The Bernoulli’s Equation

1. The Bernoulli’s Equation Pressure/density + (velocity)2 /2 + g (height) = constant (2) The three terms are known, respectively, as ‘pressure head’, ‘velocity head’, ‘gravity head’ (following practices in hydraulics). (3) Theoretical significance: ‘first integral’ of the equations of motion.

2. The Bernoulli’s Equation Conservation of energy - steady, inviscid, incompressible flow (Energy per unit mass) Gravity head Pressure head Velocity head

3. A simple harmonic oscillator Consider a particle of mass m hanging vertically from a spring of constant k. The equation of motion is m(d2x/dt2) = – kx . (2) Multiply of dx/dt, integrate and recognize that d( [dx/dt]2)/dt = 2(dx/dt)(d2x/dt2) will lead to (3) m(dx/dt)2/2 + k x2/2 = constant (4) or kinetic energy + potential energy = constant. (5) Similar, slightly involved, calculations for fluids.

4. The Momentum Theorems • Linear and angular momentum theorems • Control surfaces – conceptual (imaginary) surfaces fixed in space

5. Review… Advanced Level physics (?) • mass*velocity = momentum • Vector quantity! (v, u = new, old velocities) • Physical meaning: net force acting on the fluid allows this mass of fluid to change its momentum.

6. Ideas behind the momentum theorem • Newton’s second law F = ma applies only to the SAME (!!!) system of particle (Lagrangian approach), but in an Eulerian frame / control surface fixed in space, different particles flow into the control surface at different time instants and will leave the control surface some time later.

7. Ideas (cont’d) • Consider a control volume of moving fluid at time t. • Apply Newton’s law to this portion of fluid between t and t + dt. • Some of the fluids have left the control volume ? – Yes, but express it as the flux of momentum out of the control volume by the Reynolds transport theorem.

8. Simple concepts behind the Reynolds Transport Theorem • Applied force = rate of change of momentum for a FIXED (or the SAME) mass of fluid = (Momentum at t + dt) – (Momentum at t) = Rate of change of momentum within the ‘bulk of fluid’ + Momentum from some portions of fluid reaching ‘new positions’ – Momentum from some portions of fluid having left ‘old positions’.

9. Linear Momentum in fluid flow Sum of external forces acting on the control volume Time rate of change of linear momentum of the system = Time rate of change of linear momentum inside control surface = + Net rate of momentum flow (or momentum flux) through the control surface

10. Control volume - example • Selection of control volume is arbitrary • Criteria: easy to locate and compute external forces Influx of momentum Outflux of momentum External forces

12. Cardiovascular System • A need to transport substances over long distance, an aim which cannot be achieved otherwise by diffusion alone (TOO SLOW!!) • Metabolic energy (O2, glucose, etc.) • Waste (CO2, urea, etc.) • H2O/salts/acids/bases for adjustment of osmosis and pH. • Heat (enzymes have a preferred operating temperature) • Information (hormones) • Immunity (cellular – e.g. white blood cells, antibodies)

13. Basic Anatomy – the Heart LA, LV: left atrium and ventricle RA, RV: right atrium and ventricle PV, PA: pulmonary vein(s) and artery SVC, IVC: superior vena cava, inferior vena cava

14. Cardiac cycle movie • http://library.med.utah.edu/kw/pharm/hyper_heart1.html

15. The pulmonary circulation • Blood flows from the right atrium to the right ventricle, which in turn pumps blood to the lung through the pulmonary artery. This step will expose/enrich blood to/with oxygen. Blood returns to the left atrium through the pulmonary vein. The next phase begins when blood flows from the left atrium to the left ventricle. • (Note : the pulmonary artery (vein) is the only artery (vein) which carries deoxygenated (oxygenated) blood.

16. Forces driving blood flow FOR: (a) Pumping action of the heart; (b) Pressure gradient; (c) Gravity (??, i.e. for roughly half of the arteries/veins only, imagine blood flow in the upper/lower half of the body). AGAINST: (a) Viscosity; (b) Turbulence; and (c) Gravity (??)

17. Flow Physics • Laminar flows – ‘slow and orderly’. • Turbulent flows – ‘fast and chaotic’. • Change from laminar to turbulent flows – ‘transition’ or transitional flows. • One perspective – Changes are brought about by the loss of stability of laminar flows.

19. Flow Physics (cont’d) • Stability – Impose a wavy disturbance; • Disturbance grows – unstable; • Disturbance decays – stable. • But what are the allowed states (laminar flows)? • (Circular) Pipe flows – or Pipe Poiseuille flows • Important practically and elegant theoretically.

20. Flow Physics (cont’d) The issue of stability for these simple flows turns out to be surprisingly difficult theoretically (personal experience) and experimentally. Experimentally, in carefully controlled experiments, pipe flow can remain laminar for Reynolds number = 5,000 or more, but in practice, transition will take place for Re somewhere between 1,000 to 3,000.

