On the Validity of the No-Slip Condition Slip or No-Slip?

1. Iran University of Science & Technology
School of Mechanical Engineering
Boundary Layer Theory Course January 2021

2. (of 142Page)
On the Validity of the No-Slip Condition January 2021
Contents
 Introduction
 Navier B.C.
 No-Slip Condition
 Slip Condition
 Conclusions

3. (of 143Page)
On the Validity of the No-Slip Condition January 2021
Introduction
 The “no-slip” boundary condition → zero fluid velocity relative
to the solid at the fluid-solid interface.
 The no-slip boundary condition at a solid–liquid interface is at
the center of our understanding of fluid mechanics.
 This condition is an assumption that cannot be derived from
first principles and could, in theory, be violated.
 Successful in describing many macroscopic flows
 how about microscopic flows?
 Is “no-slip” condition always valid?
 Comparison of Continuum and Molecular Dynamics (MD)
Conclus Slip No-Slip
Navier
B.C. Intro.

4. (of 144Page)
On the Validity of the No-Slip Condition January 2021
Introduction
 Comparison of continuum and Molecular Dynamics (MD)
?0slip
v


n
No-Slip Boundary Condition → A Paradigm
Paradigms, whether right or wrong, are the basis of many judgments.

5. (of 145Page)
On the Validity of the No-Slip Condition January 2021
Navier B.C.
Claude-Louis Navier
(1785-1836)
 History of the No-Slip Condition
 Navier introduced the linear boundary condition (also proposed later
by Maxwell).
 The component of the fluid velocity tangent to the surface,𝒖||, is
proportional to the rate of strain, (or shear rate) at the surface,
 Introduces a slip length 𝝀 and assumes that the amount of slip is
proportional to the shear rate in the fluid at the solid surface.
The strain rate tensor
𝑽 𝒔𝒍𝒊𝒑 = 𝝀 × 𝜸
Conclus Slip No-Slip
Navier
B.C.
Intro.

6. (of 146Page)
On the Validity of the No-Slip Condition January 2021
Navier B.C.
 λ has the unit of a length, and is referred to as the slip length.
 λ → slip length → distance from the fluid-solid interface to where the
linearly extrapolated tangential velocity vanishes.
 For a pure shear flow, λ can be interpreted as the fictitious distance
below the surface where the no-slip boundary condition would be
satisfied.
Interpretation of the (Maxwell–Navier) slip length λ
Conclus Slip No-Slip
Navier
B.C.
Intro.

7. (of 147Page)
On the Validity of the No-Slip Condition January 2021
No-Slip Condition
 Using the Scale Analysis Method
𝑽 𝒔𝒍𝒊𝒑 = 𝝀 × 𝜸 = 𝝀 ×
𝝏𝒖
𝝏𝒚
𝑺𝒄𝒂𝒍𝒆 𝑨𝒏𝒂𝒍𝒚𝒔𝒊𝒔
𝑽 𝒔𝒍𝒊𝒑 ∼ 𝝀 ×
𝑼∞
𝑳 𝒄𝒉𝒂𝒓
×
𝒖∗
𝒚∗ ∼
𝑶 𝟏
𝑶 𝟏
⇛ 𝑽 𝒔𝒍𝒊𝒑∼ 𝝀 ×
𝑼∞
𝑳 𝒄𝒉𝒂𝒓
÷𝑼∞ 𝑽 𝒔𝒍𝒊𝒑
𝑼∞
∼
𝝀
𝑳 𝒄𝒉𝒂𝒓
⟹ 𝑽 𝒔𝒍𝒊𝒑 ∼ 𝑼∞ ×
𝝀
𝑳 𝒄𝒉𝒂𝒓
Conclus Slip No-Slip
Navier
B.C.
Intro.

