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{{Short description|Rheological method for wall slip analysis}}
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{{Continuum mechanics|cTopic=rheology}}
In [[rheology]] and [[Rheometry|capillary rheometry]], the '''Mooney plot''' (also called the '''Mooney diagram''' or '''Mooney analysis''') is a graphical method used to detect and quantify [[Wall Slip]] during the [[pressure]]-driven flow of [[non-Newtonian fluid]]s through [[capillary]] or slit dies.<ref name=Mooney1931>Mooney, M., 1931, ''Explicit formulas for slip and fluidity'', Journal of Rheology, 2(2), pp. 210-222.</ref> The technique was introduced by [[Melvin Mooney]] in 1931 and has since become one of three standard corrections routinely applied in capillary rheometry, alongside the Bagley correction and the Weissenberg-Rabinowitsch correction.<ref name=Morrison>Morrison, F. A., 2001, ''Understanding Rheology'', Oxford University Press, pp. 393-397, {{ISBN|0-19-514166-0}}.</ref><ref name=Macosko1994>Macosko, C. W., 1994, ''Rheology: Principles, Measurements, and Applications'', Wiley-VCH, New York, {{ISBN|0-471-18575-2}}.</ref><ref name=DealyWissbrun1999>Dealy, J. M. and Wissbrun, K. F., 1999, ''Melt Rheology and Its Role in Plastics Processing: Theory and Applications'', Kluwer Academic, Dordrecht.</ref> The Mooney plot is described in all major rheology textbooks<ref name=BirdEtAl1987>Bird, R. B., Armstrong, R. C. and Hassager, O., 1987, ''Dynamics of Polymeric Liquids'', Volume 1: Fluid Mechanics, 2nd ed., Wiley, New York.</ref><ref name=Larson1999>Larson, R. G., 1999, ''The Structure and Rheology of Complex Fluids'', Oxford University Press, New York, {{ISBN|0-19-512197-X}}.</ref><ref name=Han2007>Han, C. D., 2007, ''Rheology and Processing of Polymeric Materials'', Volumes 1 and 2, Oxford University Press.</ref> and is implemented as standard functionality in commercial capillary rheometer software.<ref name=Goettfert>Goettfert, 2023, ''Corrections of capillary rheometer experiments'', Goettfert Application Note (goettfert.com).</ref><ref name=Dynisco>Dynisco, 2021, ''Capillary rheometers: wall slip phenomena'', Dynisco Application Note (azom.com/article.aspx?ArticleID=20353).</ref>

The Mooney plot is constructed by plotting the apparent [[shear rate]] <math>\dot{\gamma}_a</math> against the reciprocal capillary radius <math>1/R</math> (or reciprocal diameter <math>1/D</math>) at constant values of the [[wall shear stress]] <math>\tau_w</math>. In the presence of wall slip, the resulting data fall on straight lines whose slopes are directly proportional to the [[Velocity|slip velocity]] <math>v_s</math>.<ref name=Mooney1931/><ref name=Hatzikiriakos2012>Hatzikiriakos, S. G., 2012, ''Wall slip of molten polymers'', Progress in Polymer Science, 37(4), pp. 624-643.</ref>

==Background==

===No-slip boundary condition===
A fundamental assumption in [[fluid dynamics]] is the '''[[no-slip condition]]''', which states that the velocity of a fluid at a solid boundary equals the velocity of that boundary.<ref name=Batchelor1967>Batchelor, G. K., 1967, ''An Introduction to Fluid Dynamics'', Cambridge University Press, Cambridge.</ref> For [[Newtonian fluid]]s this assumption holds under virtually all practical conditions.<ref name=Denn2001>Denn, M. M., 2001, ''Extrusion instabilities and wall slip'', Annual Review of Fluid Mechanics, 33, pp. 265-287.</ref> However, for complex fluids, including [[polymer]] melts, concentrated suspensions, [[emulsion]]s, [[foam]]s, and [[viscoplasticity|viscoplastic]] materials, the fluid may exhibit a finite velocity at the wall when the wall shear stress exceeds a critical value <math>\tau_{w,c}</math>. This phenomenon is known as '''wall slip''' and leads to systematic errors in rheological measurements if not properly accounted for.<ref name=Barnes1995>Barnes, H. A., 1995, ''A review of the slip (wall depletion) of polymer solutions, emulsions and particle suspensions in viscometers: its cause, character, and cure'', Journal of Non-Newtonian Fluid Mechanics, 56(3), pp. 221-251.</ref><ref name=Hatzikiriakos2012/> The critical shear stress for the onset of slip in linear polyethylenes has been reported to be approximately 0.09 to 0.14 MPa.<ref name=Ramamurthy1986>Ramamurthy, A. V., 1986, ''Wall slip in viscous fluids and influence of materials of construction'', Journal of Rheology, 30(2), pp. 337-357.</ref><ref name=HatzikiriakosDealy1991>Hatzikiriakos, S. G. and Dealy, J. M., 1991, ''Wall slip of molten high density polyethylene. I. Sliding plate rheometer studies'', Journal of Rheology, 35(4), pp. 497-523.</ref>

