeable and impermeable cases and being the main agenda of the work.…”