Measurement of Flow and Viscoelastic Properties

M. Anandha Rao

doi:10.1007/978-1-4614-9230-6_3

Rheology of Fluid, Semisolid, and Solid Foods pp 63-159 | Cite as

Measurement of Flow and Viscoelastic Properties

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M. Anandha RaoEmail author

Chapter

First Online: 19 November 2013

Part of the Food Engineering Series book series (FSES)

Abstract

Techniques for measuring rheological properties of fluid foods are discussed in this chapter. First, the traditional measuring systems are considered, followed by microrheological techniques using micrometer-scale probes.

Keywords

Measurement of flow and viscoelastic properties Concentric cylinder geometry Cone-plate geometry Capillary geometry Static and dynamic yield stress Dynamic rheology Microrheology

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Appendix 1

Analysis of Flow in a Concentric Cylinder Geometry

One can derive applicable equations for shear rate starting from either the general equations of conservation of mass (continuity) and conservation of momentum (motion) or by conducting balances of mass or momentum on a differential shell. Numerous examples of using the conservation equations or shell balances can be found in Bird et al. (1960) and other texts on transport phenomena. For the concentric cylinder geometry shown in Fig. 3A.1, it will be assumed that the inner cylinder is rotating and the outer cylinder is stationary. Interested readers can derive the applicable equations for the case of outer cylinder rotating using appropriate boundary conditions.

First, from a simple force balance, the total torque (M) on the inner cylinder is

$${{\sigma }_{r}}\theta \,\,\times \,2\pi {{r}_{i}}h\,\,\times \,{{r}_{i}}=M$$

(3A.1)

In terms of torque per unit height,$\text{T}=\frac{M}{h}$ Eq. 3A.1 becomes

$${{\sigma }_{\text{r}}}\theta =\frac{T}{2\pi r_{\text{i}}^{2}}$$

(3A.2)

To describe the flow between the inner and outer cylinders, we use cylindrical coordinates (r, θ, z) and note that the fluid moves in a circular motion; the velocities in the radial and the axial directions are zero: v _θ = rΩ, v _r = 0, v _z = 0, and due to symmetry ∂/∂θ = 0.

Also, at steady state ∂ρ/∂t = 0. The equation of continuity in cylindrical coordinates is

$$\frac{\partial \rho }{\partial t}+\frac{1}{r}\frac{\partial }{\partial r}(\rho r{{v}_{r}})+\frac{1}{r}\frac{\partial }{\partial \theta }(\rho {{v}_{\theta }})+\frac{\partial }{\partial z}(\rho {{v}_{z}})=0$$

All terms in the above equation are zero.

The r-component of the equation of motion is

$$\begin{aligned} & \rho \left( \frac{\partial {{v}_{r}}}{\partial t}+{{v}_{r}}\frac{\partial {{v}_{r}}}{\partial r}+\frac{{{v}_{\theta }}}{r}\frac{\partial {{v}_{r}}}{\partial \theta }\frac{v_{\theta }^{2}}{r}+{{v}_{z}}\frac{\partial {{v}_{r}}}{\partial z} \right) \\ & =-\frac{\partial p}{{{\partial }_{r}}}+\left( \frac{1}{r}\frac{\partial }{{{\partial }_{r}}}(r{{\sigma }_{rr}})+\frac{1}{r}\frac{\partial {{\sigma }_{r\theta }}}{\partial \theta }\frac{\sigma \theta \theta }{r}+\frac{\partial {{\sigma }_{rz}}}{\partial z} \right)+\rho {{g}_{r}} \\ \end{aligned}$$

The r-component of the equation of motion reduces to

$$-\rho \frac{v_{\theta }^{2}}{r}=\frac{1}{r}\frac{\partial }{{{\partial }_{r}}}(r{{\sigma }_{rr}})-\frac{\sigma \theta \theta }{r}$$

(3A.3)

Equation 3A.3 deals with the normal stresses σ_rr and σ_θθ. For the purpose of illustration, we consider the θ-component of the equation of motion in detail and note in particular that v_θ = ƒ (r)

θ-component

$$\begin{aligned} & \rho \left( \frac{\partial v\theta }{\partial t}+{{v}_{r}}\frac{\partial v\theta }{\partial r}+\frac{v\theta }{r}\frac{\partial v\theta }{\partial \theta }+\frac{{{v}_{r}}v\theta }{r}+{{v}_{z}}\frac{\partial v\theta }{\partial z} \right) \\ & =-\frac{1}{r}\frac{\partial p}{\partial \theta }+\left( \frac{1}{{{r}^{2}}}\frac{\partial }{\partial r}\left( {{r}^{2}}{{\sigma }_{r}}\theta \right)+\frac{1}{r}\frac{\partial {{\sigma }_{\theta \theta }}}{\partial \theta }+\frac{\partial {{\sigma }_{\theta }}_{z}}{\partial z} \right)+\rho {{g}_{\theta }} \\ \end{aligned}$$

(3A.4)

Under steady flow condition ∂v_θ /∂t = 0, v_r = 0, ∂v_θ /∂θ = 0, and ∂v_θ /∂z = 0, the LHS of Eq. 3A.3 reduces to zero. On the right-hand side, ∂p/∂θ = 0 because there is no pressure gradient in the θ-direction and ρg_θ = 0 because there is no θ-component of gravity. The ∂σ_θθ /∂θ and ∂σ_θz /∂z = 0 because there are no shear gradients in the θ and z directions. There is no pressure gradient in the θ direction. We, thus, have for the θ-component

$$\frac{\partial \left( {{r}^{2}}{{\sigma }_{r}}\theta \right)}{\partial r}=0$$

(3A.5)

The z-component reduces to

$$0=-\frac{\partial P}{\partial z}+\rho {{g}_{z}}$$

(3A.6)

Equation 3A.6 simply describes the hydrostatic pressure in the gap between the two cylinders. Differentiation of Eq. 3A.5, which contains the shear stress of interest, σ_rθ , results in

$${{r}^{\text{2}}}\text{d}{{\sigma }_{r}}_{\theta }+2r{{\sigma }_{r}}_{\theta }\text{d}r=0$$

(3A.7)

Equation 3A.7 can be rearranged to:

$$\text{d}r=-\frac{r}{2}\frac{\text{d}{{\sigma }_{r}}_{\theta }}{{{\sigma }_{r}}_{\theta }}$$

(3A.8)

The boundary condition at the inner cylinder rotating at an angular velocity Ω is v _θ= r _iΩ at r = r _i and at the stationary outer cylinder is v_θ = 0 at r = r_o; both expressions are based again on the assumption of no slip condition at a solid–fluid interface. The velocity distribution is obtained from

$$ \frac{{{v_\theta }}}{r} = \int\limits_{{r_i}}^r {\frac{{{\rm{d}}({v_\theta }/r)}}{{{\rm{d}}r}}} {\rm{d}}r $$

(3A.9)

Substituting from Eq. 3.8 for dr in Eq. 3A.9 and using the appropriate integration limits

$$ \frac{{{v_\theta }}}{r} = \int\limits_{{\sigma _i}}^{{\sigma _{r\theta }}} {} \left( {\frac{{r{\rm{d}}({v_\theta }/r}}{{{\rm{d}}r}}} \right)\frac{{{\rm{d}}{\sigma _{r\theta }}}}{{{\rm{2}}{\sigma _{r\theta }}}} = \int\limits_{{\sigma _{_{r\theta }}}}^{{\sigma _i}} {} \left( {\frac{{r{\rm{d}}({v_\theta }/r)}}{{{\rm{d}}r}}} \right)\frac{{{\rm{d}}{\sigma _{r\theta }}}}{{{\rm{2}}{\sigma _{r\theta }}}} $$

(3A.10)

