2. Simple Useful Flow Configurations … P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Exact Solutions of NS Equations

3. The Curl of Stokes Flow Equation Taking the curl of Stokes flow equation: Using the vector identity With

4. Stokes' Solution for an Immersed Sphere Strokes defined a stream function in spherical coordinate system (1851) as: A solenoid condition in spherical coordinate system

5. the momentum equation as a scalar equation for ψ.

6. The boundary conditions

7. The problem appears formidable, but in fact it yields readily to a product solution  (r,  ) = f(r)g(  ). F(r) is governed by the equi-dimensional differential equation: whose solutions are of the form f(r) ∝ r n, It is easy to verify that n = −1, 1, 2,4 so that

8. The boundary conditions

9. Inside the parentheses, the first term corresponds to the uniform flow. and the second term to the doublet; together they represent an inviscid flow past a sphere. The third term is called the Stokeslet, representing the viscous correction. The velocity components follow immediately

10. Properties of Stokes Solution This celebrated solution has several extraordinary properties: 1. The streamlines and velocities are entirely independent of the fluid viscosity. This is true of all creeping flows. 2. The streamlines possess perfect fore-and-aft symmetry. There is no wake. 3. The local velocity is everywhere retarded from its freestream value. 4. The effect of the sphere extends to enormous distances: At r = 10a, the velocities are still about 10 percent below their freestream values.

11. Pressure Field for an Immersed Sphere Stokes Flow Equation Integrating with respect to r from r=a to r  ∞, we get where p, is the uniform freestream pressure. The pressure deviation is proportional to  and antisymmetric. Positive at the front and negative at the rear of the sphere With This creates a pressure drag on the sphere.

12. Stress Field There is a surface shear stress which creates a drag force. The shear-stress distribution in the fluid is given by

13. Stokes Drag on Sphere The total drag is found by integrating pressure and shear around the surface: This is the famous sphere-drag formula of Stokes (1851). Consists of two-thirds viscous force and one-third pressure force. The formula is strictly valid only for Re << 1 but agrees with experiment up to about Re = 1.

14. Other Three-Dimensional Body Shapes In principle, a Stokes flow analysis is possible for any three-dimensional body shape, providing that one has the necessary analytical skill. A number of interesting shapes are discussed in the literature. Of particular interest is the drag of a circular disk: Disk normal to the freestream: Disk parallel to the freestream:

15. 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.)

16. COUETTE FLOWS These flows are named in honor of M. Couette 1890). He performed experiments on the flow between a fixed and moving concentric cylinder. Steady Flow between a Fixed and a Moving Plate. Axially Moving Concentric Cylinders. Flow between Rotating Concentric Cylinders.

17. Steady Flow between a Fixed and a Moving Plate In above Figure two infinite plates are 2h apart, and the upper plate moves at speed U relative to the lower. The pressure is assumed constant. These boundary conditions are independent of x or z ("infinite plates"); It follows that u = u(y) and NS equations reduce to

18. Continuity Equation: NS Equations: where continuity merely verifies our assumption that u = u(y) only. NS Equation can be integrated twice to obtain

19. The boundary conditions are no slip, u(-h) = 0 and u(+h) = U. c 1 = U/2h and c 2 = U/2. Then the velocity distribution is

20. The shear stress at any point in the flow From the viscosity law: For the couette flow the shear stress is constant throughout the fluid. The strain rate-even a nonnewtonian fluid would maintain a linear velocity profile.

21. Friction coefficient The dimensionless shear stress is usually defined in engineering flows as the friction coefficient

22. 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.

23. 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.

24. 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.

25. 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.