Newtonian flow with nonlinear Navier boundary condition

Matthews, Miccal T.; Hill, James M.

doi:10.1007/s00707-007-0454-8

Cited by 58 publications

(39 citation statements)

References 32 publications

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“…An explanation for the occurrence of slip Increase in hydrophobicity leads to an increase in slip at the wall, which has been observed by several researchers for the steady case [6][7][8][9][10][11][12][13][14][15][16][17]. Strictly speaking, the no-slip boundary condition is only valid if the flow adjacent to a solid surface is in thermodynamic equilibrium [26,31]. For fluid flow in small-scale systems, the collision frequency is not high enough to ensure thermodynamic equilibrium, thus a certain degree of tangential velocity slip must be allowed [26].…”

Section: Phase Lag Of Fluid Velocity Due To Wall Hydrophobicitymentioning

confidence: 99%

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Analysis of boundary slip in a flow with an oscillating wall

Thalakkottor

Mohseni

2013

Phys. Rev. E

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Molecular dynamic (MD) simulation is used to study slip at the fluid-solid boundary in an unsteady flow based on the Stokes second problem. An increase in slip is observed in comparison to the steady flow for shear rates below the critical shear rate of the corresponding steady flow. This increased slip is attributed to fluid inertial forces not represented in a steady flow. An unsteady mathematical model for slip is established, which estimates the increment in slip at the boundary. The model shows that slip is also dependent on acceleration in addition to the shear rate of fluid at the wall. By writing acceleration in terms of shear rate, it is shown that slip at the wall in unsteady flows is governed by the gradient of shear rate and shear rate of the fluid. Non-dimensionalizing the model gives a universal curve which can be used to find the slip boundary condition at the fluid-solid interface based on the information of shear rate and gradient of shear rate of the fluid. A governing non-dimensional number, defined as the ratio of phase speed to speed of sound, is identified to help in explaining the mechanism responsible for the transition of slip boundary condition from finite to a perfect slip and determining when this would occur. Phase lag in fluid velocity relative to wall is observed. The lag increases with decreasing time period of wall oscillation and increasing wall hydrophobicity. The phenomenon of hysteresis is seen when looking into the variation of slip velocity as a function of wall velocity and slip velocity as a function of fluid shear rate. The cause for hysteresis is attributed to the unsteady inertial forces of the fluid.

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Section: Phase Lag Of Fluid Velocity Due To Wall Hydrophobicitymentioning

confidence: 99%

“…Boundary slip has been the subject of less investigation in unsteady flows. A few exceptions are in unsteady gas flows [26,27] and analytic solutions for continuum scale problems [28][29][30][31]. However, to the authors knowledge, for liquids at micro-scales the research has been limited to steady flows.…”

Section: Introductionmentioning

confidence: 99%

Analysis of boundary slip in a flow with an oscillating wall

Thalakkottor

Mohseni

2013

Phys. Rev. E

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“…We point out that the Lorentz reciprocal theorem is valid only when the slip length l s = η/β is a material constant, which makes the Navier boundary condition linear [86]. In fact, nonlinear slip boundary conditions have been reported in the literature [68,[87][88][89], where the slip length l s depends on the shear rate at the solid boundaryγ when the latter is large enough. For example, the rate-dependent slip length l s = l 0 s (1 −γ /γ c ) −1/2 has been reported [87], where l 0 s is the constant slip length at lowγ , andγ c is the critical shear rate at which l s diverges.…”

Section: Lorentz Reciprocal Theoremmentioning

confidence: 83%

Anisotropic particle in viscous shear flow: Navier slip, reciprocal symmetry, and Jeffery orbit

Zhang

Qian

2015

Phys. Rev. E

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The hydrodynamic reciprocal theorem for Stokes flows is generalized to incorporate the Navier slip boundary condition, which can be derived from Onsager's variational principle of least energy dissipation. The hydrodynamic reciprocal relations and the Jeffery orbit, both of which arise from the motion of a slippery anisotropic particle in a simple viscous shear flow, are investigated theoretically and numerically using the fluid particle dynamics method [Phys. Rev. Lett. 85, 1338 (2000)]. For a slippery elliptical particle in a linear shear flow, the hydrodynamic reciprocal relations between the rotational torque and the shear stress are studied and related to the Jeffery orbit, showing that the boundary slip can effectively enhance the anisotropy of the particle. Physically, by replacing the no-slip boundary condition with the Navier slip condition at the particle surface, the cross coupling between the rotational torque and the shear stress is enhanced, as manifested through a dimensionless parameter in both of the hydrodynamic reciprocal relations and the Jeffery orbit. In addition, simulations for a circular particle patterned with portions of no-slip and Navier slip are carried out, showing that the particle possesses an effective anisotropy and follows the Jeffery orbit as well. This effective anisotropy can be tuned by changing the ratio of no-slip portion to slip potion. The connection of the present work to nematic liquid crystals' constitutive relations is discussed.

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“…The analytic solution for velocity profile under slip flow over the horizontal surfaces is available in [24]:…”

Section: Slit Flowmentioning

confidence: 99%

Boundary integral method for Stokes flow with linear slip flow conditions in curved surfaces

Nieto¹,

Giraldo²,

Power³

2009

Mesh Reduction Methods

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The no slip boundary condition is traditionally used to predict velocity fields in macro scale flows. When the scale of the problem is about the size of the mean free path of particles, it is necessary to consider that the flow slips over the solid surfaces and the boundary condition must be changed to improve the description of the flow behaviour with continuous governing fluid flow equations. Navier's slip boundary condition states that the relative velocity of the fluid respect to the wall is directly proportionally to the local tangential shear stress. The proportionally constant is called the slip length, which represent the hypothetical distance at the wall needed to satisfy the condition of no-slip flow. Some works have misused boundary conditions derived from Navier's work to model slip flow behaviour for example by employing expressions, for diagonal and curved surfaces, that were derived for flat infinite surfaces aligned with coordinate axes. In this work, the creeping flow of a Newtonian fluid under linear slip conditions is simulated for the cases of a Slit and a Couette mixer by means of the Boundary Element Method (BEM). In the evaluation of such flows, different magnitudes of slip length from 0 (no slip) to 1.0 are analysed in an effort to understand the effect of the slip boundary condition on the physical behaviour of the simulation system. Analytic solutions for both geometries under slip flow are used to estimate L2 norm error, which is below 0.25% for Couette flow and 1.25% for Slit flow, validating the approximation applied.

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Newtonian flow with nonlinear Navier boundary condition

Cited by 58 publications

References 32 publications

Analysis of boundary slip in a flow with an oscillating wall

Analysis of boundary slip in a flow with an oscillating wall

Anisotropic particle in viscous shear flow: Navier slip, reciprocal symmetry, and Jeffery orbit

Boundary integral method for Stokes flow with linear slip flow conditions in curved surfaces

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