By using the Leibnitz rule and noting to differentiate with respect to σ_rθ, and with the boundary conditions: σ_rθ = σ_i at r = r _i and σ_rθ = σ_o at r = r _o

$$ \frac{{{\rm{d}}({v_\theta }/r)}}{{{{\rm{d}}_{\sigma {\rm{i}}}}}} = \frac{1}{{{{\rm{2}}_{\sigma {\rm{i}}}}}}\left[ {{{\left( {r\frac{{{\rm{d}}({v_\theta }/r}}{{{\rm{d}}r}}} \right)}_{\rm{i}}} - {{\left( {r\frac{{{\rm{d}}({v_\theta }/r)}}{{{\rm{d}}r}}} \right)}_{\rm{o}}}} \right] $$

(3A.11)

Noting that Ω = (v_θ/r) and $\dot \gamma $ = (r(d(v _θ/r)/dr)), Eq. 3A.9 can be written as

$$ {\rm{2}}{\sigma _{\rm{i}}}\frac{{{\rm{d}}{\Omega _{\rm{i}}}}}{{{\rm{d}}{\sigma _{\rm{i}}}}} = .{\gamma _{\rm{i}}}--.{\gamma _{\rm{o}}} $$

(3A.12)

Therefore, while it is relatively easy to calculate the shear stress at the surface of the rotating cylinder from Eq. 3A.2, one can only derive an expression for the difference in shear rates at the surfaces of the inner and outer cylinders from the basic equations of flow. Additional work is required to calculate the corresponding shear rate ${\dot \gamma _1}$ and there have been several approaches to determine it. One approach has been to apply infinite series solution to the differential equation in 3A.12.

A popular method of calculating the shear rate at the surface of the rotating cylinder is to assume that the test fluid follows the simple power law model

$$ \sigma=- K{\left[ {r\frac{{{\rm{d}}({v_\theta }/r)}}{{{\rm{d}}r}}} \right]^n} $$

(3A.13)

Equation 3A.12 can be solved after recalling Eq. 3A.2 for shear stress

$$ \frac{{{V_\theta }}}{r} =- {\left( {\frac{T}{{{\rm{2}}nK}}} \right)^{{\rm{1}}/n}}\frac{n}{{\rm{2}}}{r^{ - {\rm{2}}/n}} + C $$

(3A..14)

Using the boundary conditions v_θ = 0 at r = r_o and v_θ = r _iΩ at r = r_i, one can obtain the equation given earlier, Eq. 3.7

$$ .\gamma= \frac{{2\Omega r_i^{\rm{2}}}}{{\left[ {1 - {{\left( {\frac{{{r_i}}}{{{r_o}}}} \right)}^{\rm{2}}}} \right]}}\left\{ {\frac{{1 - {{(\frac{{{r_i}}}{{{r_o}}})}^{\rm{2}}}}}{{n{{\left[ {1--\frac{{{r_i}}}{{{r_0}}}} \right]}^{{\rm{2}}/n}}}}} \right\} $$

(3A.15)

When n = 1 (Newtonian fluid), the Margules equation for shear rate is obtained

$$ .{\gamma _{\rm{i}}} = \frac{{{\rm{2}}\Omega r_i^2}}{{[ {1 - {{({r_i}/{r_o})}^2}} ]}} $$

Fig. 3A.1
Decision tree for using a concentric cylinder geometry

Examining the limiting form of the second part of Eq. 3.7 as r_i → r_o

$$ {\rm{Limit}}\left\{ {\frac{{1 - {{({r_i}/{r_o})}^{\rm{2}}}}}{{n[ {1 - {{({r_i}/{r_o})}^{\rm{2}}}} ]}}} \right\} = _{n \to {r_o}}^{\lim }\left( {\frac{{{\rm{2}}{r_i}/{r_o}}}{{ - n({\rm{2}}/n){{({r_i}{r_o})}^{({\rm{2}}/n) - 1}}}}} \right) = 1 $$

Therefore, when ri → ro, the shear rate of a non-Newtonian fluid tends to that of a Newtonian fluid. In Table 3.1, values of the correction factor in parenthesis in Eq. 3.7 are given for several values of the flow behavior index, n, and the ratio (r_i/r_o) of the concentric cylinders, and they confirm that: (1) they are small when r _i → r _o, and (2) they may be large when the fluids are substantially non-Newtonian and when (r_i/r_o) 0.95.

Although the concentric cylinder geometry is relatively easy to use in rheological studies, some of its limitations should be recognized as shown in Fig. 3.44.

Appendix 2

Analysis of Steady Laminar Fully Developed Flow in a Pipe

The shell balance method will be used to examine steady laminar flow of a fluid in a pipe. For the geometrical system illustrated in Fig. 3B.1 and for steady laminar fully developed flow of a fluid, a shell momentum balance can be conducted (Bird et al. 1960; Geankoplis 1983) using the cylindrical coordinates, r, θ, and z. The momentum balance is conducted on a control volume shell at a radius r with dimensions Δr and Δz.

1.

The balance of two opposing forces is

$$pA{{|}_{z}}-pA{{|}_{z}}{{+}_{\Delta z}}=P(\text{2}\pi r\Delta r){{|}_{z}}-p(\text{2}\pi r\Delta r){{|}_{z}}{{+}_{\Delta z}}$$

(3B.1)

2.

The shear or drag force at radius r = σ_rz (2πrΔz) which can be considered to be the momentum flow into the cylindrical surface of the shell. The net efflux of momentum is the difference between the magnitudes the momentum out and momentum in

$${{\sigma }_{rz}}(\text{2}\pi r\Delta z){{|}_{r+\Delta r}}(\text{OUT})-{{\sigma }_{rz}}(\text{2}\pi r\Delta z){{|}_{r}}(\text{IN})$$

(3B.2)

3.

The momentum flux across the annular surface at z and z + Δz is zero because the axial velocity v_z is of the same magnitude at z and z + Δz.
4.

Recognizing that the sum of forces acting on control volume = rate of momentum out of control volume – rate of momentum into control volume

$$\begin{aligned}& p(\text{2}\pi r\Delta r){{|}_{z}}-p(\text{2}\pi r\Delta r){{|}_{z+\Delta z}}={{\sigma }_{rz}}(\text{2}\pi r\Delta z){{|}_{r+\Delta r}}(\text{OUT}) \\ & -{{\sigma }_{rz}}(\text{2}\pi r\Delta r){{|}_{r}}(\text{IN}) \\ \end{aligned}$$

(3B.3)

The above equation can be simplified to

$$r\Delta r(p{{|}_{z}}-p{{|}_{z}}{{+}_{\Delta z}})=\Delta z\left[ (r{{\sigma }_{rz}}){{|}_{r}}{{+}_{\Delta r}}-(r{{\sigma }_{rz}}){{|}_{r}} \right]$$

(3B.4)

Fig. 3B.1
Schematic diagram for force balance in laminar capillary flow

Dividing both sides by (ΔrΔz), results in

$$\frac{r\left( p{{|}_{z}}-p{{|}_{z}}{{+}_{\Delta z}} \right)}{\Delta z}=\frac{\left( r{{\sigma }_{rz}} \right){{|}_{r+\Delta r}}-\left( r{{\sigma }_{rz}} \right){{|}_{r}}}{\Delta r}$$

(3B.5)

As Δr and Δz → 0, the above equation becomes

$$r\frac{\text{d}p}{\text{d}z}=\frac{\text{d}}{\text{d}r}(r{{\sigma }_{rz}})$$

(3A.6)

After integration, one gets

$$r{{\sigma }_{rz}}=-\frac{{{r}^{2}}}{2}\frac{\text{d}p}{\text{dz}}+c1$$

(3B.7)

which can be rearranged to

$${{\sigma }_{rz}}=-\frac{r}{2}\frac{\text{d}p}{\text{dz}}+\frac{c1}{r}$$

(3B.8)

