# Branching problem of tensoring two Verma modules and its application to
differential symmetry breaking operators

Reiji Murakami Graduate School of Mathematical Science, The University of
Tokyo

###### Abstract

Kobayashi-Pevzner discovered in [Selecta Math., 2016] that the failure of the
multiplicity-one property in the fusion rule of Verma modules of
$\mathfrak{sl}_{2}$ occurs exactly when the Rankin-Cohen bracket vanishes, and
classified all the corresponding parameters. In this paper we provide yet
another characterization for these parameters, and give a precise description
of indecomposable components of the tensor product. Furthermore, we discuss
when the tensor products of two Verma modules are isomorphic to each other for
semisimple Lie algebras $\mathfrak{g}$.

Keywords— branching laws, symmetry breaking operator, tensor product, Verma
module, BGG category, Lie algebra

##  1 Introduction.

The object of this article is the tensor product of two Verma modules.
Recently, a new phenomenon has been discovered in the
$\mathfrak{sl}_{2}$-case, including the failure of the multiplicity-freeness
in the fusion rule for singular integral parameters, see [6, Sect. 9] for
details. This article provides a precise description of indecomposable
components of the tensor product in the $\mathfrak{sl}_{2}$-case (Theorem
1.3), and proves that the failure of the self-duality of indecomposable
components corresponds exactly to the multiplicity-two phenomenon (Theorem
1.2).

As an introduction, we begin with a general result of the tensor product of
two Verma modules. Let $\mathfrak{g}$ be a complex semisimple Lie algebra,
$\mathfrak{h}$ a Cartan subalgebra, and $\mathfrak{b}$ a Borel subalgebra
containing $\mathfrak{h}$. Let
$M(\mu):=U(\mathfrak{g})\otimes_{U(\mathfrak{b})}\mathbb{C}_{\mu}$ be the
Verma module for $\mu\in\mathfrak{h}^{*}$, see Section 2.1 for further
notation.

######  Theorem 1.1.

Suppose that
$\mu^{\prime},\mu^{\prime\prime},\nu^{\prime},\nu^{\prime\prime}\in\mathfrak{h}^{*}$
satisfy $\mu^{\prime}+\mu^{\prime\prime}=\nu^{\prime}+\nu^{\prime\prime}$. If
each of the two sets $\\{\mu^{\prime},\mu^{\prime\prime}\\}$ and
$\\{\nu^{\prime},\nu^{\prime\prime}\\}$ contains at least one anti-dominant
element, then one has an isomorphism as $\mathfrak{g}$-modules:

$\displaystyle M(\mu^{\prime})\otimes M(\mu^{\prime\prime})\cong
M(\nu^{\prime})\otimes M(\nu^{\prime\prime}).$

The anti-dominant assumption cannot be dropped, as we shall see in Example
1.7. In general, it is a hard problem to give a precise description of the
decomposition of $M(\mu^{\prime})\otimes M(\mu^{\prime\prime})$, in
particular, because the tensor product is not necessarily completely reducible
even when both $M(\mu^{\prime})$ and $M(\mu^{\prime\prime})$ are irreducible.

Let $\mathfrak{g}:=\mathfrak{sl}_{2}(\mathbb{C})$. We identify
$\mathfrak{h}^{*}$ with $\mathbb{C}$ such that the unique positive root is
given by $2$ as usual. Then, the Verma module $M(\mu)$ has a
$(\mu+1)$-dimensional simple quotient $L(\mu)$ if
$\mu\in\mathbb{N}:=\\{0,1,2,\cdots\\}$. Following the notation as in [6, p.
894], we define the multiplicity space of $M(-\lambda^{\prime\prime\prime})$
in $M(-\lambda^{\prime})\otimes M(-\lambda^{\prime\prime})$ by:

$\displaystyle
H(\lambda^{\prime},\lambda^{\prime\prime},\lambda^{\prime\prime\prime}):=\mathop{\mathrm{Hom}}\nolimits_{\mathfrak{g}}(M(-\lambda^{\prime\prime\prime}),M(-\lambda^{\prime})\otimes
M(-\lambda^{\prime\prime})).$

