Exact self-similarity solution of the Navier-Stokes equations for a deformable channel with wall suction or injection

2001 ◽  
Author(s):  
E. Dauenhauer ◽  
J. Majdalani
Keyword(s):  
Similarity Solution ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Self Similarity ◽  
Wall Suction ◽  
Navier Stokes Equations ◽  
Suction Or Injection ◽  
Deformable Channel

scholarly journals Asymptotically based self-similarity solution of the Navier–Stokes equations for a porous tube with a non-circular cross-section

2017 ◽  
Vol 826 ◽  
pp. 396-420 ◽  
Author(s):  
M. Bouyges ◽  
F. Chedevergne ◽  
G. Casalis ◽  
J. Majdalani
Keyword(s):  
Cross Section ◽  
Similarity Solution ◽  
Stokes Equations ◽  
Circular Cross Section ◽  
Internal Flow ◽  
Navier Stokes ◽  
Solid Rocket Motor ◽  
Mean Flow ◽  
Navier Stokes Equations ◽  
Radial Deviation

This work introduces a similarity solution to the problem of a viscous, incompressible and rotational fluid in a right-cylindrical chamber with uniformly porous walls and a non-circular cross-section. The attendant idealization may be used to model the non-reactive internal flow field of a solid rocket motor with a star-shaped grain configuration. By mapping the radial domain to a circular pipe flow, the Navier–Stokes equations are converted to a fourth-order differential equation that is reminiscent of Berman’s classic expression. Then assuming a small radial deviation from a fixed chamber radius, asymptotic expansions of the three-component velocity and pressure fields are systematically pursued to the second order in the radial deviation amplitude. This enables us to derive a set of ordinary differential relations that can be readily solved for the mean flow variables. In the process of characterizing the ensuing flow motion, the axial, radial and tangential velocities are compared and shown to agree favourably with the simulation results of a finite-volume Navier–Stokes solver at different cross-flow Reynolds numbers, deviation amplitudes and circular wavenumbers.

Re-entrant corner flows of Oldroyd-B fluids

2005 ◽  
Vol 461 (2060) ◽  
pp. 2573-2603 ◽  
Author(s):  
J.D Evans
Keyword(s):  
Boundary Layers ◽  
Newtonian Fluid ◽  
Similarity Solution ◽  
Stokes Equations ◽  
Inviscid Flow ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Outer Solution ◽  
The Core ◽  
Retardation Parameter

The method of matched asymptotic expansions is used to construct solutions for the planar steady flow of Oldroyd-B fluids around re-entrant corners of angles π / α (1/2≤ α <1). Two types of similarity solutions are described for the core flow away from the walls. These correspond to the two main dominant balances of the constitutive equation, where the upper convected derivative of stress either dominates or is balanced by the upper convected derivative of the rate of strain. The former balance gives the incompressible Euler or inviscid flow equations and the latter balance the incompressible Navier–Stokes equations. The inviscid flow similarity solution for the core is that first derived by Hinch (Hinch 1993 J. Non-Newtonian Fluid Mech. 50 , 161–171) with a core stress singularity that depends upon the corner angle and radial distance as O ( r −2(1− α ) ) and a velocity behaviour that vanishes as O ( r α (3− α )−1 ). Extending the analysis of Renardy (Renardy 1995 J. Non-Newtonian Fluid Mech. 58 , 83–39), this outer solution is matched to viscometric wall behaviour for both upstream and downstream boundary layers. This structure is shown to hold for the majority of the retardation parameter range. In contrast, the similarity solution associated with the Navier–Stokes equations has a velocity behaviour O ( r λ ) where λ ∈(0,1) satisfies a nonlinear eigenvalue problem, dependent upon the corner angle and an associated Reynolds number defined in terms of the ratio of the retardation and relaxation times. This similarity solution is shown to hold as an outer solution and is matched into stress boundary layers at the walls which recover viscometric behaviour. However, the matching is restricted to values of the retardation parameter close to the relaxation parameter. In this case the leading order core stress is Newtonian with behaviour O ( r −(1− λ ) ).

