Fundamental (First) Principles of Fluid Mechanics Dept. of Experimental Orthopaedics and Biomechanics Bioengineering Reza Abedian (M.Sc.)

Fundamental (First) Principles of Fluid Mechanics we will repeatedly apply five fundamental principles : The Control Volume (CV) concept (analogous to a free body diagram) The Continuity Equation (Mass conservation) Newton’s 2nd Law (Bernoulli’s Equation) First Law of Thermodynamics (Work - Energy Equation) Dimensional homogeneity (units)

1) The Control Volume (CV) concept similar to a free-body diagram in solid mechanics equations for flow, forces, and energy will be written for a fixed control volume of fluid (e.g., a section of pipe) a properly drawn and labeled CV is an immense help in problem-solving Qin p1A1 p2A2 Qout Ein Eout t t

2) Conservation of Mass, or the Continuity Equation changes in mass inside a CV must be balanced by the net flow of mass across its boundaries: for most of our applications the flow will be at steady-state: if the fluid is a liquid, density ~constant: (1) or (2) (3)

(Net Force = rate of change of Momentum) 3) Newton’s Second Law (provided that v << 3 x 108 m/s) modified equation used in fluids applications: (Net Force = rate of change of Momentum)

Bernoulli Equation A statement of the conservation of energy in a form useful for solving problems involving fluids. For a non-viscous incompressible fluid in steady flow the sum of pressure, potential and kinetic energies per unit volume is constant at any point

Bernoulli Equation Continued... A special form of the Euler’s equation derived along a fluid flow streamline is often called the Bernoulli Equation Equation (3) is often referred to the head because all elements has the unit of length. Both elements in the equation (4) have the unit of pressure Dynamic pressure of the fluid flow (5). Since energy is conserved along the streamline, (4) can be expressed as (6).

4) First Law of Thermodynamics net work done on system + net heat input to system = Energy of system (note: Energy = potential + kinetic + internal (heat)) this principle will be used when we consider the energy and “head losses” in fluid flows due to friction

Dimensionless quantity Sarah says, "Out of every 10 apples I gather, 1 is rotten.". The rotten-to-gathered ratio is (1 rotten apple) / (10 gathered apples) = 0.1 = 10%, which is a dimensionless quantity. the unit of "radians" the length that is compared is the length of the radius of the circle. When using the unit of "degrees" the length that is compared is 1/360 of the circumference of the circle. Dimensionless quantities are widely used in the fields of mathematics, physics, engineering, and economics. Dimensionless quantities can also carry dimensionless units like % (=0.01), ppt (=10-3), ppm (=10-6), ppb (=10-9).

Reynolds Number The Reynolds Number is a non dimensional parameter defined by the ratio of dynamic pressure (ρ u2) and shearing stress (μ u / L) Re = (ρ u2) / (μ u / L) = ρ u L / μ = u L / ν            (1) For a pipe or duct the characteristic length is the hydraulic diameter.  Re = ρ u dh / μ = u dh / ν          (2) laminar when Re < 2300 transient when 2300 < Re < 4000 turbulent when Re > 4000

5) Dimensional Homogeneity this simply means that for an equation to be correct, both sides must have the same dimensions and units: M, L, t, T we will show that this is a powerful principle – “Dimensional Analysis” Practical advice: 1. Don’t switch units (e.g., English to SI) 2. Carry units through calculations, don’t crunch the numbers and then add what you think the units should be afterwards, for example, a 55-gal drum of water weighs: 3. Check your units at the end to see that they make sense!