8. (of 148Page)
On the Validity of the No-Slip Condition January 2021
No-Slip Condition
 Typically, λ ranges from a few angstroms to a few nanometers.
 The Knudsen number is a dimensionless parameter defined as:
 Maxwell theory predicts that the slip length is related to the
mean free path as:
𝑺𝒄𝒂𝒍𝒆 𝑨𝒏𝒂𝒍𝒚𝒔𝒊𝒔 ∶ 𝑽 𝒔𝒍𝒊𝒑 ∼ 𝑼∞
𝝀
𝑳 𝒄𝒉𝒂𝒓
𝑨𝒔𝒔𝒖𝒎𝒆 ∶ 𝝀 ≪ 𝑳 𝒄𝒉𝒂𝒓 ⟹
𝝀
𝑳 𝒄𝒉𝒂𝒓
∼ 𝑶(𝟎)
𝑽 𝒔𝒍𝒊𝒑 ∼ 𝑶(𝟎)
𝑲𝒏 =
𝝀
𝑳 𝒄𝒉𝒂𝒓
=
𝑴𝒆𝒂𝒏 𝑭𝒓𝒆𝒆 𝑷𝒂𝒕𝒉
𝑪𝒉𝒂𝒓. 𝑳𝒆𝒏𝒈𝒕𝒉
𝝀∗ =
𝝀
𝑳 𝒄𝒉𝒂𝒓
= 𝜶 𝑲𝒏 ⇒ 𝝀 = 𝜶 𝝀
𝒘𝒉𝒆𝒓𝒆 𝜶 ≅ 𝟏. 𝟏𝟓 𝐢𝐬 𝐭𝐡𝐞 𝐒𝐥𝐢𝐩 𝐂𝐨𝐞𝐟𝐟𝐢𝐜𝐢𝐞𝐧𝐭
Conclus Slip No-Slip
Navier
B.C.
Intro.

9. (of 149Page)
On the Validity of the No-Slip Condition January 2021
Slip Condition
 No-slip condition is believed to be valid as far as the
characteristic length of the flow is much greater than the mean
length of the path of the fluid molecular between collisions.
 If the length scale of the fluidic system is in the same range as
the mean free path, i.e., Kn = 1, the fluid cannot be treated as a
continuum.
 The next question → Traditional Situations,Where Slip Occurs.
 The phenomenon of slip has already been encountered in three
different contexts.
𝑳 𝒄𝒉𝒂𝒓 ≫ 𝝀
𝝀 =
𝝀
𝜶
𝑳 𝒄𝒉𝒂𝒓 ≫
𝝀
𝜶
∼ 𝝀 ⇒ 𝑳 𝒄𝒉𝒂𝒓 ≫ 𝝀
𝒆. 𝒈. → 𝜹
Conclus Slip No-Slip
Navier
B.C.
Intro.

10. (of 1410Page)
On the Validity of the No-Slip Condition January 2021
Slip Condition
 Gas Flow → Gas flow, in devices with dimensions that are on the
order of the mean free path of the gas molecules shows significant
slip.
 For e.x, air under standard conditions of temperature and pressure,
𝝀 = 𝟏𝟎𝟎𝒏𝒎 and, in general, 𝝀 depends strongly on pressure and
temperature.
 Non-Newtonian Fluids → The flows of Non-Newtonian fluids
such as polymer solutions show significant apparent slip in a variety
of situations, some of which can lead to slip-induced instabilities.
 Contact Line Motion → The no-slip boundary condition is not
applicable to the moving contact line (MCL) where the fluid-fluid
interface intersects the solid wall.
𝑬𝒔𝒕𝒊𝒎𝒂𝒕𝒆 𝒐𝒇 𝑴𝒆𝒂𝒏 𝑭𝒓𝒆𝒆 𝑷𝒂𝒕𝒉 → 𝝀 = 𝟏/ 𝟐𝝅𝝈 𝟐
𝝆 (for ideal gas)
Conclus Slip No-Slip
Navier
B.C.
Intro.

