Role of Surface Forces in Fluid Flow

1. Role of Surface Forces in Fluid Flow P M V Subbarao Professor Mechanical Engineering Department I I T Delhi A Form of Force for Development of Many Real Fluid devices……

2. Surface Forces The Second term on the right-hand side of above equation represents the resultant surface force acting on the control volume. As discussed before, force is an extensive quantity and hence surface force must also proportional to extent of the control volume. surface area.

3. The Surface Forces on A Fluid Flow • The Surface forces are characterized by length scales relevant to the microscopic dynamics of fluid flow. • This would mean that such forces decay extremely rapidly with distance on the macroscopic scale. • Any infinitesimal element in a continuum description is, by definition, must be much larger than all microscopic scales. • The short-ranged forces will decay on distances of the order of a mean free path. • The effect of such forces would be negligible unless two interacting (fluid/soilid) elements are directly in contact. • These forces are expected to act across the contact surface. • In other words, the short-ranged forces manifest as surface forces, in a continuum description.

4. Interaction of Fluid Particle with Perfectly smooth surface (ideal surface) Concept of Surface & Fluid Particle Interaction Real surface Φ Φ Specular reflection Diffuse reflection • The surface forces are defined for a combined solid and fluid system. • The fluid packets close to a solid wall tend to reach mechanical equilibrium with the wall.

5. Kinetic Theory of Solid Fluid Interaction • Motion of Fluid particles near a Solid surface?

6. Kinetic Theory of Gas • The Average Speed of a Gas Molecule

7. Algebraic MVM Macro View of Molecular Interactions • Amalgamation of Micro and Macro Views thru meanfree path in kinetic gas theory.

8. The Universal Law of Nature : Equilibrium • The fluid particles will exchange maximum possible momentum flux with the solid wall. • A small layer of fluid particles close to the wall come to Mechanical, Thermal and Chemical Equilibrium With solid wall. • Fundamentally this fluid layer is in Thermodynamic Equilibrium with the solid wall.

9. Flux Nature of Surface Forces • The surface forces act to transport momentum across the boundaries of an infinitesimal element. • In dilute gases, the momentum transport occurs due to molecules randomly crossing the boundary. • Carry momentum across in the appropriate direction. • Often referred to as the kinetic contribution to the stress. • In liquids, the transport of momentum can occur without physical translation of molecules via short-ranged forces acting between pairs of molecules on either side of the boundary. • Separated by a distance comparable to the range of the inter-molecular potential. • Considered as the potential contribution to the stress. • Clearly, the total effect of short-ranged forces acting on a differential element is decided by its surface area rather than the volume.

10. The System of SurfaceForces in Fluid Flows • Mechanical forces. • Electro-kinetic forces. • Electroosomosic Forces • Electrophoresic Forces

11. Surface Forces are due to Surface of Control Volume The Second term on the right-hand side of above equation represents the resultant surface force acting on the entire control surface. As discussed before, force is an extensive quantity and hence surface force is proportional to surface area. dFs is written as a scalar product of the stress tensor and area vector acting on the surface element ds: Depending on the global surface under study, this gives a force vector which can be decomposed into a normal and a shear forces.

12. Components of Forces n is the normal unit vector that points away from the surface. t is the tangential unit vector. The negative signs of n and t have been chosen to indicate that the pressure pand the shear stress  are exerted by the surroundings on the surface S. Thus, the surface force acting on a differential surface is:

13. Dealing with Surface Forces In above equation, the integration must be carried out over the entire control surface. Fora control surface consisting of inlet, exit, and wall surfaces, the second integral on the left-hand side gives: the first integral on the right-hand side gives

14. Superiority of Volume Integral The total vector surface force can also be defined as surface force per unit volume These are the surface forces, by external agent on the sides of an unit volume element.

15. Differential Form of Momentum Conservation Equations for Fluid Flows

16. The Fluid at Rest : Clue From Pascal’s First Law From the definition of a fluid, the stresses parallel to planes must vanish if the fluid is at rest. Thus the shear stresses are zero, and the normal stresses become equal to the hydrostatic pressure: We must ensure that the stresses in general flow reduce to this special case when the velocity is zero.

17. Localized Action & Reaction in a fluid Flow Stress tensor is the more fundamental quantity, characterizing the fluid response to an imposed deformation.

18. Relationship between Stress Tensor and Deformation Tensor • The surface forces resulting from the stress tensor causes a deformation of fluid particles. • An attempt to find a functional relationship between the stress tensor and the velocity gradient is an essential hypothesis to be invented !!!