By using the condition σ_rz = 0 at r = 0 (center line), the integration constant c₁ = 0, so that the stress distribution across the radius of the pipe is

$${{\sigma }_{rz}}=-\frac{r}{2}\frac{\text{d}p}{\text{dz}}$$

(3B.9)

that for the wall of the pipe becomes: σ_w = –[(r _o/2; dp/dz)], because at r = r _o, σ_rz = σ_w. When the pressure drop, Δp, is measured for fully developed flow over a length, L, the equation for the shear stress at the wall is

$${{\sigma }_{\text{w}}}=\frac{\text{D}\Delta p}{\text{4}L}$$

(3B.10)

We will derive an expression for the volumetric flow rate, Q, and then another for the shear rate. Because the axial velocity, v _z, is dependent on the radial position

$$Q=\int\limits_{_{0}}^{_{ro}}{2\pi r{{v}_{z}}\text{d}r=}\int\limits_{_{o}}^{_{ro}}{{{v}_{z}}\text{(}r\text{)}\,\text{d}\,\text{(}{{r}^{2}}\text{)}=}\,\pi \int\limits_{_{0}}^{_{ro}}{{{v}_{z}}\text{2}r\,\text{d}r+\pi \int\limits_{_{{{v}_{z}}\,\text{at}r=0}}^{_{{{v}_{z}}\,\text{at}r=ro}}{{{r}^{2}}\,\text{d}{{v}_{z}}}}$$

(3B.11)

Assuming a no slip boundary condition, that is, v _z = 0 at r = r _o, the right-hand side of the above equation can be written as

$$\pi =\int\limits_{_{o}}^{_{ro}}{{{v}_{z}}\text{2}r\,\text{d}r\,-\pi }\int\limits_{_{\,o}}^{_{{{v}_{z}}}}{{{r}^{2}}\,\text{d}{{v}_{z}}}$$

(3B.12)

The above equation can be integrated and noting that v _z = 0 at r = r _o, and r = (r _o/σ_w)σ_rz, one can obtain the expression

$$Q=\left( -\frac{\pi r_{\text{o}}^{3}}{\sigma _{\text{w}}^{3}}\,\, \right)\int\limits_{_{0}}^{_{{{\sigma }_{\text{w}}}}}{\sigma _{rz}^{2}\left( -\frac{\text{d}{{v}_{z}}}{\text{d}r}\,\, \right)d{{\sigma }_{rz}}}$$

(3B.13)

Multiplying both sides of the above equation by 4 and rearranging, we get

$$\frac{4Q}{\pi r_{\text{o}}^{3}}\,\left( \frac{4}{\sigma _{\text{w}}^{3}}\,\, \right)\int\limits_{_{0}}^{_{{{\sigma }_{\text{w}}}}}{\sigma _{rz}^{2}\left( -\frac{\text{d}{{v}_{z}}}{\text{d}r}\,\, \right)d{{\sigma }_{rz}}}$$

(3B.14)

The above equation can be used for deriving a general solution for tube flow and specific expressions for the volumetric flow rates of fluids exhibiting different rheological behaviors. For a Newtonian fluid, noting that the shear rate, $-(\text{d}{{v}_{z}}/\text{d}r)=\frac{{{\sigma }_{rz}}}{\eta },$ it can be shown that

$$\frac{4Q}{\pi r_{\text{o}}^{3}}=\,\frac{{{\sigma }_{\text{w}}}}{\eta }=\,\frac{1}{\eta }\frac{D\Delta p}{4L}$$

(3B.15)

or that the viscosity of a Newtonian fluid

$$\eta =\frac{(D\Delta p/4L)}{(8{{\overset{\mathbf{--}}{\mathop{v}}\,}_{z}}/D)}\text{,}\,\text{that}\,\text{is,}\,\text{ }\!\!\eta\!\!\text{ }\sim \frac{1}{(8{{\overset{\mathbf{--}}{\mathop{v}}\,}_{z}})}\sim \text{flow}\,\text{time}$$

(3B.16)

Equation 3B.16 is the basis for calculation of viscosity of a Newtonian fluid using glass capillary viscometer. It should also be recognized that $(4Q/\pi r_{\text{o}}^{3})=(32Q/\pi {{D}^{3}})$ gives the shear rate for Newtonian fluids but not for non-Newtonian fluids and it is called pseudoshear rate. Additional steps are required to obtain an expression for the true shear rate.

Differentiating both sides of Eq. 3B.14 and setting the limits

$$\frac{\text{d }\!\![\!\!\text{ }\sigma _{\text{w}}^{3}(4Q/\pi r_{\text{o}}^{3})}{\text{d}{{\sigma }_{\text{w}}}}=4\sigma _{\text{w}}^{2}\left( -\frac{\text{d}{{v}_{z}}}{\text{d}r} \right)$$

(3B.17)

Further differentiating the left-hand side and rearranging, one obtains the general solution for the true shear rate in tube flow

$$\left( \frac{\text{d}{{v}_{z}}}{\text{d}r} \right)=\left( \frac{3}{4} \right)\,\frac{4Q}{\pi r_{\text{o}}^{3}}+\frac{{{\sigma }_{\text{w}}}}{4}\,\frac{\text{d(4Q/}\pi r_{\text{o}}^{3}\text{)}}{\text{d}{{\sigma }_{\text{w}}}}$$

(3B.18)

Equation 3B.18 is known as the Weissenberg–Rabinowitsch–Mooney (WRM) equation in honor of the three rheologists who have worked on this problem. An alternate equation can be derived for fluids obeying the power law model between shear stress and the pseudoshear rate

$$\sigma =K'\,{{(4Q/\pi r_{\text{o}}^{3})}^{n'}}$$

(3B.19)

$$\left( -\frac{\text{d}{{v}_{z}}}{\text{d}r} \right)=\frac{3n'+1}{4n'}\,\left( \frac{4Q}{\pi r_{\text{o}}^{3}} \right)$$

(3B.20)

To use the WRM equation, the steps involved are: (1) using a tube/pipe/capillary of known diameter (D) and length (L), several sets of volumetric flow rate (Q) versus pressure drop (ΔP) data are obtained under isothermal fully developed and no slip at the pipe wall conditions.

(2) The quantities $(4Q/\pi {{r}^{3}}_{0})$ are calculated, plotted, and a smooth curve fitted.

(3) At each value of $(4Q/\pi {{r}^{3}}_{0})$ the value of: $\text{d}(4Q/\pi {{r}^{3}}_{0})/\text{d}{{\sigma }_{\text{w}}}$ is determined and the true shear rate calculated using Eq. 3B.18. Alternatively, for many foods, plots of the quantities: log $(4Q/\pi {{r}^{3}}_{0}),$ and log (D Δp/4 L) are straight lines with slopes n′ that can be used in Eq. 3B.20 to obtain the true shear rate.

Fig. 3B.2
Decision tree for using a capillary/tube viscometer

(4) Generally, for a given fluid, values of n′ and of n of the power law model: $\sigma = K'{(4Q/\pi r_0^3)^{n'}}$ will be the same. However, the value of K′ in is related to the consistency index K by the equation

$$K' = K{\left( {\frac{{3n + 1}}{{4n}}} \right)^n}$$

(3B.21)

As an example, for n = 0.3, K′ = 1.15 K.

From Eq. 3B.19, an expression can be derived for a pipe-flow apparent viscosity (η _ap) based on the diameter D and the average axial velocity in the tube $({\bar \nu _z})$

$${\eta _{{\rm{ap}}}} = K'{\left( {\frac{{8{{\mathop v\limits^{{\bf{ - - }}} }_z}}}{D}} \right)^{n' - 1}}$$

(3B.22)

When n = n′, together with Eqs. 3B.21 and 3B.22, one can obtain the generalized Reynolds number (GRe)

$$\text{GRe}=\frac{{{D}^{n}}\overset{\mathbf{--}}{\mathop{v}}\,_{_{z}}^{2-n}\rho }{{{8}^{(n-1)}}K}{{\left( \frac{4n}{3n+1} \right)}^{n}}$$

(3B.23)

Some of the considerations in selecting a capillary/tube viscometer for viscosity measurement are shown in Fig. 3B.2.

