1. Lecture 4: Isothermal Flow. Fundamental Equations Continuity equation Navier-Stokes equation Viscous stress tensor Incompressible flow Initial and boundary conditions

2. Continuity equation expresses the mass balance in a control volume, one scalar equation V No gain or loss of mass in a fixed volume V, The change of mass is solely due to inlet/outlet fluxes. This statement leads to the above-written continuity equation.

3. Navier-Stokes equation -- stress tensor divided into pressure (short-range inter- molecular forces dependent the relative position of molecules) and viscous stress tensor (short-range forces dependent on the relative motion of molecules) -- long-range forces – volume (or body) forces, e.g. gravity force Substitution gives expresses conservation of momentum (second Newton’s law applied to a fluid particle). Vector equation = three scalar equations

4. Viscous stress tensor This expression is obtained based on the following reasoning (i) the viscous force is due the relative motion of molecules, defined by the velocity gradients, i.e. the viscous stress should be proportional to velocity gradients (ii) velocity gradients are assumed to be small: no higher derivatives and non-linear terms (iii) we must exclude the case of uniform rotation ( ), when the viscous stress should also vanish (only symmetric combinations of velocity gradients) (iv) summing up (i)-(iii) to write the most general tensor (of rank 2) from the velocity derivatives -- first and second coefficients of viscosity (phenomenological coefficients to be determined experimentally)

5. Incompressible flow Criterion: slow motion (v<<c) where v is the typical fluid velocity and c is the speed of sound (c air =343m/s and c water =1560m/s). The continuity equation is simplified to: The viscous force can be also simplified as follows: it is assumed that coefficient of viscosity is constant Or, the Navier-Stokes equation becomes: Comment: for a single-phase fluid, density is function of temperature and pressure. We consider isothermal motion, when there are no temperature variations. The incompressibility assumption is equivalent to saying that the pressure-related variations in fluid density are negligibly small.

6. Initial and boundary conditions Initial conditions: field of velocity at the initial moment Boundary conditions: a) rigid wall (‘no-slip condition’: result of molecular attraction between a fluid and the surface of a solid body), illustration: http://www.youtube.com/watch?v=cUTkqZeiMow ( ) http://www.youtube.com/watch?v=cUTkqZeiMow b) interface between two immiscible fluids: c)free interface (e.g. air/liquid): : force exerted by the first fluid acted on the second fluid equals to the force exerted by the second fluid acted on the first one -- unit normal vector to the interface

7. Navier-Stokes Claude-Louis Navier (10 February 1785 in Dijon – 21 August 1836 in Paris) was a French engineer and physicist specialized in mechanics. Sir George Stokes, 1st Baronet, (13 August 1819 in Skreen, Ireland – 1February 1903, Cambridge, England) was a mathematician and physicist, who at Cambridge made important contributions to fluid dynamics, optics, and mathematical physics. He was secretary, then president, of the Royal Society.

8. Lecture 5: Circular Poiseuille Flow Governing equations (isothermal, incompressible flow): Configuration:

9. Assumptions Steady flow:, no time-dependence Plane-parallel flow: Consequences: a)Zero non-steady term b)Continuity equation is automatically satisfied for any c) Non-linear term vanishes

10. Resultant equations r- and z- projections: Pressure: differentiation of the second equation ( z -projection) in respect to z gives Use of the boundary conditions at the left and right end of the pipe gives A and B are the constants of integration

11. Velocity profile z -projection of the Navier-Stokes equation: 1.Velocity is limited, 2.At the wall, Finally, parabolic velocity profile: Volumetric flow flux:

12. Jean Louis Marie Poiseuille \pwä-'zəi\ (22 April 1797 - 26 December 1869) was a French physician and physiologist Blood circulation What happens to the blood flow as blood viscosity changes (increased cholesterol level)? What happens to the flow as the capillary radius changes (artery blockage due to cholesterol deposits)?