Continuum Hypothesis In this course, the assumption is made that the fluid behaves as a continuum, i.e., the number of molecules within the smallest region of interest (a point) are sufficient that all fluid properties are point functions (single valued at a point). For example: Consider definition of density ρ of a fluid δV* = limiting volume below which molecular variations may be important and above which macroscopic variations may be important.

Static Fluid For a static fluid Shear Stress should be zero. For A generalized Three dimensional fluid Element, Many forms of shear stress is possible.

One dimensional Fluid Element +X ++ +Y u=0 u=U

Fluid Statics Pressure : For a static fluid, the only stress is the normal stress since by definition a fluid subjected to a shear stress must deform and undergo motion. What is the significance of Diagonal Elements? Vectorial significance : Normal stresses. Physical Significance : ? For the general case, the stress on a fluid element or at a point is a tensor X Y Z  xy  xz  yz  yx  zx  zy

Stress Tensor X Y Z  yy  zz  yz  yx  zx  zy  xx  xy  xz

First Law of Pascal Proof ?

Simple Non-trivial Shape of A Fluid Element

Pressure Variation with Elevation For a static fluid, pressure varies only with elevation within the fluid. This can be shown by consideration of equilibrium of forces on a fluid element Basic Differential Equation: Newton's law (momentum principle) applied to a static fluid ΣF = ma = 0 for a static fluid i.e., ΣFx = ΣFy = ΣFz = 0 1st order Taylor series estimate for pressure variation over dz

For a static fluid, the pressure only varies with elevation z and is constant in horizontal xy planes. The basic equation for pressure variation with elevation can be integrated depending on whether ρ = constant i.e., the fluid is incompressible (liquid or low- speed gas) or ρ = ρ(z), or compressible (high-speed gas) since g is constant.

Pressure Variation for a Uniform-Density Fluid

Reading Material Fluid Mechanics – Frank M White, McGraw Hill International Editions. Introduction to Fluid Mechanics – Fox & McDOnald, John Wiley & Sons, Inc. Fluid Mechanics – V L Streeter, E Benjamin Wylie & K W Bedfore, WCB McGraw Hill. Fluid Mechanics – P K Kundu & I M Cohen, Elsevier Inc.

Pressure Variation for Compressible Fluids Basic equation for pressure variation with elevation Pressure variation equation can be integrated for γ(p,z) known. For example, here we solve for the pressure in the atmosphere assuming ρ(p,T) given from ideal gas law, T(z) known, and g ≠ g(z).

Draft Required to Establish Air Flow Air in Flue gas out

Natural Draft H chimney T atm T gas A B p A = p ref +  p Z ref,,p ref

H chimney T atm T gas A B p A = p ref +  p Z ref,,p ref

Pressure variations in Troposphere: Linear increase towards earth surface T ref & p ref are known at Z ref.  Adiabatic Lapse rate : 6.5 K/km

Reference condition: At Z ref : T=T ref & p = p ref

Pressure at A: Pressure variation inside chimney differs from atmospheric pressure. The variation of chimney pressure depends on temperature variation along Chimney. Temperature variation along chimney depends on rate of cooling of hot gas Due to natural convection. Using principles of Heat transfer, one can calculate, T gas (Z). If this is also linear: T = T ref,gas +   (Z ref -Z). Lapse rate of gas,  gas is obtained from heat transfer analysis.

Natural Draft Natural Draft across the furnace,  p nat = p A – p B The difference in pressure will drive the exhaust. Natural draft establishes the furnace breathing by –Continuous exhalation of flue gas –Continuous inhalation of fresh air. The amount of flow is limited by the strength of the draft.

Pressure Measurement Another Application of Fluid Statics

Pressure Measurement Pressure is an important variable in fluid mechanics and many instruments have been devised for its measurement. Many devices are based on hydrostatics such as barometers and manometers, i.e., determine pressure through measurement of a column (or columns) of a liquid using the pressure variation with elevation equation for an incompressible fluid.

PRESSURE Force exerted on a unit area : Measured in kPa Atmospheric pressure at sea level is 1 atm, 76.0 mm Hg, 101 kPa In outer space the pressure is essentially zero. The pressure in a vacuum is called absolute zero. All pressures referenced with respect to this zero pressure are termed absolute pressures.

Many pressure-measuring devices measure not absolute pressure but only difference in pressure. This type of pressure reading is called gage pressure. Whenever atmospheric pressure is used as a reference, the possibility exists that the pressure thus measured can be either positive or negative. Negative gage pressure are also termed as vacuum pressures.

Manometers U Tube Inverted U Tube Enlarged Leg Two Fluid Inclined Tube

U-tube or differential manometer Right Limb fluid statics : Left Limb fluid statics : Point 3 and 2 are at the same elevation and same fluid

Gauge Pressure:

Absolute, Gauge & Vacuum Pressures System Pressure Atmospheric Pressure Gauge Pressure Absolute Pressure Absolute zero pressure

Absolute, Gauge & Vacuum Pressures System Pressure Atmospheric Pressure Vacuum Pressure Absolute Pressure Absolute zero pressure

Stress Tensor for A Static Fluid X Y Z  yy  zz  xx

An important Property of A Fluid under Motion

Shear stress (  : Tangential force on per unit area of contact between solid & fluid

Elasticity (Compressibility) Increasing/decreasing pressure corresponds to contraction/expansion of a fluid. The amount of deformation is called elasticity.

Surface Tension Two non-mixing fluids (e.g., a liquid and a gas) will form an interface. The molecules below the interface act on each other with forces equal in all directions, whereas the molecules near the surface act on each other with increased forces due to the absence of neighbors. That is, the interface acts like a stretched membrane, e.g.

Vapour Pressure When the pressure of a liquid falls below the vapor pressure it evaporates, i.e., changes to a gas. If the pressure drop is due to temperature effects alone, the process is called boiling. If the pressure drop is due to fluid velocity, the process is called cavitation. Cavitation is common in regions of high velocity, i.e., low p such as on turbine blades and marine propellers.