===Origin of apparent wall slip===
In concentrated suspensions, particles cannot occupy the space adjacent to a rigid surface as efficiently as in the bulk, producing a thin '''particle-depleted layer''' (or '''apparent slip layer''') of thickness <math>\delta</math> that consists predominantly of the continuous phase.<ref name=Kalyon2005>Kalyon, D. M., 2005, ''Apparent slip and viscoplasticity of concentrated suspensions'', Journal of Rheology, 49(3), pp. 621-640.</ref><ref name=Cloitre2017>Cloitre, M. and Bonnecaze, R. T., 2017, ''A review on wall slip in high solid dispersions'', Rheologica Acta, 56, pp. 283-305.</ref> This layer has a viscosity much lower than the bulk suspension and acts as a lubricant, producing an apparent velocity jump at the wall. The same mechanism has been documented in emulsions, foams, colloidal gels, and food pastes.<ref name=YoshimuraPrudhomme1988>Yoshimura, A. S. and Prud'homme, R. K., 1988, ''Wall slip corrections for Couette and parallel disk viscometers'', Journal of Rheology, 32(1), pp. 53-67.</ref><ref name=Bertola2003>Bertola, V., Bertrand, F., Tabuteau, H., Bonn, D. and Coussot, P., 2003, ''Wall slip and yielding in pasty materials'', Journal of Rheology, 47(5), pp. 1211-1226.</ref><ref name=WallsEtAl2003>Walls, H. J., Caines, S. B., Sanchez, A. M. and Khan, S. A., 2003, ''Yield stress and wall slip phenomena in colloidal silica gels'', Journal of Rheology, 47(4), pp. 847-868.</ref>

In polymer melts, wall slip has been attributed to chain disentanglement at the polymer-wall interface.<ref name=Denn2001/> Migler, Hervet, and Leger performed direct optical measurements using evanescent waves that confirmed a sharp transition between weak and strong slip near smooth surfaces.<ref name=MiglerEtAl1993>Migler, K. B., Hervet, H. and Leger, L., 1993, ''Slip transition of a polymer melt under shear stress'', Physical Review Letters, 70(3), pp. 287-290.</ref>

==Derivation==

===Flow in a circular capillary===
Consider the steady, fully developed, isothermal, [[creeping flow]] of an [[incompressible]] fluid through a circular capillary of radius <math>R</math> and length <math>L</math>. The [[wall shear stress]] is obtained from the [[pressure drop]] <math>\Delta P</math> across the capillary (after applying the Bagley correction<ref name=Bagley1957>Bagley, E. B., 1957, ''End corrections in the capillary flow of polyethylene'', Journal of Applied Physics, 28(5), pp. 624-627.</ref>) as
:<math>
   \tau_w = \frac{R \, \Delta P}{2L} \,.
</math>

The '''apparent shear rate''' (assuming a Newtonian [[velocity profile]] with no slip) is defined as<ref name=Macosko1994/><ref name=BirdEtAl1987/>
:<math>
   \dot{\gamma}_a = \frac{4Q}{\pi R^3} = \frac{4 \bar{v}}{R}
</math>
where <math>Q</math> is the [[volumetric flow rate]] and <math>\bar{v} = Q/(\pi R^2)</math> is the mean velocity in the capillary.