Kobayashi-Pevzner [6, Thm. 9.1] discovered that the Rankin-Cohen brackets
$\mathcal{RC}_{\lambda^{\prime},\lambda^{\prime\prime}}^{\lambda^{\prime\prime\prime}}$
(see (1) below) vanishes if and only if
$H(\lambda^{\prime},\lambda^{\prime\prime},\lambda^{\prime\prime\prime})$ is
of dimension $2$, and found all the triples
$\\{\lambda^{\prime},\lambda^{\prime\prime},\lambda^{\prime\prime\prime}\\}$
of parameters when this happens. In Theorem 1.2, we give yet another
characterization of this condition by means of the $\mathfrak{g}$-module
structure of the $Z(\mathfrak{g})$-primary component
$(M(-\lambda^{\prime})\otimes
M(-\lambda^{\prime\prime}))^{\chi_{-\lambda^{\prime\prime\prime}+1}}$
corresponding to the generalized infinitesimal character
$-\lambda^{\prime\prime\prime}+1$, see Section 2.1.

######  Theorem 1.2.

If $\lambda^{\prime\prime\prime}-\lambda^{\prime}-\lambda^{\prime\prime}\in
2\mathbb{N}$, then the following four conditions on the triple
$(\lambda^{\prime},\lambda^{\prime\prime},\lambda^{\prime\prime\prime})$ are
equivalent:
(i) $\dim
H(\lambda^{\prime},\lambda^{\prime\prime},\lambda^{\prime\prime\prime})=2$.
(ii)
$\lambda^{\prime},\lambda^{\prime\prime},\lambda^{\prime\prime\prime}\in\mathbb{Z},\
2\geq\lambda^{\prime}+\lambda^{\prime\prime}+\lambda^{\prime\prime\prime},\
and\
\lambda^{\prime\prime\prime}\geq|\lambda^{\prime}-\lambda^{\prime\prime}|+2$.
(iii)
$\mathcal{RC}_{\lambda^{\prime},\lambda^{\prime\prime}}^{\lambda^{\prime\prime\prime}}=0$.
(iv) $2\leq\lambda^{\prime\prime\prime}$ and $(M(-\lambda^{\prime})\otimes
M(-\lambda^{\prime\prime}))^{\chi_{-\lambda^{\prime\prime\prime}+1}}=M(-\lambda^{\prime\prime\prime})\oplus
M(\lambda^{\prime\prime\prime}-2)$.

The equivalences $(i)\Leftrightarrow(ii)\Leftrightarrow(iii)$ were established
earlier in [6, Sect. 9]. The operator
$\mathcal{RC}_{\lambda^{\prime},\lambda^{\prime\prime}}^{\lambda^{\prime\prime\prime}}$
in $(iii)$ is a bidifferential operator, referred to as the Rankin-Cohen
bracket of degree
$\ell:=\frac{1}{2}(\lambda^{\prime\prime\prime}-\lambda^{\prime}-\lambda^{\prime\prime})\in\mathbb{N}$,
defined by

$\mathcal{RC}_{\lambda^{\prime},\lambda^{\prime\prime}}^{\lambda^{\prime\prime\prime}}:=\sum_{j=0}^{\ell}(-1)^{j}\frac{(\lambda^{\prime}+\ell-1)_{j}(\lambda^{\prime\prime}+\ell-1)_{\ell-j}}{j!(\ell-j)!}\frac{\partial^{\ell}}{\partial{z_{1}}^{\ell-j}\partial{z_{2}}^{j}}\Bigg{\rvert}_{z_{1}=z_{2}},$
(1)

where $(x)_{n}$ stands for the descending factorial $x(x-1)\cdots(x-n+1)$. The
operator was introduced originally in number theory to produce holomorphic
modular forms of higher weight
$\lambda^{\prime\prime\prime}=\lambda^{\prime}+\lambda^{\prime\prime}+2\ell\
(\ell\in\mathbb{N})$ out of those of lower weights $\lambda^{\prime}$ and
$\lambda^{\prime\prime}$. From the viewpoint of representation theory,
$\mathcal{RC}_{\lambda^{\prime},\lambda^{\prime\prime}}^{\lambda^{\prime\prime\prime}}$
is a symmetry breaking operator from a $\tilde{G}\times\tilde{G}$-equivalent
holomorphic line bundle to a $\tilde{G}$-equivalent one over the Poincaré disk
where $\tilde{G}$ denotes the universal covering group of $SU(1,1)$. Such an
operator shows how $\tilde{G}\times\tilde{G}$-symmetry is “broken” in keeping
the diagonal symmetry of $\tilde{G}$. See [6, Sect. 1.1 and Sect. 9].