Appendix 3 Analysis of Flow in a Cone-Plate Geometry

We assume that the cone of radius r_o and cone angle θ_o is on top over the plate rotating with an angular velocity Ω while the bottom plate is stationary as shown in Fig. 3.7. We consider the equation of motion in spherical coordinates r,θh,ϕ. For steady fully developed flow, the velocity components v_θ = 0 and v_r = 0, and v _ϕ is a function of r and θ. In addition, we consider pressure variations (body forces) to be negligible and that the value of θ _o is very small, < 0.1 rad. The r-component reduces

$$ \rho \frac{{v_\phi ^{\rm{2}}}}{r}\phi= \frac{1}{{{r^{\rm{2}}}}}\frac{\partial }{{\partial r}}({r^{\rm{2}}}{\sigma _{rr}}) - \frac{{{\sigma _{\theta \theta }} + {\sigma _{\phi \phi }}}}{R} $$

(3C.1)

The θ-component reduces to

$$ \frac{1}{{r\sin \theta }}\frac{\partial }{{\partial \theta }}({\sigma _{\theta \theta }}\sin \theta ) - \frac{{{\rm{cot}}\theta }}{r}{\sigma _\phi }_\phi= 0 $$

(3C.2)

The ϕ-component of the equation of motion reduces to

$$ \frac{1}{r}\frac{{\partial {\sigma _{\theta \phi }}}}{{\partial \theta }} + \frac{{{\rm{2cot}}\theta }}{r}{\sigma _\theta }_\phi= 0 $$

(3C.3)

We note that the ϕ-component contains the shear stress σ _θϕ. The boundary conditions are: v _ϕ = 0 at θ = π/2 because the plate is stationary and v_ϕ = rΩ cos θ _o at θ = (π/2) – θ _o. First, we will derive an expression for the shear stress by conducting a simple torque balance on the plate

$$M = \int\limits_0^{2\pi } {\int\limits_0^{{r_o}} {{r^2}} {\sigma _{\theta \phi }}{\rm{d}}r{\rm{d}}\phi {\rm{ = }}\frac{{2\pi r_0^3}}{3}} {\sigma _{\theta \phi }}|\pi /2$$

(3C.4)

Integration of Eq. 3C.3 results in the expression: σ _θϕ |_(π/2) = σ _θϕ(θ) sin² θ. Therefore,

$$ {\sigma _{\theta \phi }}(\theta ) = \frac{{3M}}{{2\pi r_o^3{{\sin }^2}\theta }} $$

(3C.5)

When ${\theta _o} < 0.1\,{\rm{rad,}}\,{\rm{si}}{{\rm{n}}^{\rm{2}}}\left( {\frac{\pi }{2}{\theta _o}} \right) \sim 1,$ so that

$$ {\sigma _{\theta \phi }} = \frac{{{\rm{3}}M}}{{{\rm{2}}\pi r_o^3}} $$

(3C.6)

Defining T_cn as the torque per unit area, Eq. 3C.4 can be written in terms of the cone diameter D

$$ {\sigma _{\theta \phi }} = \frac{{3{T_{cn}}}}{D} $$

(3C.7)

From Eqs. 3C.5 and 3C.6, it follows that the shear stress in a cone-plate geometry is essentially uniform.

To obtain an expression for shear rate, we can simply say that for low values of the cone angle θ_o we need not distinguish between sin θ_o and θ_o (Whorlow 1980) when the shear rate at radius r will be

$$ .{\gamma _r} = \frac{{r\Omega }}{{r{\theta _o}}} = \frac{\Omega }{{{\theta _o}}} $$

(3C.8)

Alternatively, we can consider the velocity distribution (Brodkey 1967) for v_ϕ

$$ \frac{{{v_\phi }}}{{r\,\sin \theta }} = \int\limits_{\theta= \pi /2}^\theta{\frac{{{\rm{d(}}v\phi /r\sin \theta )}}{{{\rm{d}}\theta }}} {\rm{d}}\theta$$

(3C.9)

We can also substitute for from Eq. $3{\rm{C}}{\rm{.3,}}d\theta= \frac{1}{2}\frac{{\sin \theta }}{{\cos \theta }}\frac{1}{{{\sigma _{\theta \phi }}}}$ and also switch limits: σ_θϕ at θ = 0 and σ_plate at θ = π/2. For a Newtonian fluid, we can substitute in 3C.9 the relationship between shear rate and shear stress and get 3C.11

$$ {\sigma _{\theta \phi }} = \eta \left[ {\sin \theta \frac{{{\rm{d(}}{{\rm{v}}_\phi }/r\sin \theta )}}{{{\rm{d}}\theta }}} \right] $$

(3C.10)

$$ \frac{{{v_\phi }}}{{r\sin \theta }} = \int\limits_\theta ^{\pi /{\rm{2}}} {\frac{1}{\eta }} \frac{{{\sigma _{\theta \phi }}}}{{\sin \theta }}{\rm{d}}\theta$$

(3C.11)

After substituting for σ_θϕ in 3C.11 and performing the integration results in

$$ \frac{{{v_\phi }}}{r} = \frac{{{\rm{3}}T}}{{{\rm{4}}\eta {r_o}}}\left[ {\cot {\theta _o} - 1{\rm{n}}\left( {\tan \frac{\theta }{2}} \right)\sin \theta } \right] $$

(3C.12)

Equation 3C.12 can be written for the boundary condition at $\theta= \frac{\pi }{2}--{\theta _0}:$

$$ \Omega \cos {\theta _o} = \frac{{{\rm{3}}T}}{{{\rm{4}}\eta {r_o}}}\left[ {\tan {\theta _o} - \cos {\theta _o}1{\rm{n}}\left( {\frac{{1 - \tan {\theta _o}}}{{1 + \tan {\theta _o}}}} \right)} \right] $$

(3C.13)

For low values of the cone angle θ_o, the above equation becomes

$$ \begin{array}{l}\Omega= \frac{{{\rm{3}}T}}{{{\rm{4}}\eta {r_o}}}({\theta _o} + {\theta _o}) = \frac{{3T{\theta _o}}}{{\eta D}}\\\end{array} $$

$$ {\rm{Therefore}},{\sigma _{\theta \phi }} \cong \eta \frac{\Omega }{{{\theta _o}}}. $$