===Velocity decomposition===
When wall slip occurs, the total mean velocity is the sum of a bulk deformation contribution and a slip contribution. At the wall (<math>r = R</math>), the fluid has a finite tangential velocity <math>v_s</math> rather than zero. The volumetric flow rate then decomposes as<ref name=Mooney1931/>
:<math>
   Q = Q_{\text{bulk}} + Q_{\text{slip}} = Q_{\text{bulk}} + \pi R^2 v_s \,.
</math>

Dividing through by <math>\pi R^3 / 4</math>:
:<math>
   \frac{4Q}{\pi R^3} = \frac{4 Q_{\text{bulk}}}{\pi R^3} + \frac{4 v_s}{R}
</math>
which may be written as
:<math>
   \dot{\gamma}_a = \dot{\gamma}_{a,\text{true}}(\tau_w) + \frac{4 v_s(\tau_w)}{R} \,.
</math>

Here <math>\dot{\gamma}_{a,\text{true}}</math> is the apparent shear rate that would be observed in the absence of slip; it depends only on the wall shear stress <math>\tau_w</math> for a given fluid.<ref name=Morrison/><ref name=DealyWissbrun1999/>

===The Mooney equation===
Mooney's fundamental assumption is that the slip velocity is a function of the wall shear stress alone and is independent of the capillary geometry:<ref name=Mooney1931/>
:<math>
   v_s = f(\tau_w) \,.
</math>

Under this assumption, at a fixed value of <math>\tau_w</math>, the quantity <math>\dot{\gamma}_{a,\text{true}}</math> is a constant across capillaries of different radii, and the apparent shear rate becomes a '''linear function''' of <math>1/R</math>:
:<math>
   \dot{\gamma}_a = \dot{\gamma}_{a,\text{true}}(\tau_w) + 4\, v_s(\tau_w) \cdot \frac{1}{R} \,.
</math>

This is the '''Mooney equation'''.<ref name=Mooney1931/> When expressed in terms of the capillary diameter <math>D = 2R</math>:
:<math>
   \dot{\gamma}_a = \dot{\gamma}_{a,\text{true}}(\tau_w) + 8\, v_s(\tau_w) \cdot \frac{1}{D}
</math>
the slope of <math>\dot{\gamma}_a</math> versus <math>1/D</math> at constant <math>\tau_w</math> equals <math>8 v_s</math>, and the <math>y</math>-intercept yields the slip-corrected apparent shear rate <math>\dot{\gamma}_{a,\text{true}}</math>.<ref name=HatzikiriakosDealy1992>Hatzikiriakos, S. G. and Dealy, J. M., 1992, ''Wall slip of molten high density polyethylenes. II. Capillary rheometer studies'', Journal of Rheology, 36(4), pp. 703-741.</ref><ref name=Morrison/>

==Procedure==
The experimental procedure for constructing a Mooney plot is described in standard references<ref name=Morrison/><ref name=Macosko1994/><ref name=Hatzikiriakos2012/><ref name=Goettfert/> and comprises the following steps:

# Obtain flow curves (plots of <math>\tau_w</math> vs. <math>\dot{\gamma}_a</math>) using a set of capillary dies with at least three different diameters <math>D_1, D_2, D_3</math> but the same <math>L/D</math> ratio. A practical die combination is L/D = 40/2 mm and L/D = 20/1 mm.<ref name=Goettfert/>
# Apply the '''Bagley correction''' to each die to obtain the true wall shear stress, free of entrance and exit pressure losses.<ref name=Bagley1957/>
# At each value of constant <math>\tau_w</math>, read off the apparent shear rate from each flow curve.
# Construct the '''Mooney plot''': plot <math>\dot{\gamma}_a</math> (ordinate) against <math>1/R</math> (abscissa) at each constant <math>\tau_w</math>.
# If the data at each <math>\tau_w</math> fall on a straight line, the Mooney assumption <math>v_s = f(\tau_w)</math> is validated. The '''slip velocity''' is obtained from the slope: <math>v_s = \tfrac{1}{4} \times \text{slope}</math>. The '''slip-corrected apparent shear rate''' is the <math>y</math>-intercept.
# Repeat at other values of <math>\tau_w</math> to determine <math>v_s(\tau_w)</math>.
# Apply the '''Weissenberg-Rabinowitsch correction''' to the slip-corrected data to obtain the true wall shear rate and hence the true [[viscosity]].<ref name=Rabinowitsch1929>Rabinowitsch, B., 1929, ''Uber die Viskositat und Elastizitat von Solen'', Zeitschrift fur Physikalische Chemie A, 145, pp. 1-26.</ref>

Kalika and Denn described an alternative approach in which a power-law model is assumed for the bulk flow curve and slip velocities are computed by subtraction at each data point, yielding a master plot of slip velocity independent of capillary diameter.<ref name=KalikaDenn1987>Kalika, D. S. and Denn, M. M., 1987, ''Wall slip and extrudate distortion in linear low-density polyethylene'', Journal of Rheology, 31(8), pp. 815-834.</ref>

