# Introduction to Momentum Transfer

Couette flow.
Components of the stress tensor in a fluid.

A fluid at rest can resist a normal force but not a shear force, while fluid in motion can also resist a shear force. The fluid continuously deforms under the action of shear force. A fluid’s resistance to shear or angular deformation is measured by viscosity, which can be thought of as the internal “stickiness” of the fluid. The force and the rate of strain (i.e., rate of deformation) produced by the force are related by a constitutive equation. For a Newtonian fluid, the shear stress in the fluid is proportional to the time rate of deformation of a fluid element or particle. The figure on the right shows a Couette flow where the plate at the bottom is stationary and the fluid is driven by the upper moving plate. This flow is one-dimensional since velocity components in the x - and z - directions are zero. The constitutive relation for Couette flow can be expressed as

${\tau _{yx}} = - \mu \frac{{du}}{{dy}}\qquad \qquad(1)$

where τyx is the shear stress N / m2, μ is the dynamic viscosity (N − s / m2), which is a fluid property, and du / dy is the velocity gradient in the y - direction, also known as the rate of deformation. If the shear stress τyx and the rate of deformation du / dy have a linear relationship, as shown in eq. (1), the fluid is referred to as Newtonian and eq. (1) is called Newton’s law of viscosity. It is found that the resistance to flow for all gases and liquids with molecular mass less than 5000 is well presented by eq. (1). For non-Newtonian fluids, such as polymeric liquids, slurries, or other complex fluids, for example in biological applications, such as blood or operation of joints, drag-reducing slimes on marine animals, and digesting foodstuffs the shear stress and the rate of deformation no longer have a linear relationship. Bird et al. (2002) provided detailed information for treatment for non-Newtonian fluids.

The viscous force comes into play whenever there is a velocity gradient in a fluid. For a three-dimensional fluid flow problem, the stress ${\mathbf{\tau '}}$ is a tensor of rank two with nine components. It can be expressed as the summation of an isotropic, thermodynamic stress, $- p{\mathbf{I}}$, and a viscous stress, ${\mathbf{\tau }}$:

${\mathbf{\tau '}} = - p{\mathbf{I}} + {\mathbf{\tau }}\qquad \qquad(2)$

where p is thermodynamic pressure, and I is the unit tensor defined as

${\mathbf{I}} = \left[ {\begin{array}{*{20}{c}} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array}} \right]\qquad \qquad(3)$

In a Cartesian coordinate system, the viscous stresses are

${\mathbf{\tau }} = \left[ {\begin{array}{*{20}{c}} {{\tau _{xx}}} & {{\tau _{xy}}} & {{\tau _{xz}}} \\ {{\tau _{yx}}} & {{\tau _{yy}}} & {{\tau _{yz}}} \\ {{\tau _{zx}}} & {{\tau _{zy}}} & {{\tau _{zz}}} \\ \end{array}} \right]\qquad \qquad(4)$

where the first subscript represents the axis normal to the face on which the stress acts, and the second subscript represents the direction of the stress (see figure). The components of the shear stresses are symmetric

τxy = τyxxz = τzx, and τyz = τzy

The viscous stress tensor can be expressed by Newton’s law of viscosity:

${\mathbf{\tau }} = 2\mu {\mathbf{D}} - \frac{2}{3}\mu (\nabla \cdot {\mathbf{V}}){\mathbf{I}}\qquad \qquad(5)$

where $\nabla \cdot {\mathbf{V}}$ is the divergence of the velocity. The rate of deformation, or strain rate D, presented below for a Cartesian coordinate system in a three-dimensional flow is another tensor of rank two.

${\mathbf{D}} = \frac{1}{2}\left[ {\nabla {\mathbf{V}} + {{\left( {\nabla {\mathbf{V}}} \right)}^T}} \right] = \left[ {\begin{array}{*{20}{c}} {\frac{{\partial u}}{{\partial x}}} & {\frac{1}{2}\left( {\frac{{\partial u}}{{\partial y}} + \frac{{\partial v}}{{\partial x}}} \right)} & {\frac{1}{2}\left( {\frac{{\partial u}}{{\partial z}} + \frac{{\partial w}}{{\partial x}}} \right)} \\ {\frac{1}{2}\left( {\frac{{\partial v}}{{\partial x}} + \frac{{\partial u}}{{\partial y}}} \right)} & {\frac{{\partial v}}{{\partial y}}} & {\frac{1}{2}\left( {\frac{{\partial v}}{{\partial z}} + \frac{{\partial w}}{{\partial y}}} \right)} \\ {\frac{1}{2}\left( {\frac{{\partial w}}{{\partial x}} + \frac{{\partial u}}{{\partial z}}} \right)} & {\frac{1}{2}\left( {\frac{{\partial w}}{{\partial y}} + \frac{{\partial v}}{{\partial z}}} \right)} & {\frac{{\partial w}}{{\partial z}}} \\ \end{array}} \right] \qquad \qquad(6)$

where ${\left( {\nabla {\mathbf{V}}} \right)^T}$ is the transverse tensor of $\nabla {\mathbf{V}}$. For example, in a Cartesian coordinate system, the normal and shear viscous stresses can be expressed as

${\tau _{xx}} = 2\mu \frac{{\partial u}}{{\partial x}} - \frac{2}{3}\mu \left( {\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} + \frac{{\partial w}}{{\partial z}}} \right)\qquad \qquad(7)$
${\tau _{xy}} = \mu \left( {\frac{{\partial u}}{{\partial y}} + \frac{{\partial v}}{{\partial x}}} \right)\qquad \qquad(8)$

For one-dimensional flow in Fig. 1, eq. (8) is reduced to eq. (1). The stress-strain rate relationships in eqs. (1) and (5) are valid for laminar flow only. For turbulent flow, eqs. (1) or (5) can still be used provided that the time-averaged velocity is used and the turbulent effects are included in the viscosity (White, 1991; Kays et al., 2004).

For a multicomponent system, the viscosity of the mixture is related to the viscosity of the individual component by

$\mu = \sum\limits_{i = 1}^N {\frac{{{x_i}{\mu _i}}}{{\sum\nolimits_{j = 1}^N {{x_i}{\phi _{ij}}} }}}\qquad \qquad(9)$

where xi is the molar fraction of component i and

${\phi _{ij}} = \frac{1}{{\sqrt 8 }}{\left( {1 + \frac{{{M_i}}}{{{M_j}}}} \right)^{ - 1/2}}{\left[ {1 + {{\left( {\frac{{{\mu _i}}}{{{\mu _j}}}} \right)}^{1/2}}{{\left( {\frac{{{M_j}}}{{{M_i}}}} \right)}^{1/4}}} \right]^2}\qquad \qquad(10)$

where Mi is the molecular mass of the ith species. Equation (9) can reproduce the viscosity for the mixtures with an averaged deviation of 2%. Additional correlations to estimate the viscosities of various gases and gas mixtures as well as liquids can be found from the standard reference by Poling et al. (2000).

## References

Bird, R.B., Stewart, W.E., and Lightfoot, E.N., 2002, Transport Phenomena, 2nd edition, John Wiley & Sons, New York.

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA.

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

Kays, W.M., Crawford, M.E., and Weigand, B., 2004, Convective Heat Transfer, 4th ed., McGraw-Hill, New York, NY.

Poling, B.E., Prausnitz, J. M., and O’Connell, J.P., 2000, The Properties of Gases and Liquids, 5th edition, McGraw-Hill, New York, NY.

White, F.M., 1991, Viscous Fluid Flow, 2nd ed., McGraw-Hill, New York, NY.