11. (of 1411Page)
On the Validity of the No-Slip Condition January 2021
Slip Condition
 The static and moving contact lines:
 The distance over which the fluid velocity changes from U to
zero tends to vanish as the contact line is approached.
 In particular, this stress divergence is non-integrable.
 Cause to Singularities
𝑽𝒊𝒔𝒄𝒐𝒖𝒔 𝑺𝒕𝒓𝒆𝒔𝒔 𝒗𝒂𝒓𝒊𝒆𝒔 𝒂𝒔 ∶
𝝁𝑼
𝑿
, ⟹ lim
𝑿→𝟎
𝝁𝑼
𝑿
= ∞
Conclus Slip No-Slip
Navier
B.C.
Intro.

12. (of 1412Page)
On the Validity of the No-Slip Condition January 2021
Conclusions
 The no-slip boundary condition at a solid–liquid interface is at
the center of our understanding of fluid mechanics.
 However, this condition is an assumption that cannot be derived
from first principles and could, in theory, be violated.
 Navier B.C.→ no-slip boundary condition in macroscopic flows
 Traditional Situations,Where Slip Occurs →The phenomenon
of slip has already been encountered in three different contexts
 Gas Flow, Non-Newtonian Fluids and Contact Line Motion
 Moving contact lines → Cause to Singularities
𝒘𝒉𝒆𝒓𝒆 𝑳 𝒄𝒉𝒂𝒓 ≫ 𝝀 ⇒ 𝝀 ≪ 𝑳 𝒄𝒉𝒂𝒓
𝒇𝒐𝒓 𝒆.𝒙.
𝝀 ≪ 𝜹
Conclus Slip No-Slip
Navier
B.C.
Intro.

13. Iran University of Science & Technology
School of Mechanical Engineering
Special Thanks
For Your Listening
In contrast with the usual picture where the velocity of a liquid or gas flow on
a solid wall is zero, recent experiments have shown that simple liquids and
gases may significantly slip on solid surfaces and, consequently, the no-slip
condition should be replaced by a more general relation
Daryooshborzuei@gmail.com
www.linkedin.com/in/da-borzuei

14. Iran University of Science & Technology
School of Mechanical Engineering
[1] J. Ghorbanian and A. Beskok, "Scale effects in nano-channel liquid flows," Microfluidics and
Nanofluidics, vol. 20, p. 121, 2016.
[2] E. Lauga, M. Brenner, and H. Stone, "Microfluidics: The No-Slip Boundary Condition," in Springer
Handbook of Experimental Fluid Mechanics, C. Tropea, A. L. Yarin, and J. F. Foss, Eds., ed Berlin, Heidelberg:
Springer Berlin Heidelberg, 2007, pp. 1219-1240.
[3] Y. Li, P. Tian, and D. Li, "Effective slip length at solid–liquid interface of roughness-induced surfaces with
omniphobicity," in Ninth International Symposium on Precision Mechanical Measurements, 2019, p. 113431O.
[4] G. Nagayama, T. Matsumoto, K. Fukushima, and T. Tsuruta, "Scale effect of slip boundary condition at
solid–liquid interface," Scientific Reports, vol. 7, p. 43125, 2017.
[5] A. Noghrehabadi, R. Pourrajab, and M. Ghalambaz, "Effect of partial slip boundary condition on the
flow and heat transfer of nanofluids past stretching sheet prescribed constant wall temperature," International
Journal of Thermal Sciences, vol. 54, pp. 253-261, 2012.
[6] T. Qian, X.-P. Wang, and P. Sheng, "Molecular hydrodynamics of the moving contact line in two-phase
immiscible flows," arXiv preprint cond-mat/0510403, 2005.
[7] J. Zhang and D. Y. Kwok, "Apparent slip over a solid-liquid interface with a no-slip boundary condition,"
Physical Review E, vol. 70, p. 056701, 2004.
[8] Y. Zhu and S. Granick, "Limits of the hydrodynamic no-slip boundary condition," Physical review letters,
vol. 88, p. 106102, 2002.