References

Abdel-Khalik, S., Hasseger, O., and Bird, R. B. 1974. Prediction of melt viscosity from viscosity data. Polym. Eng. Sei. 14: 859–867.Google Scholar
Abdelrahim, K. A., Ramaswamy, H. S., and Van de Voort, F. R. 1995. Rheological properties of starch solutions under aseptic processing temperatures. Food Res. Int. 28: 473–480.Google Scholar
Alexander, M. and Dalgleish, D. G. 2006. Dynamic light scattering techniques and their applications in food science. Food Biophysics. 1: 2–33.Google Scholar
Apgar, J., Tseng, Y., Fedorov, E., Herwig, M. B., Almo, S. C. and Wirtz, D. 2000. Multiple-particle tracking measurements of heterogeneities in solutions of actin filaments and actin bundles. Biophysical Journal. 79: 1095–1106.Google Scholar
Arola, D. F., Powell, R. L., Barrall, G. A. and McCarthy, M. J. 1999. Pointwise observations for rheological characterization using nuclear magnetic resonance imaging. J. Rheol. 43: 9–30.Google Scholar
Barnes, H. A. and Carnali, J. O. 1990. The vane-in-cup as a novel rheometer geometry for shear thinning and thixotropic materials. J. Rheol. 34: 841–865.Google Scholar
Barnes, H. A., Hutton, J. F., and Walters, K. 1989. An Introduction to Rheology, Elsevier Science Publishers B.V., Amsterdam, The Netherlands.Google Scholar
Bird R. B., Armstrong, R. C., and Hasseger, O. 1977. Dynamics of Polymeric Liquids. John Wiley, New York.Google Scholar
Bird, R. B., Hasseger, O., and Abdel-Khalik S. I. 1974. Co-rotational rheological models and the Goddard expansion. AIChE J 20: 1041–1066.Google Scholar
Bistany, K. L. and Kokini, J. L. 1983. Dynamic viscoelastic properties of foods in texture control. J. Rheol 27: 605–620.Google Scholar
Bongenaar, J. J. T., Kossen, N. W. F., Metz, B., and Meijboom, F. W. 1973. A method for characterizing the rheological properties of viscous fermentation broths. Biotechnol. Bioeng. 15: 201–206.Google Scholar
Briggs, J. L. and Steffe, J. F. 1996. Mixer viscometer constant (k′) for the Brookfield small sample adapter and flag impeller. J. Texture Stud. 27: 671–677.Google Scholar
Brodkey, R. S. 1967. The Phenomena of Fluid Motions, Addison-Wesley, Reading, MA.Google Scholar
Campanella, O. H. and Peleg, M. 1987. Analysis of the transient flow of mayonnaise in a coaxial cylinder viscometer. J. Rheol. 31: 439–452.Google Scholar
Campanella, O. H., Popplewell, L. M., Rosenau, J. R., and Peleg, M. 1987. Elongational viscosity measurements of melting American process cheese. J. Food Sei. 52: 1249–1251.Google Scholar
Cannon Instrument Co. 1982. Instructions for the Use of the Cannon-Ubbelohde Dilution Viscometer, State College, PA.Google Scholar
Casiraghi, E. M., Bagley, E. B., and Christiansen, D. D. 1985. Behavior of mozzarella, cheddar and processed cheese spread in lubricated and bonded uniaxial compression. J. Texture Stud. 16: 281–301.Google Scholar
Castell-Perez, M. E., Steffe, J. F., and Morgan, R. G. 1987. Adaptation of a Brookfield (HBTD) viscometer for mixer viscometer studies. J. Texture Stud. 18: 359–365.Google Scholar
Chamberlain, E. K. 1999. Rheological properties of acid converted waxy maize starches: effect of solvent, concentration and dissolution time. Ph.D. thesis, Cornell University, Ithaca, NY.Google Scholar
Champenois, Y. C., Rao, M. A., and Walker, L. P. 1998. Influence of-amylase on the viscoelastic properties of starch-gluten pastes and gels. J. Sei. Food Agric. 127–133.Google Scholar
Chatraei, S. H., Macosko, C. W., and Winter, H. H. 1981. Anew biaxial extensional rheometer. J. Rheol. 25: 433–443.Google Scholar
Choi, Y. J., McCarthy, K. L., and McCarthy, M. J. 2002. Tomographic techniques for measuring fluid flow properties. J. Food Sei. 67(7): 2718–2724.Google Scholar
Clark, A. H. and Ross-Murphy, S. B. 1987. Structural and mechanical properties of biopoly. gels. Adv. Polym. Sei. 83: 57–192Google Scholar
Clark, R. 1997. Evaluating syrups using extensional viscosity. Food Technol. 511: 49–52.Google Scholar
Cogswell, F. N. 1972. Converging flow of polym. melts in extrusion dies. Polym. Eng. Sei. 12: 64–73.Google Scholar
Cogswell, F. N. 1978. Converging flow and stretching flow: a compilation. J. Non-Newtonian FluidMech. 4: 23–38.Google Scholar
Comby, S., Doublier, J. L., and Lefebvre, J. 1986. Stress-relaxation study of high-methoxyl pectin gels, in Gums and Stabilisers for the Food Industry 3, eds., G. O. Phillips, D. J. Wedlock, and P. A. Williams, pp. 203-212. Elsevier Science Publishers, New York.Google Scholar
Corredig, M. and Alexander, M. 2008, Food emulsions studied by DWS: recent advances. Trends in Food Science & Technology. 19:67–75Google Scholar
Corrigan, A.M. and Donald, A.M. 2009. Particle tracking microrheology of gel-forming amyloid fibril networks. European Physical Journal E 28 (4):457–462.Google Scholar
Cox, W. P. and Merz, E. H. 1958. Correlation of dynamic and steady flow viscosities. J. Polym. Sei. 28: 619–622.Google Scholar
Crocker, J. C.; Weeks, E. R. Software package for particle tracking, available from: http://www.physics.emory.edu/~weeks/idl/tracking.html, (cited in Moschakis et al. (2006).
Da Silva, P. M. S., Oliveira, J. C., and Rao, M. A. 1997. The effect of granule size distribution on the rheological behavior of heated modified and unmodified maize starch dispersions. J. Texture Stud. 28: 123–138.Google Scholar
Dahbi, L., Alexander, M., Trappe, V., Dhont, J. K. G. and Schurtenberger, P. 2010. Rheology and structural arrest of casein suspensions. Journal of Colloid and Interface Science. 342:564–570.Google Scholar
Dail, R. V. and Steffe, J. F. 1990. Rheological characterization of crosslinked waxy maize starch solutions under low acid aseptic processing conditions using tube viscometry techniques. J. Food Sei. 55: 1660–1665.Google Scholar
Dasgupta, B. R., Tee, S-Y., Crocker, J. C., Frisken, B. J. and Weitz, D. A. 2001. Microrheology of polyethylene oxide using diffusing wave spectroscopy and single scattering. Physical Review E 65: 051505.Google Scholar
Dealy, J. M. 1982. Rheometers for Molten Polymers. A Practical Guide to Testing and Property Measurement, Van Nostrand Reinhold, New York.Google Scholar
Dickie, A. M. and Kokini, J. L. 1982. Use of the Bird-Leider equation in food rheology. J. Food Process Eng. 5: 157–174.Google Scholar
Diehl, K. C., Hamann, D. D., and Whitfield, J. K. 1979. Structural failure in selected raw fruits and vegetables. J. Text. Stud. 10:371–400.Google Scholar
Dogan, N., McCarthy, M. J., and Powell, R. L. 2002. In-line measurement of rheological parameters and modeling of apparent wall slip in diced tomato suspensions using ultrasonics. J. Food Sei. 67(6): 2235–2240.Google Scholar
Dogan, N., McCarthy, M. J., and Powell, R. L. 2003. Comparison of in-line consistency measurement of tomato concentrates using ultrasonics and capillary methods. J. Food Process Eng. 25(6): 571–587.Google Scholar
Doraiswamy, D., Mujumdar, A. N., Tsao, I., Beris, A. N., Danforth, S. C., and Metzner, A. B. 1991. The Cox-Merz rule extended: a rheological model for concentrated suspensions and other materials with a yield stress. J. Rheol. 35: 647–685.Google Scholar
