Chapter 1. Viscosity and the stress (momentum flux) tensor

Chapter 1. Viscosity and the stress (momentum flux) tensor

Viscosity and the Mechanisms of Momentum Transport 1.1 Newton s law of viscosity ( molecular momentum transport) 1.2 Generalization of Newton s law of viscosity 1.3 Pressure and temperature dependence of viscosity 1.4 Molecular theory of the viscosity of gases at low density 1.5 Molecular theory of the viscosity of liquids 1.6 Viscosity of suspensions and emulsions 1.7 Convective momentum transport

- sign explained later F A V = or τ yx = Y du dy x ( Newton' s law of vis cosity)

Sir Isaac Newton (4 January 1643 31 March 1727) An English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian who is considered by many scholars and members of the general public to be one of the most influential scientists in history.

F A V = or τ yx = Y du dy x ( Newton' s law of vis cosity) Fluids that obey this law are referred to as Newtonian Fluids. Polymeric liquids, suspension, slurries, pastes and other fluids are not described by this law and are referred to as non-newtonian fluids. Often the kinematic viscosity is used defined by: ν = μ/ρ Typical units for viscosity are Pa.s (kg m 1 s 1 ) and typical values are of air at 20 o C 1.8x10-5 Pa.s to 1 Pa.s of glycerol or even higher for molten polymers. Tables 1.1-1 1.1-4 give values of viscosities for typical fluids (even liquid metals).

NEWTON S LAW IN THREE DIMENSIONS Consider a flow field where the velocity components are: ux = ux( x, y, z, t); u y = u y ( x, y, z, t); uz = uz ( x, y, z, t) What is the force on this point? By whom? From which direction? We need more than a vector We need a tensor

At any instant the cubic element can be sliced in such a way to remove half of the fluid within it. Then the question is what force has to be applied to that surface to replace the force exerted by the removed fluid. They are two contributions: 1. Pressure force always normal to the surface and compressive. On the 3 different surfaces, this will be pδ, pδ, pδ x y z 2. Viscous force when there are velocity gradients. In general we have a force is acting in 1.2-1a. This can be analyzed into three components, one normal and two tangential in the three directions. These normalized with the area of the fluids give rise to three components of the stress tensor τ x τ, τ, τ xx xy xz Doing the same in all 3 directions 9 components of the stress tensor (three normal stresses and six shear stresses): It is a symmetric tensor (why?)

Doing the same in all 3 directions 9 components of the stress tensor (three normal stresses and six shear stresses). Let s arrange them into a matrix, or the stress tensor: π + pδ τ ij ij = where i and j may be x, y, or z Such that the force (traction) on any surface with outer normal n is: (Note: this expression not in BSL, but commonly used in fluid mechanics) π i = n j π ij = n π ij Let s try this out on the 3 planes that we had drawn earlier (illustrate on board )

Sign Convention: We take τ yx The force in the positive x direction on a plane perpendicular to the y- direction, and that this force is exerted by the fluid in the region of the lesser y on the fluid of greater y. In curvilinear co-ordinates it is also easy to write Newton s law of viscosity

Now generalizing Newton s law of viscosity we have to take into account: The viscous stresses may be linear combinations of velocity gradients Time derivatives should not appear (non-newtonian fluids excluded) No viscous forces for fluid at a rest or under purely rotation Fluid is isotropic-it has no preferred direction Compressibility should be taken into account Using all these the generalized Newton law in three dimensions can be written as: ( ) ij i i i j i i j ij x u x u x u δ κ τ + + = = 3 1 3 2 ( ) + = + = j i i j ij T x u x u or τ τ ) v ( v In this course will essentially study incompressible fluid flow and therefore Newton s generalized law of viscosity is:

Let s think more about viscous normal stresses:

PRESSURE AND TEMPERATURE DEPENDENCE OF VISCOSITY Viscosity is a function of temperature and pressure. Various books include such dependencies. The plot 1.3-1 gives a global view of the pressure and temperature dependence of viscosity in terms of: Reduced viscosity r = c that is plotted as a function of Reduced temperature T = T Reduced pressure p = p r T c r p c

