The stability of plane flow for a non-Newtonian fluid obeying a rheological power law

AbstractWe study the existence of weak solutions to a Newtonian fluid∼non-Newtonian fluid mixed-type equation $$ {u_{t}}= \operatorname{div} \bigl(b(x,t){ \bigl\vert {\nabla A(u)} \bigr\vert ^{p(x) - 2}}\nabla A(u)+\alpha (x,t)\nabla A(u) \bigr)+f(u,x,t). $$ u t = div ( b ( x , t ) | ∇ A ( u ) | p ( x ) − 2 ∇ A ( u ) + α ( x , t ) ∇ A ( u ) ) + f ( u , x , t ) . We assume that $A'(s)=a(s)\geq 0$ A ′ ( s ) = a ( s ) ≥ 0 , $A(s)$ A ( s ) is a strictly increasing function, $A(0)=0$ A ( 0 ) = 0 , $b(x,t)\geq 0$ b ( x , t ) ≥ 0 , and $\alpha (x,t)\geq 0$ α ( x , t ) ≥ 0 . If $$ b(x,t)=\alpha (x,t)=0,\quad (x,t)\in \partial \Omega \times [0,T], $$ b ( x , t ) = α ( x , t ) = 0 , ( x , t ) ∈ ∂ Ω × [ 0 , T ] , then we prove the stability of weak solutions without the boundary value condition.

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Steady Motion of Ice Sheets

Journal of Glaciology ◽

10.3189/s0022143000010467 ◽

1980 ◽

Vol 25 (92) ◽

pp. 229-246 ◽

Cited By ~ 10

Author(s):

L. W. Morland ◽

I. R. Johnson

Keyword(s):

Differential Equation ◽

Linear Differential Equation ◽

Power Law ◽

Perturbation Expansion ◽

Ice Sheet ◽

Leading Order ◽

Plane Flow ◽

General Law ◽

Non Linear ◽

Linear Differential

AbstractSteady plane flow under gravity of a symmetric ice sheet resting on a horizontal rigid bed, subject to surface accumulation and ablation, basal drainage, and basal sliding according to a shear-traction-velocity power law, is treated. The surface accumulation is taken to depend on height, and the drainage and sliding coefficient also depend on the height of overlying ice. The ice is described as a general non-linearly viscous incompressible fluid, with illustrations presented for Glen’s power law, the polynomial law of Colbeck and Evans, and a Newtonian fluid. Uniform temperature is assumed so that effects of a realistic temperature distribution on the ice response are not taken into account. In dimensionless variables a small paramter ν occurs, but the ν = 0 solution corresponds to an unbounded sheet of uniform depth. To obtain a bounded sheet, a horizontal coordinate scaling by a small factor ε(ν) is required, so that the aspect ratio ε of a steady ice sheet is determined by the ice properties, accumulation magnitude, and the magnitude of the central thickness. A perturbation expansion in ε gives simple leading-order terms for the stress and velocity components, and generates a first order non-linear differential equation for the free-surface slope, which is then integrated to determine the profile. The non-linear differential equation can be solved explicitly for a linear sliding law in the Newtonian case. For the general law it is shown that the leading-order approximation is valid both at the margin and in the central zone provided that the power and coefficient in the sliding law satisfy certain restrictions.

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STABILITY ANALYSIS OF THE THIN POWER LAW LIQUID FILM FLOWING DOWN ON A VERTICAL CYLINDER

Transactions of the Canadian Society for Mechanical Engineering ◽

10.1139/tcsme-2006-0006 ◽

2006 ◽

Vol 30 (1) ◽

pp. 81-96 ◽

Cited By ~ 2

Author(s):

Po-Jen Cheng ◽

Kuo-Chi Liu

Keyword(s):

Liquid Film ◽

Power Law ◽

Film Flow ◽

Vertical Cylinder ◽

Flow Index ◽

Free Film ◽

Long Wave ◽

Lateral Curvature ◽

The Stability ◽

Film Interface

The paper investigates the stability theory of a thin power law liquid film flowing down along the outside surface of a vertical cylinder. The long-wave perturbation method is employed to solve for generalized linear kinematic equations with free film interface. The normal mode approach is used to compute the stability solution for the film flow. The degree of instability in the film flow is further intensified by the lateral curvature of cylinder. This is somewhat different from that of the planar flow. The analysis results also indicate that by increasing the flow index and increasing the radius of the cylinder the film flow can become relatively more stable as traveling down along the vertical cylinder.