Laminar Flow in a Uniformly Porous Channel

1958 ◽  
Vol 25 (4) ◽  
pp. 613-617
Author(s):  
F. M. White ◽  
B. F. Barfield ◽  
M. J. Goglia
Keyword(s):  
Free Parameter ◽  
Stokes Equations ◽  
Fluid Velocity ◽  
Wall Friction ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Laminar Channel Flow ◽  
Judicious Choice ◽  
Suction Or Injection ◽  
Uniform Fluid

Abstract The problem of laminar channel flow has been investigated for the case of uniform fluid suction or injection through the channel walls. The solution can be divided into three steps: (a) A judicious choice of stream function reduces the Navier-Stokes equations to an ordinary, fourth-order, nonlinear differential equation, which contains a free parameter R, the Reynolds number based upon fluid velocity through the wall. (b) Since general analysis of this equation is intractable, the parameter R is eliminated by a suitable transformation. (c) The transformed, nonparametric equation yields to a series solution, valid and absolutely convergent for all R. From this general solution, expressions are developed for velocity components, pressure distribution, and wall-friction coefficient.

Nonsteady Stagnation-Point Flows Over Permeable Surfaces: Explicit Solutions of the Nevier-Stokes Equations

1995 ◽  
Vol 117 (1) ◽  
pp. 189-191 ◽  
Author(s):  
G. I. Burde
Keyword(s):  
Time Dependence ◽  
Stagnation Point ◽  
Stokes Equations ◽  
Similarity Solutions ◽  
Navier Stokes ◽  
Explicit Solutions ◽  
Navier Stokes Equations ◽  
New Approach ◽  
The Common ◽  
Suction Or Injection

Some new explicit solutions of the unsteady two-dimensional Navier-Stokes equations describing nonsteady stagnation-point flows with surface suction or injection are presented. The solutions have been obtained using a new approach for finding explicit similarity solutions of partial differential equations. As distinct from the common Birkhoff’s similarity transformation, which permits only one form of an unsteady potential flow field and only one form of time dependence for suction (or blowing) velocity, the transforms obtained permit consideration of a variety of special solutions differing in the forms of time dependence.

Exact vortex solutions of the Navier–Stokes equations with axisymmetric strain and suction or injection

2009 ◽  
Vol 626 ◽  
pp. 291-306 ◽  
Author(s):  
ALEX D. D. CRAIK
Keyword(s):  
Stokes Equations ◽  
Navier Stokes ◽  
Vortex Flows ◽  
Navier Stokes Equations ◽  
Axisymmetric Vortex ◽  
Vortex Solutions ◽  
Suction Or Injection

New solutions of the Navier–Stokes equations are presented for axisymmetric vortex flows subject to strain and to suction or injection. Those expressible in simple separable or similarity form are emphasized. These exhibit the competing roles of diffusion, advection and vortex stretching.

Peristaltic Flow in a Deformable Channel

2011 ◽  
Vol 66 (1-2) ◽  
pp. 24-32 ◽  
Author(s):  
Dil Nawaz ◽  
Khan Marawat ◽  
Saleem Asghar
Keyword(s):  
Stokes Equations ◽  
Expansion Ratio ◽  
Peristaltic Flow ◽  
Wave Number ◽  
Navier Stokes ◽  
Physical Parameters ◽  
Peristaltic Motion ◽  
Navier Stokes Equations ◽  
Small Wave Number ◽  
Deformable Channel

The effects of wall contraction or expansion on the characteristics of the peristaltic flow have been considered in this paper. For that, we present a theoretical model of laminar incompressible viscous peristaltic flow in a deformable channel. The problem is modeled in terms of unsteady twodimensional Navier Stokes equations and the solution is obtained using the perturbation method. The physical parameters appearing due to deformation and the peristaltic motion are the wall expansion ratio (α) and the wave number (δ ), respectively. Analytic perturbation results are obtained for small wave number and small wall expansion ratio. Basically the study is undertaken to examine the peristaltic motion along with the deformation of the channel. This will enhance our understanding of deformation/squeezing and peristalsis phenomena independently and jointly. Deformation effects are shown on the otherwise peristaltic fluid flow. The results of peristaltic flow [Shapiro et al., J. Fluid Mech. Digit. Archive 37, 799 (1969)] can be recovered for the limiting case of α equal to zero.

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