PRESSURE AND TEMPERATURE DEPENDENCE OF VISCOSITY The reduced viscosity is either read off Figure 1.3-1, or calculated if the value of viscosity is known at a given T and p. If critical p-v-t data are available then: c ~ 1/ 2 2 / 3 1/ 2 2/3 2 / 2 1/ 6 = 61.6( MT c ) ( Vc ) and c = 7.70M pc Tc where p c in atm, T c in K, and For mixtures use: V ~ c in cm 3 g-mole. p ' c = N a= 1 x a p ca T ' c = N a= 1 x a T ca ' c = N a= 1 x a ca

What produces viscosity? Molecular collision that transports momentum 1.1 Newton s law of viscosity ( molecular momentum transport) 1.2 Generalization of Newton s law of viscosity 1.3 Pressure and temperature dependence of viscosity 1.4 Molecular theory of the viscosity of gases at low density 1.5 Molecular theory of the viscosity of liquids 1.6 Viscosity of suspensions and emulsions 1.7 Convective momentum transport

VISCOSITY OF SUSPENSIONS AND EMULSIONS The first major contribution to this area of viscosity of suspensions of spheres was that of Einstein. He considered a suspension of spheres so dilute that the movement of one sphere does not influence the fluid flow in the neighborhood of any other sphere. He calculated the effective viscosity of a suspension as: eff 0 where = 1+ 5 φ 2 o φ is the viscosity of the suspending medium and is the volume fraction of spheres. For suspensions greater than 5% interactions are significant and then other equations have developed. One such is the Mooney equation given by: eff 0 = exp 1 5 2 φ ( φ / φ ) o

Another approach is the cell theory developed by Graham, + + + + = ) )(1 (1 1 4 9 2 5 1 2 2 1 0 ψ ψ ψ φ eff For concentrated suspensions of non-spherical particles, the Krieger-Doughrty equation applies: max max 0 1 φ φ φ A eff = For emulsions or suspensions of tiny droplets in which the suspended material undergo internal circulation the Taylor equation can be used. φ + + = 1 1 2 5 0 1 o o eff Finally for dilute suspensions of charged spheres the Einstein equation may be replaced by the Smoluchowski equation, + + = e o eff k R D π ζ φ 2 0 ) 2 / ( 1 2 5 1

CONVECTIVE MOMENTUM TRANSPORT So far we have discussed molecular transport of momentum. However, momentum can be transferred by another mechanism known as the convective transport. See Figure 1.7-1 for an explanation. The volume flow rate across the shaded area in (a) is v x. This carries with it momentum ρv. Thus the momentum flux across the shaded area is v x ρv. Similarly the momentum flux across the shaded area in (b) is v y ρv and in (c) is v z ρv. Each of these momenta has three componenets, so total we have nine components.

For example: ρ v x v y is the y-momentum across a surface perpendicular to the x-direction. [Recall: τ xy is the molecular flux of the y-momentum across a surface normal to x-dir.] All nine components are: ρvv = i j δ i δ j ρ v i v j To calculate the momentum flux through a plane of arbitrary orientation see Fig. 1.7-2.

The rate of flow of momentum flux across the surface is: (n. v)ρv or [n. ρvv ] Similarly the molecular momentum flux is: [n. π ] = pn +[n. τ] In general, a combined (molecular + convective) momentum flux tensor: φ=π + ρvv=pδ + τ + ρvv φ xy = the combined flux of the y-momentum across a surface perpendicular to the x direction by molecular and convective mechanisms.

Example (Prob. 1D1): uniform rotation of a fluid. (a)verify that the velocity distribution v = [w x r] represents solid-body rotation, where w is the constant angular velocity and r is the positional vector (x, y, z). (b)what are v + ( v) T and v for this flow field? (c) What is the viscous stress tensor for this flow?

Solution to Prob. 1D1: r δ r = cos θδ 1 + sin θδ 2 ; δ θ = sin θδ 1 + cos θδ 2.

Solution to Prob. 1D1 (cont d):