数理解析研究所講究録
1495 巻 2006 年 26-31

26

Self-similar scaling in power-law fluid models
BY JOSEF MA’
Charles University in Prague, Faculty of Mathematics and Physics,
Mathematical Institute, Sokolovsk\’a 83, 18675 Praha 8, Czech Republic$\mathrm{L}\mathrm{E}\mathrm{K}\uparrow$

Abstract. Navier-Stokes equations represents a key reference model both in the analysis of nonlinear
partial diffcrcntial systerns, and in non-Newtonian fluid mcchanics. The power-law fluid, one of the mostpopular non-Newtonian models, generalizes the Navier-Stokes fluid in the fact that its non-constantviscosity is a power of the shear rate. A modification of the power-law fluids that consists in multiplying
the power of the shear ratc by the pressure is also considered. Such a model fits to the .ss of fluids withthe viscosity depending on the shear rate and on the the pressure. We look at all these models from thepoint of view of invariance with respect to suitable scaling, and present some relevant results and open
problems.
$\mathrm{c}\mathrm{l}\mathrm{a}$Dedicated to Professor $Tai$ -Ping$Li\mathrm{u}$on the occasion of his sixtieth birthday.

1. Power-law fluids
the velocity and the mean normal stress (the pressure), we consider an
Denoting
depends on the pressure and on theincompressible homogeneous fluid whose (generalized) viscosity
velocity
gradient Vv. Unsteady flows
partof therepresenting the symmetricshear rate
at a constant temperature, areof such fluids, that take place in a three-dimensional container St
equations
form
of
the
partialdiffercntial
described by thc system of$\mathrm{v}=(v_{1}, v_{2}, v_{3})$$p$$\nu_{g}$$|\mathrm{D}(\mathrm{v})|^{2},$$\mathrm{D}(\mathrm{v})$$\subset \mathbb{R}^{3}$$\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{v}=0$,$\mathrm{v}_{t},+\mathrm{d}\mathrm{i}\mathrm{v}(\mathrm{v}\otimes \mathrm{v})=-\nabla p+\mathrm{d}\mathrm{i}\mathrm{v}(2\nu_{g}(p, |\mathrm{D}(\mathrm{v})|^{2})\mathrm{D}(\mathrm{v}))$

.

(1.1)

This systcm includes as special subsystems$\bullet$Schaeffer’s model, derived in [19] to capturc flows of certain granular materials, with$\nu_{\mathit{9}}(p, |\mathrm{D}(\mathrm{v})|^{2})=\frac{\alpha \mathrm{p}}{|\mathrm{D}(\mathrm{v})|}$$0$$(a>0)$ .

$|\mathrm{D}(\mathrm{v})|\neq 0$

is independent of the shear rate, but depends
Fluids with pressure dependent viscosity, where. Such models, ffequently considered with an exponential viscositypressure relationship, are very important in elastohydrodynamics and in processes taking place
under high pressures, see for example [2] or [22].
$\nu_{\mathit{9}}$on the prcssurc, i.e.,.

valid if$\nu_{\mathit{9}}=\nu_{g}(p)$Fluids utzth shear rate dependent viscosity, where, including$\nu_{g}$is indepcndent of the pressure, i.e.,$\nu_{\mathit{9}}=$$\nu_{g}(|\mathrm{D}(\mathrm{v})|^{2})$, where $r>2$ is fixed, and-Ladyzhenskaya’s fluids with
$r=3$
this system of PDEs is frcqucntly called Smagorinski’s modelarc positive numbers. For
is the turbulent viecosity.
is the molecular viscosity andof turbulence, see [21]; thenwhere $r\in(1, \infty)$ is fixed and is a positive-Power-law fiuids withnumbcr.
( being a positive number).
-Navier-Stokes fluids with$\nu_{g}(|\mathrm{D}(\mathrm{v})|^{2})=\nu_{0}+\nu_{1}|\mathrm{D}(\mathrm{v})|^{r-2}$$\nu_{1}$$\nu_{1}$$\nu_{0}$$\nu_{\mathit{9}}(|\mathrm{D}(\mathrm{v})|^{2})=\nu_{1}|\mathrm{D}(\mathrm{v})|^{\mathrm{r}-2}$$\nu_{g}(p, |\mathrm{D}(\mathrm{v})|^{2})=\nu_{0}$$\nu_{0}$$\nu_{1}$$\nu_{0}$\dag er The contribution of J. M\’alek to this work is a part of the research project MSM 0021620839 financed by MSMT. The
support of CSF, project GACR 201/05/0164, is also acknowlcdgcd.

