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Transport Phenomena 3 Unit 2: Momentum Transport Topics • • • • Definitions Properties at a point Molecular Transport of Momentum Convective Momentum Transport Definitions • Fluid: as a substance which deforms continuously under the action of a shear stress. • Then when a fluid is at rest, there can be no shear stress Concept of a continuum • Fluids, like matter , are composed of molecules whose numbers stagger the imagination. • In a cubic inch of air at room conditions there are some 1020 molecules. Any theory to predict the individual motions of this many molecules would be extremely complex, far beyond out present abilities Concept of a continuum (Cont.) • Most engineering work is concerned with the macroscopic or bulk behavior of a fluid rather than with the microscopic or molecular behavior. • In most cases is convenient to think as a continuous distribution of matter or a continuum. • Validity of this concept is seem to be dependent upon the type of information desired rather than the nature of the fluid. Definitions • Steady state flow – physical properties of the fluid does not change. • Unsteady state flow – the properties change with time in any point of the system Definitions • Uniform flow – when the velocity at any point, across transversal section in the direction that motion occurs is constant. In other words when the velocity vector are parallel and preserved their intensity • Non uniform – when the velocity vectors are not parallel and did not maintain their intensity constant. Example: The flow fluid through a tube of constant diameter is uniform, but the flow through a tube of variable diameter (conic form) is not uniform. Properties At a Point • Density: It is defined as the mass per unit volume. Under flow conditions, particularly in gases, the density may vary greatly throughout the fluid. • The density, ρ, at a particular point in the fluid is defined as m lim V V V • Where Δm: mass contained in the volume ΔV • δV is the smallest volume. Properties At a Point (Cont.) • Stress: • Consider the force ΔF acting on an element ΔA of the body shown in the figure. • The force ΔF is resolved into components normal and parallel to the surface of the element. • The force per unit area or stress ata point is defined as the limit of ΔF/ΔA as ΔA→δA Force on an element of fluid Fn lim ii A A A Fs lim ij A A A ΔF ΔFn ΔFs ΔA • Where σii is called the normal stress and the shear stress (Flux tensor) ij is Molecular Transport Momentum • The buildup to the steady, laminar velocity profile for a fluid contained between two plates. • Each of the plates have an area A, separated by a distance Y. • In the space let’s consider there is a fluid (gas or liquid). Build-up to the steady-state laminar velocity profile Y t<0 vx(y,t) Small t vx(y) t=0 V Large t • In this Figure 1: • First: The system is at rest, but a time t= 0 • Second: At t= 0 the lower plate is set in motion in the direction at a constant velocity • Third: At time proceeds (Small t), the fluid gain momentum • Fourth: Ultimately the linear steady-state velocity profile shown in the figure is established. • At state-steady motion, a constant force F is required to maintain the motion of the lower plate. F V (1) A Y • The force is proportional to the area and to the velocity and inversely proportional to the distance between plates. • Where F/A is , which is the force in the direction x per unit area perpendicular to the y direction. • It is the force exerted by the fluid of lesser y on the fluid of greater y, therefore V/Y can be replaced by –dvx/dy. • The equation can also be written as yx dvx dy Newton’s Law of viscosity (2) • Where: • μ: viscosity of fluid, (Pa.s) • yx : Flux tensor (shear stress) in the positive y direction , [(N/m2)= Pa] • vx: velocity in x direction, (m/s) • y: distance, (m) Generalization of Newton’s Law of viscosity • The equation (2) was defined only in terms of a simple steady state shearing flowing in which vx as a function of y alone, and vx and vz are zero. • This situation is not really so common. Usually the system is composed by a flow in which the three velocity components may depend on all three coordinates and possible on time. Consider • A general flow pattern in which the velocity shows in various directions depends on the time. • The velocity components are given by vx vx ( x, y, z, t ); v y v y ( x, y, z, t ), vz v2 ( x, y, z, t ) • There will be then, 9 stress components ij (where i and j may be taken on the designations (x,y,z), instead of the yx component that appears in equation (2) Pressure and viscous forces acting on planes z x, y, z y x • The volume element can be cut to each of three coordinates in turn. • Find the forces that have been exerted on that surface by the fluid that was removed. • There will be two forces that contribute – Associated with the pressure – Associated with the viscous forces • Pressure force – Always perpendicular to the exposed surface • For example in x direction the force will be a vector pδx that is the pressure (a scalar) multiplied by the unit vector δx in the x direction. Similarly for the other sections. – Pressure forces will be exerted when the fluid is stationary as well as when it is in motion • Viscous forces • They exist only when the are velocity gradients within the fluid. • They are neither perpendicular nor parallel to the surface element, rather at some angle to the surface. • Those forces represent in the Figure are vectors with scalar components. x – For example, x has components xx , xy , xz • Pressure and Viscous Forces acting on planes y x x pδy pδx z pδz • For convenient a new variable is defined that include both ij p ij ij • Where δij is the kronecher delta, which is 1 if i = j and zero if i ≠ j. Two Ways to Interpret the definition ij p ij ij • force in the j direction on a unit area to the i direction, where it is understood that the fluid in the region of lesser xi is exerting the force on the fluid of a greater xi. • flux of j-momentum in the positive (+) i direction, It is understood that is, from the region of lesser xi to that of greater xi. Generalizing Newtons’ Law (Equation 2) • How are the stresses related to the velocity gradients in the fluid? • Restrictions – The viscous stresses may be linear combinations of all velocity gradients x ij k ijkl where i,j,k and l may be 1,2,3 x1 – Time derivatives or time integrals should not appear in the expression. – Do no expect viscous forces of the fluid is in a state of pure rotation. No viscous forces present are only those one that are symmetric linear combinations of velocity gradients j i xi xi x y z and y z x ij – If the fluid is isotropic (not preferred direction) the coefficient in front of the two expressions must be scalar so that j i x y z ij A B ij y z x xi xi – Common agreement among fluid dynamicists the scalar constant B is set equal (3/2μ – k), where k is the identically zero for monoatomic gases at low density. j i ij xi xi x y z 2 ij k k 3 x y z – These relations for the stresses in a Newtonian fluid are associated with the names of Navier, Poisson and Stokes • Writing these set of equations more concise in vector-tensor notation 2 v (v) k .v 3 • Where δ is the unit tensor with components δij, v is the velocity gradient tensor with components / xi j ,(v) is the transpose of the velocity gradient tensor with components / xi j , and (.v) is the divergence of the velocity vector. Importance of this generalization • This equation involve two coefficients characterizing the fluid: – The viscosity μ – The dilatational viscosity k. Convective Momentum Transport • The momentum can be transported by bulk flow of the fluid. This process is called convective transport. Use the same figure on three planes • The fluid vector at the centre is v • The volume rate of flow across the area in the first figure is vx. The fluid carries with it momentum ρv per unit volume. • Hence the momentum flux across this area is then vxρv (vectors) • Convective momentum fluxes through planes of unit area perpendicular to the coordinates direction x pvyv pvxv pvzv END