==Detection of wall slip==
Wall slip is detected by observing '''divergence of the flow curves''' obtained from capillaries of different diameters.<ref name=Ramamurthy1986/><ref name=HatzikiriakosDealy1991/><ref name=Denn2001/> In the absence of slip, all flow curves collapse onto a single master curve, since for a given fluid
:<math>
   \dot{\gamma}_a = f(\tau_w) \quad \text{(independent of } R \text{)}
</math>
When slip is present, the flow curves shift apart. At a given <math>\tau_w</math>, smaller-diameter capillaries yield higher apparent shear rates, because the surface-to-volume ratio <math>2/R</math> is larger and the relative contribution of slip is greater:<ref name=Hatzikiriakos2012/>
:<math>
   \dot{\gamma}_a \bigg|_{R_1} > \dot{\gamma}_a \bigg|_{R_2} \quad \text{for} \quad R_1 < R_2 \quad \text{at fixed } \tau_w \,.
</math>

The '''critical wall shear stress''' <math>\tau_{w,c}</math> at which the flow curves begin to diverge marks the '''onset of wall slip'''.<ref name=Ramamurthy1986/><ref name=KalikaDenn1987/> Ramamurthy demonstrated this divergence for several LLDPE and HDPE resins and showed that the critical stress is relatively insensitive to molecular weight, molecular weight distribution, and temperature.<ref name=Ramamurthy1986/>

In rotational rheometry, the analogous diagnostic involves comparing flow curves measured at different gap heights in a parallel plate geometry; if the curves do not superimpose, wall slip is indicated.<ref name=YoshimuraPrudhomme1988/><ref name=YoshimuraPrudhomme1988b>Yoshimura, A. S. and Prud'homme, R. K., 1988, ''Wall slip effects on dynamic oscillatory measurements'', Journal of Rheology, 32(6), pp. 575-584.</ref>

==Slip velocity models==

===Navier linear slip===
The simplest model, due to Navier (1827),<ref name=Navier1827>Navier, C. L. M. H., 1827, ''Sur les lois du mouvement des fluides'', Memoires de l'Academie Royale des Sciences de l'Institut de France, 6, pp. 389-440.</ref> assumes a linear relationship:
:<math>
   v_s = \beta \, \tau_w
</math>
where <math>\beta</math> is the Navier slip coefficient. For this model, the Mooney equation becomes
:<math>
   \dot{\gamma}_a = \dot{\gamma}_{a,\text{true}}(\tau_w) + \frac{4 \beta \, \tau_w}{R} \,.
</math>

===Nonlinear (power-law) slip===
For many practical systems, particularly those with shear-thinning phases, the slip velocity follows a power-law dependence:<ref name=Hatzikiriakos2012/><ref name=Kalyon2005/>
:<math>
   v_s = \beta' \, \tau_w^{s_\beta}
</math>
where <math>\beta'</math> is a slip coefficient and <math>s_\beta</math> is the slip exponent. Hatzikiriakos and Dealy found experimentally that the slip velocity of HDPE melts follows an approximate power-law relationship with wall shear stress above the critical stress.<ref name=HatzikiriakosDealy1992/><ref name=HatzikiriakosDealy1991/> A Newtonian binder (<math>s_\beta = 1</math>) recovers the linear Navier model.

==Corrections in capillary rheometry==
The Mooney slip analysis is one of three standard corrections applied in capillary rheometry. The recommended order is:<ref name=Morrison/><ref name=Macosko1994/><ref name=LaunReview>Laun, H. M., 2004, ''Capillary rheometry for polymer melts revisited'', Rheologica Acta, 43(5), pp. 509-528.</ref><ref name=Syrjala2012>Syrjala, S. and Aho, J., 2012, ''Capillary rheometry of polymer melts: simulation and experiment'', Korea-Australia Rheology Journal, 24(3), pp. 241-247.</ref>

# '''Bagley correction''' for entrance and exit pressure losses:<ref name=Bagley1957/>
#:<math>\tau_w = \frac{R(\Delta P - \Delta P_{\text{end}})}{2L} = \frac{R \, \Delta P}{2(L + e R)}</math>
#:where <math>e</math> is the Bagley correction factor (end correction).
# '''Mooney correction''' for wall slip (as derived above).
# '''Weissenberg-Rabinowitsch correction''' for non-Newtonian velocity profiles:<ref name=Rabinowitsch1929/>
#:<math>\dot{\gamma}_{\text{true}} = \frac{3n'+1}{4n'}\,\dot{\gamma}_{a,\text{true}}</math>