27

A Ladyzhenskaya’s fluid reduces to the Navicr-Stokes fluid by putting $\nu_{1}=0$ , and to power-law fluids
by setting $\nu_{0}=0$ . Note also that taking $r=2$ in the constitutive equation for the Ladyzhenskaya’s fluid. Wc refer to [16] and [17] forleads again to the Navicr-Stokes fluid with the constant viscosity
more details and views on these models.
$2(\nu_{0}+\nu_{1})$As said above, we will primarily deal with three models. Time-dependent flows of fluids with theviscosity depending linearly on the prcssure and polynornially on the shcar rate arc described by$\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{v}=0$

.

,

(1.2)

$\mathrm{v}_{t},+\mathrm{d}\mathrm{i}\mathrm{v}(\mathrm{v}\copyright \mathrm{v})-2\nu_{1}\mathrm{d}\mathrm{i}\mathrm{v}(p|\mathrm{D}(\mathrm{v})|^{\mathrm{r}-2}\mathrm{D}(\mathrm{v}))=-\nabla p$in (1.2) leads to Schaeffer’s model.
Note that setting $r=1$ andAlso, ifin (1.2) does not depend on one obtains the system describing unsteady flows of thepower-law fluids$\alpha=2\nu_{1}$$p$$\nu_{\mathit{9}}$$\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{v}=0$

,

.

(1.3)

$\mathrm{v}_{t},+\mathrm{d}\mathrm{i}\mathrm{v}(\mathrm{v}\otimes \mathrm{v})-2\nu_{1}\mathrm{d}\mathrm{i}\mathrm{v}(|\mathrm{D}(\mathrm{v})|^{r-2}\mathrm{D}(\mathrm{v}))=-\nabla p$

Taking $f=2$ in (1.3), this system finaUy reduces to the evolutionary Navier-Stokes system
divv $=0$ ,.

$\mathrm{v}_{t},+\mathrm{d}\mathrm{i}\mathrm{v}(\mathrm{v}\otimes \mathrm{v})-\nu_{1}\Delta \mathrm{v}=-\nabla p$(1.4)

2. Self-similar scaling
It is known that if$(\mathrm{v},p)$solves the Navier-Stokes system (1.4) then$\mathrm{v}^{\lambda}(t, x)=\lambda \mathrm{v}(\lambda^{2}t, \lambda x)$,$(\mathrm{v}^{\lambda},p^{\lambda})$defined through(2.1)$(\lambda>0)$$p^{\lambda}(t, x)=\lambda^{2}p(\lambda^{2}t.\lambda x)$,

solve (1.4) too.
Similarly, solutions of the equations for power-law fluids (1.3), considered for $r\in(1,3)$ , are invariant
with respect to the scaling$\mathrm{v}^{\lambda}(t, x):=\lambda^{\frac{1}{Sr}=}’ \mathrm{v}(\lambda^{\mp_{rt,\lambda X)}},$

.

(2.2)

$p^{\lambda}(t, x):=\lambda^{2\frac{r-1}{3r}}\mathrm{P}(\lambda\neq_{-\overline{r}t,\lambda X)}$

solves (1.3) as well. Note that the scaling (2.1) of thesolves (1.3), thenIt means that if
Navier-Stokes system is also included by setting $r=2$ in (2.2).
$(\mathrm{v}^{\lambda},p^{\lambda})$$(\mathrm{v},p)$values ofInterestingly, looking for
respect to the scaling of the form$\mathrm{v}\mathrm{u}\mathrm{c},\mathrm{h}$$\alpha,$and 7 that provide solutions of (1.2) invariant with$\beta$,$\mathrm{v}^{\lambda}(t, x)=\lambda^{\alpha}\mathrm{v}(\lambda^{\beta}t, \lambda x)$$p^{\lambda}(t, x)=\lambda^{\gamma}p(\lambda^{\beta}t, \lambda x)$

