Engr. Fazal-E-Jalal FLUID MECHANICS-I INTRODUCTION (Contd…) Lecture # 02 CONTENTS OF TODAY’S LECTURE: Compressibility Viscosity Newton’s equation of viscosity Units of Viscosity FLUID MECHANICS-I CE-224 Engr. Fazal-E-Jalal Prepared by: Engr. Fazal-E-Jalal Fluid Mechanics-I

Compressible and Incompressible Fluids Fluid mechanics deals with both compressible and in compressible fluids, that is with liquids and gases, of either constant or variable density. No such thing in reality as “Incompressible fluid”, the term is used when the change in density with pressure is NEGLIGIBLE. AND THIS IS THE CASE WITH “LIQUIDS”. We may also consider GASES as incompressible when P variation is small compared with absolute pressure. Prepared by: Engr. Fazal-E-Jalal Fluid Mechanics-I

Compressible and Incompressible Fluids Evidence of Elasticity of fluids is that sound waves (which really are pressure waves) travel through liquids. Ordinarily liquids are considered to be incompressible fluids. In WATER HAMMER problems, we must consider the compressibility of fluids. Prepared by: Engr. Fazal-E-Jalal Fluid Mechanics-I

Compressible and Incompressible Fluids Flow of air in a ventilating system: Gas or steam flowing at high velocity through long pipeline: Gas is treated as Incompressible Because, P variation is so small that the change in density is of no importance. P variation is great that change in density cannot be ignored. Prepared by: Engr. Fazal-E-Jalal Fluid Mechanics-I

Compressible and Incompressible Fluids High up in air!!! An airplane flying below 250 mph , density of air may be considered as constant. But an object moving at 760 mph (approaching velocity of sound), then P & Density adjacent to body is different from distant air. TREAT AIR AS COMPRESSIBLE FLUID.. Prepared by: Engr. Fazal-E-Jalal Fluid Mechanics-I

Compressibility of Liquids Compressibility is the change in volume due to change in pressure. The compressibility of liquid is inversely related to its volume modulus of elasticity (also known as bulk modulus). Eν = - ν(dp/dν) = - (ν/dν)dp Where; ν = Specific Volume. (ν/dν) = Dimensionless ratio Prepared by: Engr. Fazal-E-Jalal Fluid Mechanics-I

Compressibility of Liquids In most engineering problems, the bulk modulus at or near atmospheric pressure is one of the interest. The BULK MODULUS is a property of fluid. And for liquids, is a function of temperature and pressure. Eν is directly related to temperature. It is maximum at 50 ͦC. Thus water has minimum compressibility at this temperature. Prepared by: Engr. Fazal-E-Jalal Fluid Mechanics-I

Compressibility of Liquids We often specify applied pressures in terms of absolute terms, because atmospheric pressure varies. Absolute pressure is the actual pressure on fluid relative to absolute zero. The standard atmospheric pressure at sea level is about 14.7 psia or 101.3 kn/m2 abs or 1013 mb. Prepared by: Engr. Fazal-E-Jalal Fluid Mechanics-I

Compressibility of Liquids Bars and milli bars were previously used in metric systems to express pressure. 1 mb = 100 N/m2 Most pressures are measured relative to the atmosphere and are known as GAUGE PRESSURES. Prepared by: Engr. Fazal-E-Jalal Fluid Mechanics-I

Volume modulus of mild steel is about 26,000,000 psi Volume modulus of mild steel is about 26,000,000 psi. And that of cold water is 320,000 psi. Mercury is approximately 8% as compressible as water. WESEE THAT WATER IS ABOUT 80 TIMES AS COMPRESSIBLE AS STEEL. Compressibility of Nitric acid is nearly 6 times greater than that of water. Eν =  ν/ ν = p/ Eν Eν = Mean value of modulus for pressure range (1,2 refer to before and after condition) At any temperature, the bulk modulus of water does not vary great deal for moderate range in pressure. Prepared by: Engr. Fazal-E-Jalal Fluid Mechanics-I

Compressibility of Liquids Assuming Eν to have value of 32,000 psi, we see the increasing pressure of water by 1000 psi will compress it only 0.3% of it’s original volume. Therefore, we find that the usual assumption regarding water as being INCOMPRESSIBLE is justified. Sample problem 2.3 Page 18 Chapter. 2 Properties of fluids) Prepared by: Engr. Fazal-E-Jalal Fluid Mechanics-I

Fluid Mechanics with Engineering Applications Practice Problems 2.5.1 2.5.2 2.5.3 2.5.4 2.5.5 Fluid Mechanics with Engineering Applications Prepared by: Engr. Fazal-E-Jalal Fluid Mechanics-I

IDEAL FLUID The fluid in which there is no friction; it is INVISCID (it’s viscosity is zero). The internal forces at any section within it are always normal to the section, even during motion. So, these forces are purely pressure forces. This does not exist in reality, many fluids approximate frictionless flow at sufficient distances from solid boundaries and hence we can analyze their behavior by assuming an ideal fluid. Prepared by: Engr. Fazal-E-Jalal Fluid Mechanics-I

Viscosity In real fluids, either liquid or gas, tangential or shearing forces are developed always whenever there is motion relative to a body, thus creating fluid friction, because these forces oppose the motion of one particle past another. These frictional forces give rise to a fluid property called Viscosity. Prepared by: Engr. Fazal-E-Jalal Fluid Mechanics-I