21. Pipe Poiseuille flows • Velocity profile parabolic (quadratic in terms of the radius). • One way is to establish the profile from the exact differential equations of motion (Year 3). • Year 2 – First principle: Consider a ‘cylindrical shell’. Assume the fluid flow is steady and ‘fully developed’ – i.e. parabolic profile reached.

22. Pipe flows (cont’d) In solid mechanics, a particle / mass will move only if an external force is applied. In fluid mechanics, a similar reasoning applies, but now we talk about an applied external pressure differential (or pressure gradient) as the basic force driving the flow. Friction, or viscosity, opposes the fluid and dissipates the energy. Hence an external pressure must be sustained.

23. Pipe flows (cont’d) Imagine water released from a large reservoir is allowed to enter a long, circular pipe. At the entrance of the pipe, the velocity profile will be initially fairly ‘flat’. Further downstream, friction at the wall will reduce the fluid velocity there to zero. To maintain the same mass flow, the central portion of the pipe will see an increase in velocity.

24. Pipe flows (cont’d) Eventually, the force due to pressure will be balanced by the shear stress from fluid friction (viscosity), and a steady state (dynamic equilibrium) will be reached. The flow is then said to be ‘fully developed’. The velocity profile will be bulging at the center but attains a value of zero at the wall. Challenge now is to show that this profile is parabolic.

25. Pipe flows (cont’d) Before we present the details of the analytical derivations, let us look at the ‘shell analogies’ of laminar versus turbulent flows. It will demonstrate that the shear stress at the wall will be larger for turbulent flows.

26. Sliding cylinders analogy For laminar flows, we shall establish shortly that the velocity profile will be parabolic. This year we shall use a macroscopic approach, and next year, in an elective, more advanced, course in fluid mechanics, from a set of partial differential equations. For turbulent flows, the profile is ‘flatter’, having less curvature, in most of the flow. To maintain the same mass flow, the maximum velocity for turbulent flows than to be slightly smaller than that of laminar flows.

28. Sliding cylinders analogy In the presence of viscosity (friction), the velocity at the wall must be zero (the ‘no slip’ condition). Hence the spatial gradient, or rate of change of the velocity profile, must be greater for turbulent flows, i.e. laminar flows are usually associated with smaller drag force.

29. Modeling of blood flow In large blood vessels like arteries, it is plausible / reasonable (but obviously not exact) to model blood flow as the transport of a fluid slightly denser than water, with a certain prescribed viscosity, and to ignore the presence red blood cells, white blood cells and other suspended particles. This approximation will fail for ‘micro-circulation’ – flow in capillaries, where a different dynamics must be employed.

30. Velocity Profile for Fully Developed Flow As the flow is steady, i.e. has no acceleration, the net force on the element is zero. The force due to pressure must balance the net force due to the difference in shear stress on the inner and outer surfaces of the cylindrical shell. Net force due to pressure: (∆p) 2 π r dr

31. Pipe flows (cont’d) Net force due to shear stress [∂(τ 2 π r dx)/∂r] dr Now τ = shear stress (by definition) = (coefficient of viscosity)X(velocity gradient); Hence a differential equation in the velocity: (1/r) d[ μ r du/dr ]/dr = dp/dx

32. Pipe flows (cont’d) Solve (or just integrate twice) → parabolic velocity profile. Details: dp/dx must be < 0 for the flow to goes from left to right (i.e. must have high pressure on the left). Hence write dp/dx as = - ∆p/L where ∆p is the drop in pressure and L is the length of the pipe.

33. Pipe flows (cont’d) If the fluid occupies the central portion of the pipe as well, then the term involving log r must be suppressed to avoid singularities. The no slip conditions must also be applied at r = a (radius of the pipe). Combining these two conditions: u = ∆p(a2 – r2)/(4 μL)

34. Pipe flows (cont’d) Flow Rate: Flow rate in the shell = (velocity of the fluid)X(cross sectional area of the shell) Total volume flow rate = integral of the flow rate of the shell over the whole circle, r: 0 → a

35. Poiseuille Flow • Assumptions • Rigid, straight pipe • Incompressible, Newtonian fluid • Fully developed, steady flow • Laminar flow Tube radius = a Flow rate = [volume/time] Analagous to V = R*I in electricity Pressure gradient = pressure loss per unit length

36. Pipe flows (cont’d) Flow rate larger if a larger pipe is used, intuitively correct, but Q is actually proportional to (radius)4, i.e. if the radius reduced by 50%, the flow rate is only 1/16 of the original. Flow rate larger if a larger pressure gradient is applied – OK. Flow rate smaller if the viscosity is increased – OK, as the effect of friction is larger, or the flow needs to overcome a larger friction.

37. Poiseuille Flow Which is analogous to… (In electricity) ‘Driving force’ = resistance * flow

38. Laminar vs turbulent Friction comes in two parts: • Molecular transport – smaller in magnitude, (b) Turbulent eddies – larger part. (c) Velocity profile ‘flatter’ around the center of the pipe and hence stress greater at the wall.