Reynoldification of N-S Equations

1. Reynoldification of N-S Equations P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Creation of High degree of Contrast among Viscous Flows ….

2. Honor of Osborne Reynolds : Engineering of Mathematics • Consider the Navier-Stokes equations with constant density it their dimensional form: Define dimensionless variables as: • Here U, L are assumed to be a velocity and length characteristic of the problem being studied. • In the case of flow past a body, L might be a body diameter and U the flow speed at infinity.

3. In low Mach number region with constant viscosity: Non-dimensionalization of NS Equation

4. Non-dimensional form of NS Equation

5. Importance of Reynolds Number • The Reynolds number Re is the only dimensionless parameter which is always important in simplifying the equations of fluid motion for various applications. • Reynoldsification helps in simplifying NS equations by ingnoring less important terms. • This simplification helps in obtaining analytical solutions to engineering parameters like friction factor. • Better understanding of the complexity of real fluid flow is achieved by fitting the situation into Reynolds frame work.

6. Academic Classification of Fluid Flows using Reynolds Number

7. Reynolds Classification of Real World Flows

8. All Engineering Applications Reynolds Classification of Flying Systems

9. Micro Classification of Specific Fluid flowFire Dynamics & Fuel Jet Quality

10. Comprehensive Nature of Research

11. Wiser Innovations

12. Very Low Reynolds Number Flows • These are flows with Reynolds number lower than unity, Re<< 1. • Since Re = UL/, the smallness of Re can be achieved by considering • extremely small length scales, or • by dealing with a highly viscous liquid, or • by treating flows of very small velocity, so-called creeping flows.

13. The Creeping Flows • The choice Re << 1 is an very interesting and important assumption. • It is relevant to many practical problems, especially in a world where fluid devices are shrinking in size. • A particularly interesting application is to the swimming of micro-organisms. • This assumption, unveils a special dynamical regime which is usually referred to as Stokes flow. • To honor George Stokes, who initiated investigations into this class of fluid problems. • We shall also refer to this general area of fluid dynamics as the Stokesian realm. • This is of extreme contrast to the theories of ideal inviscid flow, which might be termed the Eulerian realm.

14. The Principle Characteristic Creeping Flows • Re is indicative of the ratio of inertial to viscous forces. • The assumption of small Re means that viscous forces dominate the dynamics. • That suggests that to drop entirely the term Dv/Dt from the Navier-Stokes equations. • This renders the linear system. • The linearity of the problem will be a major simplification.

15. Simplification of NS Equations for Stokes Realm It is tempting to say that the smallness of Re means that the left-hand side of the first equation can be neglected. This leads to the reduced (linear) system

16. Stoksification of Creeping flow Equations • Redefine the dimensionless pressure as pL/(2μU) instead of p/(U2).

17. Stokes Flow • The basic assumption of creeping flow was developed by Stokes (1851) in a seminal paper. • This states that density (inertia) terms are negligible in the momentum equation. • Under non-gravitational field. In dimensional form with

18. The Curl of Stokes Flow Equation With Taking the curl of Stokes flow equation: Using the vector identity Any viscous flow, whose curl of curl of vorticity is zero is called as stokes flow or creeping flow

19. Some Applications • How does sedimentation vary with the size of the sediment particles? • What electric field is required to move a charged particle in electrophoresis? • What g force is required to centrifuge cells in a given amount of time. • What is the effect of gravity on the movement of a monocyte in blood? • How rapidly do enzyme-coated beads move in a bioreactor? • The flow geometry of all above mentioned applications is flow past a sphere. • Define the term vorticity in spherical coordinate system.