Stability of the poiseuille flow in a longitudinal magnetic field

Proskurin, Alexander V.; Sagalakov, A. M.

doi:10.1134/s1063784212050234

Cited by 4 publications

(3 citation statements)

References 10 publications

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“…The magnetic Prandtl number P m = R m /R e ∝ ν/η and measures the ratio of the dissipative effect of flow (viscosity) and the magnetic field (resistivity). Previous studies indicate that P m plays an important role in both the instabilities of shear flow and a shear magnetic field [24]. Thus, the roles of P m in the unstable spectra of PF are numerically calculated in this subsection.…”

Section: Roles Of the Magnetic Prandtl Numbermentioning

confidence: 97%

“…In the threedimensional calculation, it is found that the assumption used to obtain the modified Orr-Sommerfeld equation [20] is not suitable [22]. Even in the two-dimensional (2D) calculation, if the assumption of a small Reynolds number is abandoned, the calculation of the stability equation of the sixth-order ODE for the incompressible MHD flow indicates a new instability branch for a large Reynolds number [23,24]. However, in [23,24], the stability curves are obtained by fixing the magnetic Prandtl number P m = R m /R e ∝ ν/η, where ν and η are the viscosity and the resistivity of the fluid medium, respectively, which only infers that the new instability branch is excited in the region when the Reynolds number and magnetic Reynolds number R m = P m R e are both large.…”

Section: Introductionmentioning

confidence: 99%

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Unstable spectra of plane Poiseuille flow with a uniform magnetic field

Lai¹,

Yu²,

Ren³

et al. 2022

Plasma Phys. Control. Fusion

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The unstable spectra of plane Poiseuille ﬂow (PF) in the present of a longitudinal magnetic ﬁeld are numerically investigated using an eigenvalue solver of an incompressible magnetohydrodynamic equations. It is found that the strength of the magnetic ﬁeld and the dissipative eﬀect of the magnetic perturbation have played diﬀerent roles in diﬀerent parameter regions. The magnetic ﬁeld has strong suppression eﬀect on the classical plane PF instability with large Reynolds number Rein the region with magnetic Prandtl number Pm = [0.1, 1] or magnetic Reynolds number Rm = [103, 106]. Here, the Reynolds number and the magnetic Reynolds number are defined as Re=aV0/ν and Rm=aV0μ/η with a, V0, ν and η are the typical length, velocity, viscosity and resistivity respectively. The magnetic Prandtl number is defined as Pm = Rm/Re∝ν/η, which is proportional to the ratio of the viscosity and the resistivity of the fluid medium. As the strength of magnetic ﬁeld increases, the PF instability can be completely stabilized in the limit of Pm → ∞ or/and Rm → ∞. It is interestingly found that a new instability branch is excited in the small magnetic Prandtl number (Pm → 0) or moderate magnetic Reynolds number (Rm = 104∼ 106) and large Reynolds number (Re →∞) regions. The new type instability is veriﬁed to be driven by the magnetic Reynolds stress and modulated by the dissipative eﬀect of the magnetic perturbation. The wavelength of the original PF instability gradually shifts to the long wavelength region, but the wavelength of the new branch is almost unchanged, as Re increases with ﬁxed Rm. However, the wavelength of the original instability branch is almost unchanged, but the wavelength of the new instability branch shifts to the long wavelength region, as Rm increases with ﬂexed Re.

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Section: Roles Of the Magnetic Prandtl Numbermentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Unstable spectra of plane Poiseuille flow with a uniform magnetic field

Lai¹,

Yu²,

Ren³

et al. 2022

Plasma Phys. Control. Fusion

View full text Add to dashboard Cite

show abstract

“…It is well known that, in solving problems of magnetohydrodynamics, the analog of the Poiseuille solution is the Hartmann flow [20,23]. The superposition of the Poiseuille and Hartmann flows was studied in [14,32,37,38]. In addition, the exact Poiseuille solution was used in the hydrodynamics of non-Newtonian fluids in [1,10,12,13,15,24,25,30,46].…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Gradient Flow of a Vertical Vortex Fluid in a Thin Layer

Привалова¹,

Prosviryakov²,

Simonov³

2019

Nelin. Dinam.

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A new exact solution to the Navier -Stokes equations is obtained. This solution describes the inhomogeneous isothermal Poiseuille flow of a viscous incompressible fluid in a horizontal infinite layer. In this exact solution of the Navier -Stokes equations, the velocity and pressure fields are the linear forms of two horizontal (longitudinal) coordinates with coefficients depending on the third (transverse) coordinate. The proposed exact solution is two-dimensional in terms of velocity and coordinates. It is shown that, by rotation transformation, it can be reduced to a solution describing a three-dimensional flow in terms of coordinates and a two-dimensional flow in terms of velocities. The general solution for homogeneous velocity components is polynomials of the second and fifth degrees. Spatial acceleration is a linear function. To solve the boundaryvalue problem, the no-slip condition is specified on the lower solid boundary of the horizontal fluid layer, tangential stresses and constant horizontal (longitudinal) pressure gradients specified on the upper free boundary. It is demonstrated that, for a particular exact solution, up to three points can exist in the fluid layer at which the longitudinal velocity components change direction. It indicates the existence of counterflow zones. The conditions for the existence of the zero points of the velocity components both inside the fluid layer and on its surface under nonzero tangential stresses are written. The results are illustrated by the corresponding figures of the velocity component profiles and streamlines for different numbers of stagnation points.

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Stability of a Pressure-Driven Flow Between Coaxial Cylinders in a Longitudinal Magnetic Field

Proskurin

2020

J Appl Mech Tech Phy

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Stability of the poiseuille flow in a longitudinal magnetic field

Cited by 4 publications

References 10 publications

Unstable spectra of plane Poiseuille flow with a uniform magnetic field

Unstable spectra of plane Poiseuille flow with a uniform magnetic field

Nonlinear Gradient Flow of a Vertical Vortex Fluid in a Thin Layer

Stability of a Pressure-Driven Flow Between Coaxial Cylinders in a Longitudinal Magnetic Field

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