The true [[shear viscosity]] after all corrections is<ref name=Morrison/>
:<math>
   \eta(\dot{\gamma}_{\text{true}}) = \frac{\tau_w}{\dot{\gamma}_{\text{true}}} \,.
</math>

==Extension to other geometries==
Kalyon (2005) extended the Mooney analysis to [[viscoplasticity|viscoplastic]] ([[Herschel-Bulkley fluid|Herschel-Bulkley]]) fluids and showed that the same slip coefficient is obtained from Couette, capillary, and slit flow geometries, validating the physical basis of the method.<ref name=Kalyon2005/>

For flow through a rectangular slit die of height <math>2H</math> and width <math>W \gg 2H</math>:<ref name=Kalyon2005/><ref name=Laun1983>Laun, H. M., 1983, ''Polymer melt rheology with a slit die'', Rheologica Acta, 22(2), pp. 171-185.</ref>
:<math>
   \dot{\gamma}_{a,\text{slit}} = \dot{\gamma}_{a,\text{true}}(\tau_w) + \frac{6 v_s(\tau_w)}{2H}
</math>
so that plotting <math>\dot{\gamma}_{a,\text{slit}}</math> versus <math>1/H</math> at constant <math>\tau_w</math> yields a straight line with slope <math>3 v_s</math>.

Yoshimura and Prud'homme developed the analogous correction for parallel plate and Couette geometries in rotational rheometers, extending the applicability of the Mooney approach beyond capillary flow.<ref name=YoshimuraPrudhomme1988/>

==Limitations and modifications==

===Limitations of the classical Mooney analysis===
Despite its status as a standard method, the original Mooney analysis has several known limitations that have been discussed extensively in the literature:

* '''Geometry-independent slip assumption''': The fundamental assumption <math>v_s = f(\tau_w)</math> can fail when particle migration, pressure-dependent effects, or gap-dependent mechanisms are significant.<ref name=Wilms2021>Wilms, P., Wieringa, J., Blijdenstein, T. and Kohlus, R., 2021, ''On the difficulty of determining the apparent wall slip of highly concentrated suspensions in pressure driven flows'', Journal of Non-Newtonian Fluid Mechanics, 298, 104691.</ref><ref name=MartinWilson2005>Martin, P. J. and Wilson, D. I., 2005, ''A critical assessment of the Jastrzebski interface condition for the capillary flow of pastes, foams and polymers'', Chemical Engineering Science, 60(11), pp. 3471-3475.</ref>

* '''Negative slip velocities''': For some filled compounds and complex polymer systems, the Mooney analysis yields physically impossible (negative) values of <math>v_s</math>, indicating breakdown of the underlying assumptions.<ref name=Leblanc2001>Leblanc, J. L., 2001, ''Wall slip and compressibility like effects in capillary rheometer tests on complex polymer systems'', Plastics, Rubber and Composites, 30(6), pp. 277-283.</ref><ref name=Mourniac1992>Mourniac, P., Agassant, J. F. and Vergnes, B., 1992, ''Determination of the wall slip velocity in the flow of a SBR compound'', Rheologica Acta, 31(6), pp. 565-574.</ref>

* '''Pressure-dependent slip''': Hatzikiriakos and Dealy demonstrated experimentally that the slip velocity of HDPE depends on the wall normal stress (and hence on pressure), producing an apparent <math>L/D</math> dependence that violates the Mooney assumption. They developed a modified Mooney technique to account for this effect.<ref name=HatzikiriakosDealy1992/>

* '''Viscous heating''': At high shear rates, viscous dissipation can raise the temperature within the capillary, altering both the bulk viscosity and the slip behavior. Rosenbaum and Hatzikiriakos developed a nonisothermal model that corrects the slip velocity for this effect.<ref name=Rosenbaum1997>Rosenbaum, E. E. and Hatzikiriakos, S. G., 1997, ''Wall slip in the capillary flow of molten polymers subject to viscous heating'', AIChE Journal, 43(3), pp. 598-608.</ref>