Dzuy, N. Q. and Boger, D. V. 1983. Yield stress measurement for concentrated suspensions. J. Rheol. 27: 321–349.Google Scholar
Dzuy, N. Q. and Boger, D. V. 1985. Direct yield stress measurement with the vane method. J. Rheol. 29: 335–347.Google Scholar
Elliott, J. H. and Ganz, A. J. 1971. Modification of food characteristics with cellulose hydrocolloids, I. Rheological characterization of an organoleptic property. J. Texture Stud. 2: 220–229.Google Scholar
Elliott, J. H. and Ganz, A. J. 1977. Salad dressings-preliminary rheological characterization. J. Texture Stud. 8: 359–371.Google Scholar
Ferry, J. D. 1980. Viscoelastic Properties of Polymers, John Wiley, New York.Google Scholar
Gardel, M. L., Valentine, M.T. and Weitz, D. A. 2005. Microrheology, in Microscale Diagnostic Techniques, ed. K. Breuer., p. 1–54, New York: Springer-Verlag.Google Scholar
Genovese, D. B. and Rao, M. A. 2003a. Vane yield stress of starch dispersions. J. Food Sei. 68(7): 2295–2301.Google Scholar
Genovese, D. B. and Rao, M. A. 2003b. Apparent viscosity and first normal stress of starch dispersions: role of continuous and dispersed phases, and prediction with the Goddard-Miller model. Appl. Rheol. 13(4): 183–190.Google Scholar
Genovese, D. B., Acquarone, V. M., Youn, K.-S., and Rao, M. A. 2004. Influence of fructose and sucrose on small and large deformation rheological behavior of heated Amioca starch dispersions. Food Sei. Technol. Int. 10(1): 51–57.Google Scholar
Giboreau, A., Cuvelier, G., and Launay, B. 1994. Rheological behavior of three biopolymer/water systems with emphasis on yield stress and viscoelastic properties. J. Texture Stud. , 25: 119–137.Google Scholar
Gosal WJ, Clark AH, Ross-Murphy SB. 2004. Fibrillar β-lactoglobulin gels: Part 2. Dynamic mechanical characterization of heat-set systems. Biomacromolecules 5(6):2420–9.Google Scholar
Grikshtas, R. and Rao, M. A. 1993. Determination of slip velocities in a concentric cylinder viscometer with Mooney and Kiljanski methods. J. Texture Stud. 24: 173-184. Grosso, C. R. F. and Rao, M. A. 1998. Dynamic rheology of structure development in low-methoxyl pectin + Ca2 + + sugar gels. Food Hydrocolloids 12: 357–363.Google Scholar
Hamann, D. D. 1983. Structural failure in solid foods, in Physical Properties of Foods, eds. M. Peleg, and E. B. Bagley, pp. 351-383 AVI Publ., Westport, CT.Google Scholar
Hamann, D. D. 1987. Methods for measurement of rheological changes during thermally induced gelation of proteins. Food Technol. 41(3): 100, 102–108.Google Scholar
Hansen, L. M., Hoseney, R. C., and Faubion, J. M. 1990. Oscillatory probe rheometry as a tool for determining the rheological properties of starch-water systems. J. Texture Stud. 21: 213–224.Google Scholar
James, A. E., Williams, D. J. A., and Williams, P. R. 1987. Direct measurement of static yield properties of cohesive suspensions. Rheol. Acta 26: 437–446.Google Scholar
Jao, Y. C., Chen, A. H., Lewandowski, D., and Irwin, W. E. 1978. Engineering analysis of soy dough rheology in extrusion. J. Food Process Eng. 2: 97–112.Google Scholar
Keentok, M. 1982. The measurement of the yield stress of liquids. Rheol. Acta 21: 325–332.Google Scholar
Khagram, M., Gupta, R. K., and Sridhar, T. 1985. Extensional viscosity of xanthan gum solutions. J. Rheol. 29: 191–207.Google Scholar
Kokini, J. L. and Dickie, A. 1981. An attempt to identify and model transient viscoelastic flow in foods. J. Texture Stud. 12: 539–557.Google Scholar
Komatsu, H. and Sherman, P. 1974. A modified rigidity modulus technique for studying the rheological properties of w/o emulsions containing microcrystalline wax. J. Texture Stud. 5: 97–104.Google Scholar
Kulicke, W.-M. and Porter, R. S. 1980. Relation between steady shear flow and dynamic rheology. Rheol. Acta 19: 601–605.Google Scholar
Lai, K. P., Steffe, J. F, and Ng, P. K. W. 2000. Average shear rates in the Rapid Visco Analyser (RVA) mixing system. Cereal Chem. 77(6): 714–716.Google Scholar
Larson, R. G. 1985. Constitutive relationships for polymeric materials with power-law distributions of relaxation times. Rheol. Acta 24: 327–334.Google Scholar
Leider, P. J. and Bird, R. B. 1974. Squeezing flow between parallel disks-I. Theoretical analysis. Ind. Eng. Chem. Fundam. 13: 336–341.Google Scholar
Leppard, W. R. and Christiansen, E. B. 1975. Transient viscoelastic flow of polymer solutions. Am. Inst. Chem. Engrs. J. 21: 999–1006.Google Scholar
Liao, H.-J. 1998. Simulation of continuous sterilization offluid food products: theroleofthermorheological behavior of starch dispersion and process, Ph.D. thesis, Cornell University, Ithaca, NY.Google Scholar
Lopes da Silva, J. A. L., Gonpalves, M. P., and Rao, M. A. 1993. Viscoelastic behavior of mixtures of locust bean gum and pectin dispersions. J. Food Eng. 18: 211–228.Google Scholar
Lopes da Silva, J. A. L., Gonsalves, M. P., and Rao, M. A. 1994. Influence of temperature on dynamic and steady shear rheology of pectin dispersions. Carbohydr. Polym. 23: 77–87.Google Scholar
Lopes da Silva, J. A., Rao, M. A., and Fu, J.-T. 1998. Rheology of structure development and loss during gelation and melting, in Phase/State Transitions in Foods: Chemical, Rheological and Structural Changes, eds. M. A. Rao and R. W. Hartel, pp. 111–156, Marcel Dekker, Inc., NY.Google Scholar
Macosko, C. W. 1994. Rheology: Principles, Measurements and Applications, VCH Publishers, New York.Google Scholar
Maranzano, B. J. and Wagner, N. J. 2002. Flow-small angle neutron scattering measurements of colloidal dispersion microstructure evolution through the shear-thickening transition. J. Chem. Phys. 117: 10291–10302.Google Scholar
Mason, P. L., Bistany, K. L., Puoti, M. G., and Kokini, J. L. 1982. A new empirical model to simulate transient shear stress growth in semi-solid foods. J. Food Process Eng. 6: 219–233.Google Scholar
Mason, T. G. 2000. Estimating the viscoelastic moduli of complex fluids using the generalized Stokes-Einstein equation. Rheol. Acta 39:371–378.Google Scholar
Mason, T. G. and Weitz, D. A. 1995. Optical measurements of frequency-dependent linear viscoelastic moduli of complex fluids. Physical Review Letters. 74(7): 1250–1253.Google Scholar
Mason, T. G., Ganesan, K., van Zanten, J. H., Wirtz, D. and Kuo, S. C. 1997. Particle Tracking Microrheology of Complex Fluids. Physical Review Letters. 79 (17):3282–3285.Google Scholar
Matsumoto, T., Hitomi, C., and Onogi, S. 1975. Rheological properties of disperse systems of spherical particles in polystyrene solution at long time-scales. Trans. Soc. Rheol. 194: 541.Google Scholar
McCarthy, K. L. and Seymour, J. D. 1993. A fundamental approach for the relationship between the Bostwick measurement and Newtonian fluid viscosity. J. Texture Stud. 24(1): 1–10.Google Scholar
McCarthy, K. L. and Seymour, J. D. 1994. Gravity current analysis of the Bostwick consistometer for power law foods. J. Texture Stud. 25(2): 207–220.Google Scholar
