An Unified Analysis of Macro & Micro Flow Systems… P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Wall Bound Flows

Selected Analytical Solutions to NS Equations Couette (wall-driven) steady flows Poiseuille (pressure-driven) steady duct flows Unsteady duct flows Unsteady flows with moving boundaries Duct flows with suction and injection Wind-driven (Ekman) flows Similarity solutions (rotating disk, stagnation flow, etc.)

The No-Slip Boundary Condition in Viscous Flows The Riddle of Fluid Sticking to the Wall in Flow. Consider an isolated fluid particle. When the particle hits a wall of a solid body, its velocity abruptly changes. Φi Φi ui ui vi vi Φr Φr ur ur vr vr This abrupt change in momentum of the ball is achieved by an equal and opposite change in that of the wall or the body.

Video of the Event At the point of collision we can identify normal and tangential directions nˆ and tˆto the wall. The time of impact t 0 is very brief. It is a good assumption to conclude that the normal velocity v n will be reversed with a reduction in magnitude because of loss of mechanical energy. If we assume the time of impact to be zero, the normal velocity component v n is seen to be discontinuous and also with a change in sign.

Time variation of normal and tangential velocity components of the impinging particle. Whether it is discontinuous or not, the fact that it has to change sign is obvious, since the ball cannot continue penetrating the solid wall.

The Tangential Component of Velocity The case of the tangential component v t is far more complex and more interesting. First of all, the particle will continue to move in the same direction and hence there is no change in sign. If the wall and the ball are perfectly smooth (i.e. frictionless). v t will not change at all. In case of rough surfaces v t will decrease a little. It is important to note that v t is nowhere zero. Even though the ball sticks to the wall for a brief period t 0, at no time its tangential velocity is zero! The ball can also roll on the wall.

Incident of A Single Fluid Particle on a Smooth Wall When the particle hits a smooth wall of a solid body, its velocity abruptly changes. This sudden change due to perfectly smooth surface (ideal surface) was defined by Max Well as Specular Reflection in The Slip is unbounded Φi Φi ui ui vi vi Φr Φr ur ur vr vr

Incident of A Single Fluid Particle on a Rough Wall Φi Φi ui ui vi vi Φr Φr ur ur vr vr When the particle hits a rough wall of a solid body, its velocity also changes. This sudden change due to a rough surface (real surface) was defined by Max Well as Diffusive Reflection in The Slip is finite.

The condition of fluid flow at the Wall This is fundamentally different from the case of an isolated particle since a flow field has to be considered now. The difference is that a fluid element in contact with a wall also interacts with the neighbouring fluid. The problem of velocity boundary condition demands the recognition of this difference. During the whole of 19th century extensive work was required to resolve the issue. The idea is that the normal component of velocity at the solid wall should be zero to satisfy the no penetration condition. What fraction of fluid particles will partially/totally lose their tangential velocity in a fluid flow?

Thermodynamic equilibrium Thermodynamic equilibrium implies that the macroscopic quantities need have sufficient time to adjust to their changing surroundings. In motion, exact thermodynamic equilibrium is impossible as each fluid particle is continuously having volume, momentum or energy added or removed. Fluid flow heat transfer can at the most reach quasi- equilibrium. The second law of thermodynamics imposes a tendency to revert to equilibrium state. This also defines whether or not the flow quantities are adjusting fast enough.

Knudsen Layer In the region of a fluid flow very close to a solid surface, the occurrence of quasi thermodynamic equlibrium is also doubtful. This is because there are insufficient molecular-molecular and molecular-surface collisions over this very small scale. This fails to justify the occurrence of quasi thermodynamic- equilibrium. Two defining characteristics of this near-surface region of a gas flow are the following: First, there is a finite velocity of the gas at the surface ( velocity slip). Second, there exists a non-Newtonian stress/strain-rate relationship that extends a few molecular dimensions into the gas. This region is known as the Knudsen layer or kinetic boundary layer.

λ u s u w Wall Surface at a distance of one mean free path/lattice spacing. Knudsen Layer

Molecular Flow Dimensions Mean Free Path is identified as the smallest dimensions of gaseous Flow. MFP is the distance travelled by gaseous molecules between collisions. Lattice Spacing is identified as the smallest dimensions of liquid Flow. Mean free path : d : diameter of the molecule n : molar density of the fluid, number molecules/m 3 Lattice Dimension: V is the molar volume N A : Avogadro’s number.

Velocity Extrapolation Theory Knudsen defined a non-dimensional distance as the ratio of mean free path of the gas to the characteristic dimension of the system. This is called “Knudsen number”

Boundary Conditions Maxwell was the first to propose the boundary model that has been widely used in various modified forms. Maxwell’s model is the most convenient and correct formulation. Maxwell’s model assumes that the boundary surface is impenetrable. The boundary model is constructed on the assumption that some fraction (1-  r ) of the incident fluid molecules are reflected form the surface specularly. The remaining fraction  r are reflected diffusely with a Maxwell distribution.  r  gives the fraction of the tangential momentum of the incident molecules transmitted to the surface by all molecules. This parameter is called the tangential momentum accommodation coefficient.

Slip Boundary conditions Maxwell proposed the first order slip boundary condition for a dilute monoatomic gas given by: Where : -- Tangential momentum accommodation coefficient ( TMAC ) Velocity of gas adjacent to the wall Velocity of wall -- Velocity gradient normal to the surface-- Mean free path

Non-dimensional form ( I order slip boundary condition) Using the Taylor series expansion of u about the wall, Maxwell proposed second order terms slip boundary condition given by: Second order slip boundary condition Maxwell second order slip condition Modified Boundary Conditions

Regimes of Engineering Fluid Flows Conventional engineering flows: Kn < Micro Fluidic Devices : Kn < 0.1 Ultra Micro Fluidic Devices : Kn <1.0

 Based on the Knudsen number magnitude, flow regimes can be classified as follows : Continuum Regime:Kn 10  In continuum regime no-slip conditions are valid.  In slip flow regime first order slip boundary conditions are applicable.  In transition regime (according to the literature present) higher order slip boundary conditions may be valid.  Transition regime with high Knudsen number and free molecular regime need molecular dynamics. Flow Regimes