,

(2.3)

we come to thc conclusion that this is possible only ifcither$r=1$ (Schacffcr’s modcJ)

and

or

$\alpha=-1$$\Rightarrow\beta=0$, $\beta=\alpha+1$ ,$\gamma=2\alpha$and $\gamma=-2$ .

(2.4)
(2.5)

If (2.4) happens then we obtain the whole family of selhimilar solutions of (1.2) of the form$\mathrm{v}^{\lambda}(t, \tau,)=\lambda^{\alpha}\mathrm{v}(\lambda^{\alpha+1}t, \lambda x)$,$p^{\lambda}(t, x)=\lambda^{2\alpha}p(\lambda^{\alpha+1}t, \lambda x)$

,

(2.6)

while in the case (2.5) the self-similar solutions are of the form$\mathrm{v}^{\lambda}(t, x)=\frac{1}{\lambda}\mathrm{v}(t, \lambda x)$,$p^{\lambda}(t, x)= \frac{1}{\lambda^{2}}p(t, \lambda x)$

.

(2.7)

We use these scaling in two ways. First, in Section 3, studying the behavior of the rate of dissipation
under these scalings, we classify the difficulty of the problems, and we make a relation between such
behavior and the results achieved in dependence on the value of the power-law index . Then, in Section
4, we come to the task of a construction of singular solutions to the governing systems of PDEs in a
sclf-sirnilar form,
$r$

28

3. Classification of the difficulty of the problems
We use above scaling to magnify the flow near the point of interest located inside the fluid domain.
Studying the behavior of the averaged rate of dissipation$d(p, \mathrm{v})$defined through(3.1)$d(p, \mathrm{v}):=\int_{-1}^{0}\int_{B_{1}(0)}\nu_{g}(p, |\mathrm{D}(\mathrm{v})|^{2})|\mathrm{D}(\mathrm{v})|^{2}dxdt$

for

as

$d(p^{\lambda}, \mathrm{v}^{\lambda})$$\lambdaarrow\infty$, we can give the following classification of the problem:
supercritical,
critical,$arrow 0$

if$d(p^{\lambda}, \mathrm{v}^{\lambda})\{$$arrow A\in(0, \infty)$

asthen the problem is$\lambdaarrow\infty$$\{$

subcritical.

$arrow\infty$

Roughly speaking, we may say that for a subcritical problem the zooming (near a possible singularity) is
as A– , while for supercritical case the energy dissipated out of the system ispenalized byan insensitivc mea.sure of this magnific.ation.
Since for the power-law fluids$d(\mathrm{p}^{\lambda}, \mathrm{v}^{\lambda})$$\infty$$d( \mathrm{v}):=d(p, \mathrm{v})=2\nu_{1}\int_{-1}^{0}\int_{\mathcal{B}_{1}(0)}|\mathrm{D}(\mathrm{v})|^{r}dxdt$

,

(3.2)

and consequently$d( \mathrm{v}^{\lambda})=2\nu_{1}\lambda^{5r-11}\int_{-\lambda^{\mp_{r}}}^{0}\int_{\mathcal{B}_{\lambda}(0)}|\mathrm{D}_{y}(\mathrm{v})\}^{r}dyd\tau$

,

(3.3)

we observe that$arrow 0$$d(\mathrm{v}^{\lambda})\{$$r< \frac{11}{6}$$arrow A\in(0, \infty)$$arrow\infty$as$\lambdaarrow\infty$

if$\{$ $r= \frac{11}{5}$$r> \frac{11}{5}$,,,implying that problem (1.3) is$\{$supercritical.
critical.
subcritical.

