ME 254

Chapter I Integral Relations for a Control Volume An engineering science like fluid dynamics rests on foundations comprising both theory and experiment. With fluid dynamics, progress has been especially dependent upon an intimate cross- fertilization between the analytical and empirical branches; the experimental results being most fruitfully interpreted in terms of theoretical reasoning, and the analyses in turn suggesting critical and illuminating experiments which further amplify and strengthen the theory. All analyses concerning the motion of compressible fluids must necessarily begin either directly or in- directly, with the statements of the four basic physical laws governing such motions. These laws, which are independent of the nature of the particular fluid are Prof. Dr. MOHSEN OSMAN2

(i) The law of conservation of mass (ii) Newton’s second law of motion (iii) The first law of thermodynamics (iv) The second law of thermodynamics In addition to certain subsidiary laws such as equation of state of a perfect gas, the proportionality law between shear stress and rate of shear deformation in a Newtonian fluid, the Fourier law of heat conduction, etc. Definition of a Fluid. A substance which deforms continuously under the action of shearing forces. “ a fluid cannot withstand shearing stresses ”. Liquids vs. Gases. Both of which are fluids. For most practical purposes the words “liquid” and “gas” are of value insofar as the former denotes a fluid which generally exhibits only small percentage changes in density. Prof. Dr. MOHSEN OSMAN3

The Concept of a Continuum In most engineering problems our primary interest lies not in the motions of molecules, but rather in the gross behavior of the fluid thought of a continuous material. The treatment of fluids as continua may be said to be valid whenever the smallest volume of fluid of interest contains so many molecules as to make statistical average meaningful. This course concerns the motion of compressible fluids which may be treated as continua; (continuum approach). Properties of the Continuum Density at a Point. The “density” as calculated from molecular mass within given volume is plotted versus the size of the unit volume. There is a limiting value below which mole- cular variations may be important and above which aggregate variations may be important. The density of a fluid is best defined as Prof. Dr. MOHSEN OSMAN4

for all liquids and gases. of air contains approximately molecules at standard conditions. This definition illuminates the idea of a continuum and shows the true nature of a continuum property “at a point” as fictitious but highly useful concept. Number of molecules per unit volume n = where P =pressure T = temperature & σ = Boltzmann constant J/kmol Fluid Dynamic Description Lagrangian Technique An identifiable mass is followed as it moves through space ( microscopic formulation) Eulerian Technique Flow through an identifiable region is monitored ( macroscopic formulation). [flow has confined surface] Prof. Dr. MOHSEN OSMAN5

Properties of the Velocity Field The Velocity Field Velocity is a vector function of position and time and thus has three components u, v, and w, each of which is itself a scalar field: The acceleration of a particle: The acceleration is fundamental to fluid mechanics since it occurs in Newton’s law of dynamics (local acc.) (convective acceleration) The gradient operator del Prof. Dr. MOHSEN OSMAN6

Vector Product Scalar Product The general expression for Eulerian time-derivative operator following a particle This operator can be applied to any fluid property, scalar or vector, e.g., the pressure. Prof. Dr. MOHSEN OSMAN7

Volume and Mass Rate of Flow Volume Rate of Flow Consider volume rate of flow through an arbitrary surface. The volume swept out of dA in time dt is the volume of the slanted parallelepiped in the figure The integral of is the total volume rate of flow Q through the surfase S. Q = Mass flux is given by the following relation where is the normal component of velocity For constant density An average velocity passing through the surface is given by: Prof. Dr. MOHSEN OSMAN8

System versus Control Volume Control Volume Terminology A system is defined as an arbitrary quantity of mass of fixed identity. Everything external to this system is denoted by the term surroundings and the system is separated from its surroundings by its boundaries. The control volume is defined as an arbitrary volume, fixed in space, and through which fluid flows. The surface which bounds the control volume is called the control surface. The Reynolds Transport Theorem In order to convert a system analysis into a control volume analysis we must convert our mathematics to apply to a specific region rather than to individual masses. This conversion, called the Reynolds transport theorem, can be applied to all basic laws. Prof. Dr. MOHSEN OSMAN9

One-Dimensional Fixed Control Volume Example of inflow and outflow as three systems pass through a control volume Now let B be any (extensive) property of the fluid (mass, energy, momentum, etc.) and let β = be the intensive value or the amount of B per unit mass in any small portion of the fluid. The total amount of B in the control volume is thus & where is the differential mass of the fluid The time derivative of is defined by the calculus limit Prof. Dr. MOHSEN OSMAN10

Prof. Dr. MOHSEN OSMAN11 For any Arbitrary Fixed Control Volume The rate of change of B for a given mass as it is moving around is equal to the rate of change of B inside the control volume plus the net efflux (flow out minus flow in) of B from the control volume.

By letting the property B be the mass For steady flow within the control volume Conservation of mass principle reduces to = 0 Prof. Dr. MOHSEN OSMAN12