McKelvey, J. N. 1962. Polymer Processing, John Wiley and Sons, New York.Google Scholar
Metz, B., Kossen, N. W. F., and van Suijdam, J. C. 1979. The rheology of mould suspensions in Advances in Biochemical Engineering, eds. Ghose, T. K. A. Fiechter, and N. Blakebrough, Vol. 2, pp. 103–156, New York: Springer Verlag.Google Scholar
Metzner, A. B. and Otto, R. E. 1957. Agitation of non-Newtonian fluids. Am. Inst. Chem. Eng. J. 3: 3–10.Google Scholar
Michaels, A. S. and Bolger, J. C. 1962. The plastic flow behavior of flocculated kaolin suspensions. Ind. Eng. Chem. Fund. 1: 153–62.Google Scholar
Mills, R and Kokini, J. L. 1984. Comparison of steady shear and dynamic viscoelastic properties of guar and karaya gums. J. Food Sei. 49: 1-4 and 9.Google Scholar
Mitchell, J. R. 1984. Rheological techniques, in Food Analysis: Principles and Techniques, eds. D. W. Gruenwedel and J. R. Whitaker, pp. 151–220, Marcel Dekker, New York.Google Scholar
Mooney, M. 1931. Explicit formulas for slip and fluidity. J. Rheol. 2: 210–222.Google Scholar
Morris, E. R. 1981. Rheology of hydrocolloids, in Gums and Stabilisers for the Food Industry 2, eds. G. O. Philips, D. J. Wedlock, and P. A. Williams, pp. 57-78, Pergamon Press Ltd., Oxford, Great Britain.Google Scholar
Morris, E. R., Cutler, A. N., Ross-Murphy, S. B. and Rees, D. A. 1981. Concentration and shear rate dependence of viscosity in random coil polysaccharide solutions. Carbohydr. Polym. 1: 5–21.Google Scholar
Moschakis, T., Murray B. S. & Dickinson, E. 2006. Particle tracking using confocal microscopy to probe the microrheology in a phase-separating emulsion containing nonadsorbing polysaccharide. Langmuir 22:4710–4719.Google Scholar
Moschakis, T., Murray B. S. & Dickinson, E. 2010. On the kinetics of acid sodium caseinate gelation using particle tracking to probe the microrheology. J. Colloid and Interface Science 345(2: 278–285.Google Scholar
Nicolas, Y. and Paques, M. 2003. Microrheology: an experimental technique to visualize food structure behavior under compression-extension deformation conditions. J Food Sei. 68(6): 1990–1994.Google Scholar
Nussinovitch, A., Kaletunc, G., Normand, M. D., and Peleg, M. 1990. Recoverable work versus asymptotic relaxation modulus in agar, carrageenan and gellan gels. J. Texture Stud. 21: 427–438.Google Scholar
Oakenfull, D. 1984. A method for using measurements of shear modulus to estimate the size and thermodynamic stability of junction zones in non-covalently cross-linked gels. J. Food Sei. 49: 1103-1104, 1110.Google Scholar
Oakenfull, D. G., Parker, N. S., and Tanner, R. I. 1989. Method for determining absolute shear modulus of gels from compression tests. J. Texture Stud. 19: 407–417.Google Scholar
Okechukwu, P. E., Rao, M. A., Ngoddy, P. O., and McWatters, K. H. 1991. Rheology of sol-gel thermal transition in cowpea flour and starch slurry. J. Food Sei. 56: 1744–1748.Google Scholar
Owen, S. R., Tung, M. A., and Paulson, A. T. 1992. Thermorheological studies of food polymer dispersions. J. Food Eng. 16: 39–53.Google Scholar
Padmanabhan, M. 1995. Measurement of extensional viscosity of viscoelastic liquid foods. J. Food Eng. 25:311–327.Google Scholar
Padmanabhan, M. and Bhattacharya, M. 1993. Planar extensional viscosity of corn meal dough. J. Food Eng. 18: 389–411.Google Scholar
Peleg, M. 1980. Linearization of relaxation and creep curves of solid biological materials. J. Rheol. 24:451–463.Google Scholar
Perkins, T. T., Smith, D. E., and Chu, S. 1997. Single polymer dynamics in an elongational flow. Science 276:2016–2021.Google Scholar
Pine, D. J., Weitz, D. A., Chaikin, P. M. and Herbolzheimer, E. 1988. Diffusing-wave spectroscopy. Physical Review Letters 60(2): 1134–1137.Google Scholar
Plazek, D. J. 1996. 1995 Bingham medal address: Oh, thermorheological simplicity, wherefore art thou? J. Rheol. 40: 987–1014.Google Scholar
Qiu, C.-G. and Rao, M. A. 1988. Role of pulp content and particle size in yield stress of apple sauce. J. Food Sei. 53: 1165–1170.Google Scholar
Qiu, C.-G. and Rao, M. A. 1989. Effect of dispersed phase on the slip coefficient of apple sauce in a concentric cylinder viscometer. J. Texture Stud. 20: 57–70.Google Scholar
Rao, M. A. 1975. Measurement of flow properties of food suspensions with a mixer. J. Texture Stud. 6: 533–539.Google Scholar
Rao, M. A. 1977a. Rheology of liquid foods-a review. J. Texture Stud. 8: 135–168.Google Scholar
Rao, M. A. 1977b. Measurement of flow properties of fluid foods-developments, limitations, and interpretation of phenomena. J. Texture Stud. 8: 257–282.Google Scholar
Rao, M. A. 1992. Measurement of viscoelastic properties of fluid and semisolid foods, in Viscoelastic Properties of Food, eds. M. A. Rao and J. F. Steffe, pp. 207-232, Elsevier Applied Science Publishers, London.Google Scholar
Rao, M. A. 2005. Rheological properties of fluid foods, in Engineering Properties of Foods, eds. M. A. Rao and S. S. H. Rizvi, and A. K. Datta, 3rd ed., pp. 41–99, CRC Press, Boca Raton, FL.Google Scholar
Rao, M. A. 2013. Food microstructure and rheology, Chapter 13, in Food Microstructures: Microscopy, Measurement and Modelling, eds: Vic Morris and Kathy Groves, Woodhead Publishing Ltd., Cambridge, UK.Google Scholar
Rao, M. A. and Cooley, H. J. 1984. Determination of effective shear rates of complex geometries. J. Texture Stud. 15: 327–335.Google Scholar
Rao, M. A. and Cooley, H. J. 1992. Rheology of tomato pastes in steady and dynamic shear. J. Texture Stud. 23:415–425.Google Scholar
Rao, M. A. and Cooley, H. J. 1993. Dynamic rheological measurement of structure development in high-methoxyl pectin/fructose gels. J. Food Sei. 58: 876–879.Google Scholar
Rao, M. A., Cooley, H. J., and Liao, H.-J. 1999. High temperature rheology of tomato puree and starch dispersion with a direct-drive viscometer. J. Food Process Eng. 22: 29–40.Google Scholar
Rao, V. N. M., Delaney, R. A. M., and Skinner, G. E. 1995. Rheological properties of solid foods, in Engineering Properties of Foods, eds. M. A. Rao and S. S. H. Rizvi, 2nd ed., pp. 55–97, Marcel Dekker, Inc., New York.Google Scholar
Rayment, P., Ross-Murphy, S. B., and Ellis, P. R. 1998. Rheological properties of guar galactomannan and rice starch mixtures. II. Creep measurements. Carbohydr. Polym. 35: 55–63.Google Scholar
Rieger, F. and Novak, V. 1973. Power consumption of agitators in highly viscous non-Newtonian liquids. Trans. Inst. Chemi. Eng. 51: 105–111.Google Scholar
Roberts, I. 2003. In-line and on-line rheology measurement of food, in “Texture in Food, Volume 1: SemiSolid Foods,” pp. 161-182, edited by Brian M. McKenna, Woodhead Publishing Ltd., Cambridge, UK.Google Scholar
Saunders, P. R. and Ward, A. G. 1954. An absolute method for the rigidity modulus of gelatine gel, in Proceedings of the Second International Congress on Rheology, ed. V. G. W. Harrison, pp. 284–290. Academic Press, New York.Google Scholar
Schlichting, H. 1960. Boundary Layer Theory, McGraw-Hill, New York.Google Scholar