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The Steady Profile of an Axisymmetric Ice Sheet

Journal of Glaciology ◽

10.3189/s0022143000011205 ◽

1981 ◽

Vol 27 (95) ◽

pp. 25-37 ◽

Cited By ~ 1

Author(s):

I. R. Johnson

Keyword(s):

Aspect Ratio ◽

Power Law ◽

Viscous Incompressible Fluid ◽

Ice Sheet ◽

Plane Flow ◽

The Third ◽

Basal Sliding ◽

Steady Plane ◽

Third Dimension ◽

Special Case

AbstractSteady plane flow under gravity of an axisymmetric ice sheet resting on a horizontal rigid bed, subject to surface accumulation and ablation, basal drainage, and basal sliding is treated according to a power law between shear traction and velocity. The surface accumulation is taken to depend on height, and the drainage and sliding coefficient also depend on the height of overlying ice. The ice is described as a general non-linearly viscous incompressible fluid, and temperature variation through the ice sheet is neglected. Illustrations are presented for Glen’s power law (including the special case of a Newtonian fluid), and the polynomial law of Colbeck and Evans. The analysis follows that of Morland and Johnson (1980) where the analogous problem for an ice sheet deforming under plane flow was considered. Comparisons are made between the two models and it is found that the effect of the third dimension is to reduce (or leave unchanged) the aspect ratio for the cases considered, although no general formula can be obtained. This reduction is seen to depend on both the surface accumulation and the sliding law.

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Stability of Flexibly Supported Oil Journal Bearings Using Non-Newtonian Lubricants: Linear Perturbation Analysis

Journal of Tribology ◽

10.1115/1.1340632 ◽

2000 ◽

Vol 123 (3) ◽

pp. 651-654 ◽

Cited By ~ 3

Author(s):

K. Raghunandana ◽

B. C. Majumdar, and ◽

R. Maiti

Keyword(s):

Power Law ◽

Perturbation Analysis ◽

Perturbation Technique ◽

Stability Margin ◽

Journal Bearings ◽

Diameter Ratio ◽

Linear Perturbation ◽

Flexible Support ◽

The Stability ◽

Power Law Index

The purpose of this paper is to study the effect of non-Newtonian lubricant on the stability of oil film journal bearings mounted on flexible support using linear perturbation technique. The model of non-Newtonian lubricant developed by Dien and Elrod is taken into consideration. The dynamic co-coefficients are calculated for different values of power law index and length to diameter ratio. These are then used to find stability margin for different support parameters to study the effect of the non-Newtonian lubricant.

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Mass transfer from solid particles to power law non-Newtonian fluid in granular bed at low Reynolds numbers

Chemical Engineering Journal ◽

10.1016/s1385-8947(01)00295-9 ◽

2002 ◽

Vol 88 (1-3) ◽

pp. 119-125 ◽

Cited By ~ 7

Author(s):

G Peev

Keyword(s):

Mass Transfer ◽

Newtonian Fluid ◽

Power Law ◽

Reynolds Numbers ◽

Solid Particles ◽

Granular Bed ◽

Low Reynolds Numbers ◽

Low Reynolds

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Steady Motion of Ice Sheets

Journal of Glaciology ◽

10.1017/s0022143000010467 ◽

1980 ◽

Vol 25 (92) ◽

pp. 229-246 ◽

Cited By ~ 102

Author(s):

L. W. Morland ◽

I. R. Johnson

Keyword(s):

Differential Equation ◽

Linear Differential Equation ◽

Power Law ◽

Perturbation Expansion ◽

Ice Sheet ◽

Leading Order ◽

Plane Flow ◽

General Law ◽

Non Linear ◽

Linear Differential

AbstractSteady plane flow under gravity of a symmetric ice sheet resting on a horizontal rigid bed, subject to surface accumulation and ablation, basal drainage, and basal sliding according to a shear-traction-velocity power law, is treated. The surface accumulation is taken to depend on height, and the drainage and sliding coefficient also depend on the height of overlying ice. The ice is described as a general non-linearly viscous incompressible fluid, with illustrations presented for Glen’s power law, the polynomial law of Colbeck and Evans, and a Newtonian fluid. Uniform temperature is assumed so that effects of a realistic temperature distribution on the ice response are not taken into account. In dimensionless variables a small paramterνoccurs, but theν= 0 solution corresponds to an unbounded sheet of uniform depth. To obtain a bounded sheet, a horizontal coordinate scaling by a small factorε(ν) is required, so that the aspect ratioεof a steady ice sheet is determined by the ice properties, accumulation magnitude, and the magnitude of the central thickness. A perturbation expansion inεgives simple leading-order terms for the stress and velocity components, and generates a first order non-linear differential equation for the free-surface slope, which is then integrated to determine the profile. The non-linear differential equation can be solved explicitly for a linear sliding law in the Newtonian case. For the general law it is shown that the leading-order approximation is valid both at the margin and in the central zone provided that the power and coefficient in the sliding law satisfy certain restrictions.

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