* '''Shear-induced particle migration''': In concentrated suspensions, particles migrate radially under shear gradients, altering the near-wall composition along the die length. This transient process introduces length-dependent apparent slip that the Mooney analysis cannot capture.<ref name=Phillips1992>Phillips, R. J., Armstrong, R. C., Brown, R. A., Graham, A. L. and Abbott, J. R., 1992, ''A constitutive equation for concentrated suspensions that accounts for shear-induced particle migration'', Physics of Fluids A, 4(1), pp. 30-40.</ref><ref name=Wilms2021/>

* '''Die material and surface effects''': The material of construction of the capillary die and its surface finish can influence the onset and magnitude of wall slip. Ramamurthy showed that using brass dies or adhesion promoters can substantially reduce slip in polyethylene extrusion.<ref name=Ramamurthy1986/> Chen and coworkers further investigated the role of surface roughness and chemical composition.<ref name=Chen1993>Chen, Y., Kalyon, D. M. and Bayramli, E., 1993, ''Effects of surface roughness and the chemical structure of materials of construction on wall slip behavior of linear low density polyethylene in capillary flow'', Journal of Applied Polymer Science, 50(7), pp. 1169-1177.</ref>

===Jastrzebski modification===
Jastrzebski (1967)<ref name=Jastrzebski1967>Jastrzebski, Z. D., 1967, ''Entrance effects and wall effects in an extrusion rheometer during the flow of concentrated suspensions'', Industrial and Engineering Chemistry Fundamentals, 6(3), pp. 445-454.</ref> proposed that the slip velocity depends inversely on the capillary radius:
:<math>
   v_s = \frac{\beta_J \, \tau_w}{R}
</math>
so that plotting <math>\dot{\gamma}_a</math> versus <math>1/R^2</math> at constant <math>\tau_w</math> yields straight lines (the '''Jastrzebski plot'''). Martin and Wilson argued that this condition lacks physical grounding and recommended against its use without further justification.<ref name=MartinWilson2005/>

===Tikhonov regularization===
Owens, Corfield, and coworkers developed a Tikhonov regularization approach to the Mooney analysis that treats the extraction of slip velocity and bulk flow curves as an inverse problem, yielding more robust results when the standard graphical method produces noisy or inconsistent data.<ref name=Owens2003>Owens, L. N., Corfield, G. M., Elvidge, J. A. and Sherwood, J. D., 2003, ''A new method of processing capillary viscometry data in the presence of wall slip'', Journal of Rheology, 47(2), pp. 337-352.</ref><ref name=Corfield2006>Corfield, G. M., Adams, M. J., Briscoe, B. J., Fryer, P. J. and Lawrence, C. J., 2006, ''A critical examination of capillary rheometry for foods (exhibiting wall slip)'', Food and Bioproducts Processing, 84(C4), pp. 302-312.</ref>

==Applications==

===Polymer melts===
The Mooney analysis is routinely applied to polymer melts, particularly high molecular weight linear polymers such as [[LLDPE]], [[HDPE]], [[polypropylene]], and [[polystyrene]], which exhibit wall slip above a critical wall shear stress.<ref name=Ramamurthy1986/><ref name=KalikaDenn1987/><ref name=HatzikiriakosDealy1991/><ref name=Hatzikiriakos2012/> In such systems, the slip velocity typically follows a power law in <math>\tau_w</math> and the transition from '''weak slip''' (partial disentanglement at the wall) to '''strong slip''' (nearly plug flow) can be tracked on the Mooney plot.<ref name=Denn2001/><ref name=Hatzikiriakos2012/> Hatzikiriakos and coworkers demonstrated good agreement between slip velocities obtained from the Mooney technique on HDPE and independent sliding plate rheometer measurements.<ref name=HatzikiriakosDealy1992/> Studies on metallocene-catalyzed LLDPE resins have used the Mooney plot to show that nearly complete plug flow occurs at high wall shear stresses.<ref name=Hatzikiriakos2003>Hatzikiriakos, S. G., Kazatchkov, I. B. and Vlassopoulos, D., 2000, ''Interfacial phenomena in the capillary extrusion of metallocene polyethylenes'', Journal of Rheology, 44(2), pp. 427-456.</ref>

===Rubber compounds===
Rubber compounds (SBR, EPDM) are known to exhibit wall slip, and the Mooney method is commonly attempted. However, Mourniac, Agassant, and Vergnes found that the standard Mooney analysis failed to produce physically acceptable results for SBR compounds and developed alternative characterizations based on gap-dependent slip.<ref name=Mourniac1992/> Leblanc reported similar difficulties with filled rubber compounds, observing negative slip velocities from the Mooney analysis.<ref name=Leblanc2001/>