Senouci, A. and Smith, A. C. 1988. An experimental study of food melt rheology. I. Shear viscosity using a slit die viscometer and a capillary rheometer. Rheol. Acta 27: 546–554.Google Scholar
Sestak, J., Zitny, R., and Houska, M. 1983. Simple rheological models of food liquids for process design and quality assessment. J. Food Eng. 2: 35–49.Google Scholar
Shama, F. and P. Sherman. 1969. The influence of work softening on the viscoelastic properties of butter and margarine. J. Texture Stud. 1: 196–205.Google Scholar
Shama, F. and Sherman, P. 1973. Identification of stimuli controlling the sensory evaluation of viscosity. II. Oral methods. J. Texture Stud. 4:111–118.Google Scholar
Sharma, S. K., Hill, A. R., and Mittal, G. S. 1992. Evaluation of methods to measure coagulation time of ultrafiltered milk. Milchwissenschaft 47(11): 701–704.Google Scholar
Sharma, S. K., Hill, A. R., Goff, H. D., and Yada, R. 1989. Measurement of coagulation time and curd firmness of renneted milk using a Nametre viscometer. Milchwissenschaft 44(11): 682–685Google Scholar
Sherman, P. 1966. The texture of ice cream 3. Rheological properties of mix and melted ice cream. J. Food Sei. 31: 707–716.Google Scholar
Sherman, P. 1970. Industrial Rheology, Academic Press, New York.Google Scholar
Sherman, P. and Benton, M. 1980. Influence of skim milk powder/recodan R S ratio on the viscoelasticity of groundnut oil-in-water imitation milks. J. Texture Stud. 11: 1–13.Google Scholar
Shomer, I., Rao, M. A., Bourne, M. C., and Levy, D. 1993. Rheological behavior of potato tuber cell suspensions during temperature fluctuations and cellulase treatments. J. Sei. Food. Agric. 63: 245–250.Google Scholar
Smith, T. L., Ferry, J. D., and Schremp, F. W. 1949. Measurement of the mechanical properties of polymer solutions by electromagnetic transducers. J. App. Phys. 20: 144–153.Google Scholar
Sridhar, T., Tirtaatmadja, V., Nguyen, D. A., and Gupta, R. K. 1991. Measurement of extensional viscosity of polymer solutions. J. Non-Newtonian Fluid Mech. 40: 271–280.Google Scholar
Stainsby, G., Ring, S. G., and Chilvers, G. R. 1984. A static method for determining the absolute shear modulus of a syneresing gel. J. Texture Stud. 15: 23–32.Google Scholar
Steffe, J. F. 1996. Rheological Methods in Food Process Engineering, Freeman Press, East Lansing, Michigan.Google Scholar
Steiner, E. H. 1958. A new rheological relationship to express the flow properties of melted chocolate. Revue Internationale de la Chocolaterie 13: 290–295.Google Scholar
Tamura, M. S., Henderson, J. M., Powell, R. L., and Shoemaker, C. F. 1989. Evaluation of the helical screw rheometer as an on-line viscometer. J. Food Sei. 54: 483–484.Google Scholar
Tanner, R. I. 1988. Recoverable elastic strain and swelling ratio, in Rheological Measurements, eds. A. A. Collyer and D. W. Clegg, pp. 93-118, Elsevier Applied Science, New York.Google Scholar
Tattiyakul, J. 1997. Studies on granule growth kinetics and characteristics of tapioca starch dispersion during gelatinization using particle size analysis and rheological methods. M. S. thesis, Cornell University, Ithaca, NY.Google Scholar
Tattiyakul, J. and Rao, M. A. 2000. Rheological behavior of cross-linked waxy maize starch dispersions during and after heating. Carbohydrate Polymers 43: 215–222.Google Scholar
Truong, V. D. and Daubert, C. R. 2000. Comparative study of large strain methods for assessing failure characteristics of selected food gels. J. Texture Stud. 31: 335–353.Google Scholar
Truong, V. D. and Daubert, C. R. 2001. Textural characterization of cheeses using vane rheometry and torsion analysis. J. Food Sei. 66: 716–721.Google Scholar
Van Wazer, J. R., Lyons, J. W., Kim, K. Y., and Colwell, R. E. 1963. Viscosity and Flow Measurement, Interscience Publishers, New York.Google Scholar
Vernon Carter, E. J. and Sherman, P. 1980. Rheological properties and applications of mesquite tree Prosopis juliflora gum 2. Rheological properties and stability of o/w emulsions containing mesquite gum. J. Texture Stud. 11:351–365.Google Scholar
Vitali, A. A. and Rao, M. A. 1982. Flow behavior of guava puree as a function of temperature and concentration. J. Texture Stud. 13:275–289.Google Scholar
Weitz, D. A. and Pine, D. J. 1992. Dynamic Light Scattering, edited by W. Brown, Oxford University Press, Oxford.Google Scholar
Whorlow, R. W. 1980. Rheological Techniques, Halsted Press, New York.Google Scholar
Wood, F. W. and Goff, T. C. 1973. The determination of the effective shear rate in the Brabender Viscograph and in other systems of complex geometry. Die Starke 25: 89–91.Google Scholar
Wu, M. C., Lanier, T. C. and Hamman, D. D. 1985b. Thermal transitions of admixed starch/fish protein systems during heating. J. Food Sei. 50: 20–25.Google Scholar
Wu, M. C., Lanier, T. C., and Hamann, D. D. 1985a. Rigidity and viscosity changes of croacker actomyosin during thermal gelation. J. Food Sei. 50: 14–19.Google Scholar
Xu, J., Tseng, Y., Carriere, C. J. and Wirtz, D. 2002. Microheterogeneity and micro-rheology of wheat gliadin suspensions studied by multiple particle tracking. Biomacromolecules, 3(1): 92–99.Google Scholar
Xu, J., Chang, T., Inglett, G. E., Kim, S., Tseng, Y. and Wirtz, D. 2007. Micro-heterogeneity and micro-rheological properties of high-viscosity oat β-glucan solutions. Food Chemistry 103: 1192–1198.Google Scholar
Yang, W. H. and Rao, M. A. 1998. Complex viscosity-temperature master curve of cornstarch dispersion during gelatinization. J. Food Proc. Eng. 21: 191–207.Google Scholar
Yoo, B. and Rao, M. A. 1995. Yield stress and relative viscosity of tomato concentrates: effect of total solids and finisher screen size. J. Food Sei. 60: 777-779, 785.Google Scholar
Yoo, B. and Rao, M. A. 1996. Creep and dynamic rheological behavior of tomato concentrates: effect of concentration and finisher screen size. J. Texture Studies 27: 451–459.Google Scholar
Yoo, B., Rao, M. A., and Steffe, J. F. 1995. Yield stress of food suspensions with the vane method at controlled shear rate and shear stress. J. Texture Stud. 26: 1–10.Google Scholar
Yoshimura, A. and Prud’homme, R. K. 1988a. Wall slip corrections for Couette and parallel disk viscometers. J. Rheol. 32: 53–67.Google Scholar
Yoshimura, A. and Prud’homme, R. K. 1988b. Wall slip effects on dynamic oscillatory measurements. J. Rheol. 32: 575–584.Google Scholar
Youn, K.-S. and Rao, M. A. 2003. Rheology and relationship among rheological parameters of cross-linked waxy maize starch dispersions heated in fructose solutions. J. Food Sei. 68: 187–194.Google Scholar
Zhou, Z., Solomon, M. J., Scales, P. J., and Boger, D. V. 1999. The yield stress of concentrated flocculated suspensions of size distributed particles. J. Rheol. 43: 651–71.Google Scholar

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M. Anandha Rao
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1.Department of Food ScienceCornell UniversityGenevaUSA

Measurement of Flow and Viscoelastic Properties

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