Viscosity The viscosity of a fluid is a measure of its resistance to shear or angular deformation. The friction forces in flowing fluid result from cohesion and momentum interchange between molecules. For Example Motor oil has high viscosity, and resistance to shear, and feels “sticky” whereas gasoline has low viscosity. Prepared by: Engr. Fazal-E-Jalal Fluid Mechanics-I

Viscosity In Liquids: T inversely related to Viscosity In Gases: T directly related to Viscosity Gases Liquids Viscosity Temperature Prepared by: Engr. Fazal-E-Jalal Fluid Mechanics-I

Viscosity NEWTON’S EQUATION OF VISCOSITY:  = F/A = (U/Y) =  (dU/dY) This was first suggested by Sir Isaac Newton (1642-1727) first suggested it. He is better known for his formulation of the fundamental laws of motion and gravity and for the development of differential calculus, NEWTON, an English mathematician and natural philosopher, also made many pioneering studies in FLUID MECHANICS. Prepared by: Engr. Fazal-E-Jalal Fluid Mechanics-I

Newton’s Equation of Viscosity Consider two plates, sufficiently large so that end conditions may be neglected, placed on small distance Y apart, the space between being filled with the fluid. The lower surface is assumed to be stationary, while the upper one is moved parallel to it with a velocity U by the application of force F corresponding to some area A of the moving plate. Prepared by: Engr. Fazal-E-Jalal Fluid Mechanics-I

Newton’s Equation of Viscosity .. U F dy Y y u du Prepared by: Engr. Fazal-E-Jalal Fluid Mechanics-I

Newton’s Equation of Viscosity Particles in the fluid in contact with each plate will adhere to it. And, if Y is not too great or the velocity U too high, the velocity gradient will be a straight line. The action is much as if the fluid were made up of a series of thin sheets, each of which would slip a little relative to the next. Prepared by: Engr. Fazal-E-Jalal Fluid Mechanics-I

Newton’s Equation of Viscosity Experiment has shown that for a large class of fluids: F α (A.U)/Y It may be seen from similar triangles in figure that U/Y can be replaced by the velocity gradient du/dy. If a constant of proportionality μ is now introduced, the shear stress  between any two thin sheets of fluid may be expressed by:  = F/A = (U/Y) =  (dU/dY) Newton’s equation of Viscosity Prepared by: Engr. Fazal-E-Jalal Fluid Mechanics-I

Viscosity From Newton’s equation of viscosity we have,  =  / (dU/dY) This is known as Coefficient of viscosity, the absolute viscosity, the dynamic viscosity or simply the viscosity of fluid. The distinction between solids and fluid lies in the manner in which each can resist SHEARING STRESS. Further distinction among various kinds of fluids and solids is as: Prepared by: Engr. Fazal-E-Jalal Fluid Mechanics-I

Viscosity … Elastic solid Ideal plastic Non Newtonian fluid  = Shear stress Newtonian fluid Ideal fluid (dU/dY) Prepared by: Engr. Fazal-E-Jalal Fluid Mechanics-I

Viscosity In case of solids, shear stress depends on magnitude of deformation but according to Newton’s equation of viscosity the shear stress is proportional to time rate of (angular) deformation. A fluid for which absolute viscosity does not change with rate of deformation is called NEWTONIAN FLUID. The slope of this line is “Absolute Viscosity” A fluid for which absolute viscosity changes with rate of deformation is called NON-NEWTONIAN FLUID. Prepared by: Engr. Fazal-E-Jalal Fluid Mechanics-I

Viscosity Non Newtonian fluids are relatively uncommon in engineering use (examples are paints, printer’s ink, gels and emulsions, sludges and slurries, and certain plastics). So, we will use fluids that obey Newton’s equation of viscosity under normal conditions. Prepared by: Engr. Fazal-E-Jalal Fluid Mechanics-I

Viscosity In BG System: Dimensions of  = (lb/ft2)/(fps/ft) = lb.sec/ft2 In S.I System: Dimensions of  = (N/m2)/(s-1) = N.s/m2 Poise (P) is a widely used unit for viscosity in Metric system. IT IS NAMED AFTER JEAN LOUIS POISEILLE (1799-1869). HE WAS ONE OF THE FIRST INVESTIGATORS OF VISCOSITY. 1 poise = 0.10 N.s/m2 1 Cp = 0.01 poise = 1 Mn.s/m2 (Frequently a more Convenient unit) VISCOSITY OF WATER AT 68.4 ͦF IS 1 Cp. So viscosity of fluid in cPs is viscosity of fluid relative to that of water at 68.4 ͦF. Prepared by: Engr. Fazal-E-Jalal Fluid Mechanics-I

ν =  / ƿ Viscosity Kinematic Viscosity = Absolute Viscosity / Density Is called so because force is not involved, the only dimensions being length and time, as in Kinematics. UNITS: In BG: ft2/sec In S.I: m2/s In Metric system it had units cm2/s, also known as STOKE(St). Name given after Sir George Stoke, an English Physicist and pioneering investigator of viscosity. 1 cSt = 0.01 St = 10-6 m2/s Prepared by: Engr. Fazal-E-Jalal Fluid Mechanics-I

Viscosity DISTINCTION BETWEEN  & ν :  of most fluids is virtually INDEPENDENT of pressures encountered ordinarily in engineering work. ν of gases varies strongly with pressure because of change in density. Sample problem 2.8 Page 34 Chapter. 2 (Properties of fluids) Prepared by: Engr. Fazal-E-Jalal Fluid Mechanics-I

Prepared by: Engr. Fazal-E-Jalal Fluid Mechanics-I