Here, we discuss what is known on mathematical consistency of the system (1.3). We shall see that theplays an significant bound when formulating available results. Note that the Navierparameter
Stokes equations in three spatial dimensions falls into the class of supercritical problems. Let us also ,all(see [16]) that by mathematical consistency we mean long-time and large-data existence of (suitable) weaksolution\dagger , its uniqueness, regularity and further qualitative properties.
mathematically self-consistent; long-time
Evolutionary power-law fluid model (1.3) is forexistence of weak solution has been proved by Ladyzhenskaya (see [7], [8]), using the monotone operator
theory and Minty’s trick to idcrltify the “ correct” limit of the viscosity term, and the compact embcdding. Regularity and consequently uniqueness for a
to obtain the correct limit in quadratic termsmooth initial velocity is addressed in [15]. Note that the non-degenerate case (i.e. the Ladyzhenskaya’s
the results on rcgularity and umiqucncss forand onfluids) has becn trcated before and forinwere put into place by Bellout, Bloom and Ne\v{c}as in [1], M\’alek, Ne\v{c}as andexistence forin [12], see also [16] for more details.
[13] and by M\’alek, Ne\v{c}as, Rokyta and(under more general assumptions
The long-time and large-data existence of weak solution foris achieved by Bul\’i\v{c}ek et al. in [3] for theon smoothness of data than in [12] for example) and for
spatially periodic and Navicr’s slip boundary value problcms\ddagger . Thc same task to establish long-time andlarge-data existence result for the no-slip boundary conditions is non-trivial and it is addressed in the
recent work by Wolf [24].
It is worth noting that thc rcsult in [3] holds also for a ccrtain class of incompressible fluids withpressure and shear rate dependent viscosity for Navier’s slip boundary conditions, extending thus the$r= \frac{11}{5}$$\mathrm{r}\mathrm{e}\mathrm{c}$$r \geq\frac{11}{5}$$\mathrm{d}\mathrm{i}\mathrm{v}(\mathrm{v}\otimes \mathrm{v})$$r \geq\frac{11}{5}$$r \geq\frac{11}{5}$$\mathrm{R}\mathrm{u}’\check{\mathrm{z}}\mathrm{i}\check{\mathrm{c}}\mathrm{k}\mathrm{a}$$r> \frac{9}{6}$$\mathrm{R}^{\mathrm{o}}\mathrm{u}\check{\mathrm{z}}\mathrm{i}6\mathrm{k}\mathrm{a}$$r \leq\frac{11}{5}$$r> \frac{8}{5}$\dag er Since the balance equations of continuum physics are formulated over any measurable sets and such formulations aretantamount to the weak forms of the balance equations that are formulated pointwise, the notion of (suitable) weak solutions
seems to primary, while the notion of smooth solution is rather secondary. This point of view has been obvious to Oseen,

and corlsequently to Leray [9].
\d ag er This solution is suitablc weak solution in thc scnse of Caffarcli, Kohn and Nircnbcrg, see [4], if$r> \frac{9}{\mathrm{s}}$.

29thcory developcd in [11] for spatially periodic problem. On the othcr hand, it is not at all clear if theresult by Wolf can be extended to time-dependent flows of fluids with shear rate and pressure dependent
viscosity that are subject to no-slip boundary conditions.
. Supported by the
Another question of interest is the long-time and large-data existence for. The main
analysis (see [6] and [5]) of steady flows one may conjecture that it may be true for
ncw ingrcdience incorporated by Frehsc ct al. in [6], see also [5], is based on an improvement of properties
regarding Lipschitz truncations of Sobolev functions.
We complete this section by providing a classification of the difficulties of the problen (1.2) according
solving (1.2) andfor
defined through (3.1), andasto the bchavior of
fulfilling either (2.6) or (2.7).
Considering firstof the form (2.6), we compute $(r=1)$$r< \frac{8}{5}$$r \in(\frac{6}{5}, \frac{8}{5}]$$d(p^{\lambda}, \mathrm{v}^{\lambda})$$\lambdaarrow\infty$$(\mathrm{v}^{\lambda},p^{\lambda})$$d(p, \mathrm{v})$$(\mathrm{v}^{\lambda},p^{\lambda})$

(3.4)

dr.