===Concentrated suspensions===
Wall slip is particularly important in concentrated suspensions of rigid particles (ceramic pastes, pharmaceutical formulations, solid propellant simulants). Yilmazer and Kalyon measured slip layer thicknesses on the order of a few percent of the particle diameter using the Mooney technique in highly filled suspensions.<ref name=YilmazerKalyon1989>Yilmazer, U. and Kalyon, D. M., 1989, ''Slip effects in capillary and parallel disk torsional flows of highly filled suspensions'', Journal of Rheology, 33(8), pp. 1197-1212.</ref> Wilms and coworkers provided a systematic comparison of the accuracy of the original Mooney analysis and various modifications for concentrated suspensions, and proposed a best-practice approach.<ref name=Wilms2021/>

===Food and consumer products===
The Mooney analysis has been applied to food pastes and consumer products. Corfield and coworkers critically examined capillary rheometry of starch-based pastes exhibiting wall slip, and assessed the robustness of the Mooney method for such materials.<ref name=Corfield2006/>

===Other materials===
Further applications of the Mooney method have been reported for solder pastes,<ref name=Durairaj2009>Durairaj, R., Mallik, S., Seman, A., Marks, A. and Ekere, N. N., 2009, ''The effect of wall-slip formation on the rheological behaviour of lead-free solder pastes'', Materials and Design, 30(10), pp. 3786-3791.</ref> wood-polymer composites,<ref name=KalyonGevgilili2003>Kalyon, D. M. and Gevgilili, H., 2003, ''Wall slip and extrudate distortion of three polymer melts'', Journal of Rheology, 47(3), pp. 683-699.</ref> powder injection moulding compounds,<ref name=FilipEtAl2019>Filip, P., Hausnerova, B., Sanetrnik, D. and Hnatkova, E., 2019, ''Wall slip of highly filled powder injection moulding compounds in dependence on capillary die geometry'', Powder Technology, 347, pp. 29-35.</ref> abrasive flow machining media,<ref name=AFM2023>Sharma, A. K. et al., 2023, ''On characterization of shear viscosity and wall slip for concentrated suspension flows in abrasive flow machining'', International Journal of Advanced Manufacturing Technology.</ref> and natural fluid muds.<ref name=TalmonMeshkati2022>Talmon, A. M. and Meshkati, E., 2022, ''Rheology, rheometry and wall slip'', in ''Sediment Transport: Hydrodynamics and Fluvial Processes'', IntechOpen.</ref>

==Distinction from the Mooney-Rivlin plot==
The Mooney plot for wall slip should not be confused with the '''Mooney-Rivlin plot''' used in [[rubber]] [[elasticity (physics)|elasticity]]. In the latter, the reduced stress is plotted against <math>1/\alpha</math> (where <math>\alpha</math> is the [[finite strain theory|stretch ratio]]) for [[uniaxial tension]] data of a [[Mooney-Rivlin solid]], and a linear fit yields the material constants <math>C_1</math> and <math>C_2</math>. Both plots are named after the same [[Melvin Mooney]], who contributed to both [[rubber]] mechanics and capillary [[rheometry]].<ref name=Mooney1931/>

==References==
<references/>

==Further reading==
* Dealy, J. M. and Wang, J., 2013, ''Melt Rheology and its Applications in the Plastics Industry'', 2nd ed., Springer, Dordrecht. {{ISBN|978-94-007-6394-4}}.
* Tadmor, Z. and Gogos, C. G., 2006, ''Principles of Polymer Processing'', 2nd ed., Wiley, Hoboken. {{ISBN|0-471-38770-3}}.
* Dealy, J. M., 1982, ''Rheometers for Molten Plastics'', Van Nostrand Reinhold, New York.

==See also==
* [[Capillary breakup rheometry]]
* [[Weissenberg effect]]
* [[Mooney-Rivlin solid]]
* [[Herschel-Bulkley fluid]]
* [[Non-Newtonian fluid]]
* [[Navier-Stokes equations#Boundary conditions|Navier slip condition]]
* [[Viscoplasticity]]

[[Category:Rheology]]
[[Category:Fluid mechanics]]
[[Category:Non-Newtonian fluids]]
[[Category:Polymer physics]]
[[Category:Viscosity]]
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