Self-similar solutions of nonstationary equations of a plane laminar magnetohydrodynamic boundary layer

1969 ◽  
Vol 7 (1) ◽  
pp. 8-11
Author(s):  
A. G. Boev
Keyword(s):  
Boundary Layer ◽  
Nonstationary Equations ◽  
Self Similar ◽  
Similar Solutions

Natural convection flow far from a horizontal plate

1999 ◽  
Vol 387 ◽  
pp. 227-254 ◽  
Author(s):  
VALOD NOSHADI ◽  
WILHELM SCHNEIDER
Keyword(s):  
Boundary Layer ◽  
Prandtl Number ◽  
Stokes Equations ◽  
Axisymmetric Flow ◽  
The Self ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Boundary Layer Equations ◽  
Self Similar ◽  
Similar Solutions

Plane and axisymmetric (radial), horizontal laminar jet flows, produced by natural convection on a horizontal finite plate acting as a heat dipole, are considered at large distances from the plate. It is shown that physically acceptable self-similar solutions of the boundary-layer equations, which include buoyancy effects, exist in certain Prandtl-number regimes, i.e. 0.5<Pr[les ]1.470588 for plane, and Pr>1 for axisymmetric flow. In the plane flow case, the eigenvalues of the self-similar solutions are independent of the Prandtl number and can be determined from a momentum balance, whereas in the axisymmetric case the eigenvalues depend on the Prandtl number and are to be determined as part of the solution of the eigenvalue problem. For Prandtl numbers equal to, or smaller than, the lower limiting values of 0.5 and 1 for plane and axisymmetric flow, respectively, the far flow field is a non-buoyant jet, for which self-similar solutions of the boundary-layer equations are also provided. Furthermore it is shown that self-similar solutions of the full Navier–Stokes equations for axisymmetric flow, with the velocity varying as 1/r, exist for arbitrary values of the Prandtl number.Comparisons with finite-element solutions of the full Navier–Stokes equations show that the self-similar boundary-layer solutions are asymptotically approached as the plate Grashof number tends to infinity, whereas the self-similar solution to the full Navier–Stokes equations is applicable, for a given value of the Prandtl number, only to one particular, finite value of the Grashof number.In the Appendices second-order boundary-layer solutions are given, and uniformly valid composite expansions are constructed; asymptotic expansions for large values of the lateral coordinate are performed to study the decay of the self-similar boundary-layer flows; and the stability of the jets is investigated using transient numerical solutions of the Navier–Stokes equations.

CONVERGENCE OF LOCALLY SELFSIMILAR SOLUTIONS TO EXACT NUMERICAL SOLUTIONS OF BOUNDARY LAYER EQUATIONS FOR A PLATE

2021 ◽  
pp. 49-62
Author(s):  
Yu.N. Grigoriev ◽  
◽  
A.G. Gorobchuk ◽  
I.V. Ershov ◽  
◽  
...  
Keyword(s):  
Boundary Layer ◽  
Finite Difference ◽  
Linear Stability ◽  
Numerical Solutions ◽  
Linear Stability Theory ◽  
Plane Boundary ◽  
Transfer Coefficients ◽  
Vibrationally Excited ◽  
Self Similar ◽  
Similar Solutions

This paper considers a possibility of using locally self-similar solutions for a stationary boundary layer in linear stability problems. The solutions, obtained at various boundary conditions for a vibrationally excited gas, are compared with finite-difference calculations of the corresponding flows. An initial system of equations for a plane boundary layer of the vibrationally excited gas is derived from complete equations of two-temperature relaxation aerodynamics. Relaxation of vibrational modes of gas molecules is described in the framework of the Landau – Teller equation. Transfer coefficients depend on the static flow temperature. In a complete problem statement, the flows are calculated using the Crank – Nicolson finite-difference scheme. In all the considered cases, it is shown that the locally self-similar velocity and temperature profiles converge to the corresponding profiles for a fully developed boundary-layer flow calculated in a finite-difference formulation. The obtained results justify the use of locally self-similar solutions in problems of the linear stability theory for boundary-layer flows of a vibrationally excited gas.

Rotationally symmetric flow of Cu-Al2O3/water hybrid nanofluid over a heated porous boundary

2021 ◽  
pp. 095440622110231
Author(s):  
Sahreen Tahira ◽  
M Mustafa ◽  
Ammar Mushtaq
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Viscous Dissipation ◽  
Transfer Process ◽  
Vortex Flow ◽  
Heat Transfer Process ◽  
Hybrid Nanofluid ◽  
Dissipation Term ◽  
Self Similar ◽  
Similar Solutions

This paper endeavours to provide an analysis for nanofluid boundary layer that develops beneath a generalized vortex flow subjected to viscous dissipation effects. Tangential flow high above the disk is assumed to vary with radial distance [Formula: see text] as [Formula: see text]. Hybrid nanofluid comprises alumina-[Formula: see text] and copper-[Formula: see text] nanoparticles with water as base liquid. A generalized version of von Kármán relations proposed in a recent paper is opted to present self-similar solutions. In presence of viscous dissipation term, self-similar solutions are possible only when temperature difference is proportional to [Formula: see text]. Otherwise, the solutions are only locally similar. Present model reduces to two special situations namely rigid body rotation [Formula: see text] and potential vortex [Formula: see text]. It is perceived that inclusion of nanoparticles markedly affects the boundary layer development under the prescribed vortex flow, and the associated heat transfer process. Also, viscous dissipation term has important implications on the resulting heat transfer process.

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