$d( \mathrm{p}^{\lambda},\mathrm{v}^{\lambda})=2\nu_{1}\int_{-1}^{0}\int_{\mathcal{B}_{1}(0)}p^{\lambda}|\mathrm{D}_{x}(\mathrm{v}^{\lambda})|dxdt=\lambda^{2\alpha-3}2\nu_{1}\int_{-\lambda^{1+\alpha}}^{0}\int_{B_{\lambda}(0)}p|\mathrm{D}_{y}(\mathrm{v})|dy$),), subc,ritical (formodel can be classified as supcrcritical (forIt implies that
despitefact
thatdifferently,
the
Sayingscaling.
dependence)
inof the consideredor critical (forsolutions to the time-dependent Schaeffer’s model ((2.1) with $r=1$ ) are invariant with respect to various
changes of scale, this invariance cannot be used to assign to the problem one of the labels: critical,$\alpha>\frac{3}{2}$$\alpha<\frac{3}{2}$$\mathrm{S}\mathrm{c}\mathrm{h}\mathrm{a}\epsilon \mathrm{f}\mathrm{f}\mathrm{e}\mathrm{r}’ \mathrm{s}$$\alpha=\frac{3}{2}$subcritical, supercritical.
On thc other hand, solutions of the system (1.2) for $r\geq 1$ are invariant with rcspect to the$(2,7)$ and then$d(p^{\lambda}, \mathrm{v}^{\lambda})=2\nu_{1}\int_{-1}^{0}\int_{B_{1}(0)}p^{\lambda}|\mathrm{D}_{x}(\mathrm{v}^{\lambda})|^{r}dxdt=\frac{2\nu_{1}}{\lambda^{5}}\int_{-1}^{0}\int_{B_{\lambda}(0)}p|\mathrm{D}_{y}(\mathrm{v})|^{r}dyd\tau$Since$d(p^{\lambda}, \mathrm{v}^{\lambda})arrow 0\mathrm{a}‘ \mathrm{s}\lambdaarrow\infty$.

$\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{i}\iota$(3.5), the problem is supercritical for all $r>1$ .

4. Singular self-similar solutions
In his article [9], Leray used the property (2.1) and proposed a construction of a self-simlar weak solution
exhibiting a singularity at a critical time $T>0$ . Leray’s program can be easily generalized to power-law
fluids (1.3), which wc will do next.
represents a nontrivial (weak\dagger ) solution of the system
Assume that$(\mathrm{U},Q)$$\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{U}=0$,$\frac{3-r}{2}y_{k}\frac{\theta \mathrm{U}}{\theta y_{k}}+\frac{r-1}{2}\mathrm{U}+U_{k}\frac{\partial \mathrm{U}}{\partial y_{k}}-\mathrm{d}\mathrm{i}\mathrm{v}(|\mathrm{D}(\mathrm{U})|^{r-2}\mathrm{D}(\mathrm{U}))=-\nabla Q$

.

(4.1)

Then
$\mathrm{v}_{*}(t, x)==^{1}(T-t)^{\mathrm{r}-1}\mathrm{U}((T-=^{X}t)^{3-r})$,$p_{*}(t, x)= \frac{1}{(T-t)^{r-1}}Q(_{(T-t)^{3-r}}=^{X})$(4.2)

form a weak solution to (1.3) that develops the singularity at as $tarrow T-$ .
Thus, in this appraoch the construction of singular weak solutions to (1.3) “reduces” to the questionof existcnce of nontrivial solutions to (4.1).
For $r=2$ , Ne\v{c}as et al. in [18] completed the original Leray’s program by showing that if. The proof relies on the fact that the scalar quantity
is a weak solution of (4.1) (with $r=2$ ) thensatisfics the maximum principlc. This resltlt wa‘s generalizcd in [23]; a simple proof
under stronger assumption is given in [10].
$T$$\mathrm{U}\in L^{3}(\mathrm{R}^{3})^{3}$$\mathrm{U}\equiv 0$$\frac{1}{2}|\mathrm{U}|^{2}+Q+y\cdot \mathrm{U}$\dag er We rnay for example assume t,hat$\mathrm{U}\in L^{2}(\mathbb{R}^{3})^{S}\cap W^{1,r}(\mathrm{R}^{3})^{3}$.

30

If $r\neq 2$ , one observes from how wcak solutions to (4.1) arc introduced that if
we have$r> \frac{11}{5}$then$\mathrm{U}\equiv 0$,For$r< \frac{11}{5}$$||\mathrm{v}_{\delta}(t)||_{2}^{2}=(T-t)^{11-5r}||\mathrm{U}||_{2}^{2}arrow 0$$|| \nabla \mathrm{v}_{\epsilon}(t)||_{r}^{r}=\frac{||\nabla \mathrm{U}||_{r}^{r}}{(T-t)^{(5r-9)/2}}arrow+\infty$

as $tarrow T-$ ,as $tarrow T-$ for$|| \nabla \mathrm{v}_{\mathit{8}}(t)||_{s}^{s}=\frac{||\nabla \mathrm{U}||_{2}^{2}}{(T-t)^{(3\mathrm{r}+2s-9)/2}}arrow+\infty$$r> \frac{9}{5}$

as $tarrow T$ –for,$s> \frac{9-3r}{2}$.

The last line just indicates that for $rarrow 1+$ , the higher norms of Vv blow-up.
We end up this section by preliminary results dealing with explicit constructions of nontrivial solutions
to (4.1). We refer to [14] for details.
Set $\Psi(y):=(y_{3}-y_{2})^{2}+(y_{1}-y_{3})^{2}+(y_{2}-y_{1})^{2}$ and consider solutions to (4.1) of the form$\mathrm{U}(y)=\frac{A}{[\Psi(y)]^{\alpha}}(y\mathrm{s}-y_{2}, y_{1}-y_{3}, y_{2}-y\iota)$,$P(y)=Q(\Psi(y))$ for $Q$ smooth.

(4.3)

As shown in [14], solutions of (4.1) having the structure (4.3) exist if$1<r< \frac{3}{2}$and$\alpha=\frac{1}{2-r}$.

Note that if we consider time-indcpendent flows, singular solutions of thc form (4.3) exist for all $r>1$
for example) for the. This suggestv to look for counterexamples to regularity (asandgeneralized Stokes-like systems, whereas “ generalized”’ means that is non-constant, and depends on, etc.
or$C^{1,\alpha}$$\alpha=\frac{1}{r-1}$$p$$\nu_{g}$$|\mathrm{D}(\mathrm{v})|$

5. Conclusion
The main objectives of this contribution are twofold.
First, we wished to point it out that there are interesting models of non-Newtonian mechanics forincompressible fluids that deserve the attention of mathematical analysts. Since the pressure is mostly
eliminated from the analysis of the Navier-Stokes equations (despite the fact that the pressure can play asignificant role in understanding of the whole system, see [20] for example), most of the studies of models
for incompressible fluids with thc viscosity depending on the pressurc (and the shear rate) requires todeal with the pressure since the beginning. Here, we restrict ourselves to a few special models (power-law
fluid, Schaeffer’s fluid and their combinations) of one class of non-Newtonian fluids, namely fluids with
non-constant viscosity, that share the property that their solutions are invariant with respect to a suitablescaling.
Second, using this scale invariance we show that powcr-law fluids can be classified in dcpendence on
the power-law index . Interestingly, this standard classification is not applicable to Schaeffer’s model.
We also documented that power-law models are well understood if the power-law index satisfies theand the problem is subcritical or critical, whilc there is a lot of open problems if
condition(the Navier-Stokes system falls to this range).
$r$

$r$

$r< \frac{11}{6}$$r \geq\frac{11}{5}$Acknowledgments. I thank organizers, especially to Yoshiyuki Kagei, for inviting me to Kyoto meetingand giving me the opportunity to meet Japan mathematical community dealing with analysis of various
fluid models. I also wish to thank to Professor Tetsumura Miyakawa.

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