The novelty of Theorem 1.2 is the equivalence of $(ii)\Leftrightarrow(iv)$,
which is derived from Theorem 1.3 about a description of the
$\mathfrak{g}$-module structure of the tensor product of two Verma modules.
Let $P(a)$ denote the projective covering of $M(a)$, see Definition 2.1 and
Example 2.2.

######  Theorem 1.3.

For $\mu^{\prime},\mu^{\prime\prime}\in\mathbb{C}$, one has an isomorphism as
$\mathfrak{g}$-modules:

$M(\mu^{\prime})\otimes M(\mu^{\prime\prime})\cong\bigoplus_{a\in
A}P(a)\oplus\bigoplus_{b\in B}M(b).$

We set $X:=\mu^{\prime}+\mu^{\prime\prime}-2\mathbb{N}$. The finite (possibly
empty) set $A\equiv A(\mu^{\prime},\mu^{\prime\prime})$ and a countable set
$B\equiv B(\mu^{\prime},\mu^{\prime\prime})$ are defined as follows:

$\displaystyle A$ $\displaystyle:=\begin{cases}\emptyset&if\
\mu^{\prime}+\mu^{\prime\prime}\notin\mathbb{N},\\\ X\ \cap\
[-|\mu^{\prime}-\mu^{\prime\prime}|,-2]&if\
\mu^{\prime},\mu^{\prime\prime}\in\mathbb{N},\\\ X\ \cap\
[-\mu^{\prime}-\mu^{\prime\prime}-2,-2]&otherwise,\end{cases}$ $\displaystyle
B$ $\displaystyle:=X\setminus(A\cup A^{*}),$

where $A^{*}:=\\{-a-2\in\mathbb{Z}:\ a\in A\\}$.

See [5, Thm. 4.10] for analogous results on the restriction of parabolic Verma
modules with respect to $\mathfrak{so}_{n+1}\supset\mathfrak{so}_{n}$.

######  Corollary 1.4.

For the tensor product $M(\mu^{\prime})\otimes M(\mu^{\prime\prime})$, the
following four conditions on
$(\mu^{\prime},\mu^{\prime\prime})\in\mathbb{C}^{2}$ are equivalent.
(i)
$\dim\mathop{\mathrm{Hom}}\nolimits_{\mathfrak{g}}(N,M(\mu^{\prime})\otimes
M(\mu^{\prime\prime}))\leq 1$ for any simple $\mathfrak{g}$-module
$N\in\mathcal{O}$;
(ii)
$\dim\mathop{\mathrm{Hom}}\nolimits_{\mathfrak{g}}(M(\nu),M(\mu^{\prime})\otimes
M(\mu^{\prime\prime}))\leq 1$ for all $\nu\in\mathbb{C}$;
(iii) the tensor product $M(\mu^{\prime})\otimes M(\mu^{\prime\prime})$ is
self-dual;
(iv) it does not contain any reducible Verma module as a direct summand.

We shall use Remark 1.6 in the proof of Theorem 1.2.

######  Remark 1.5.

The finite set $A$ is contained in $\\{-2,-3,-4,\cdots\\}$, see Example 2.2.
It is non-empty if and only if
$(\mu^{\prime},\mu^{\prime\prime})\in\mathbb{C}^{2}$ satisfies
$\mu^{\prime}+\mu^{\prime\prime}\in\mathbb{N}$ and
$\mu^{\prime}-\mu^{\prime\prime}\notin\\{0,\pm 1\\}$.

######  Remark 1.6.

For $\nu\in\mathbb{N}$, the Verma module $M(\nu)$ occurs in
$M(\mu^{\prime})\otimes M(\mu^{\prime\prime})$ as a direct summand if and only
if $\mu^{\prime},\mu^{\prime\prime}\in\mathbb{N}$ and
$|\mu^{\prime}-\mu^{\prime\prime}|\leq\nu\leq\mu^{\prime}+\mu^{\prime\prime}$.
See [6, Thm. 9.1]. This condition is an analogue of the Clebsch-Gordan
formula.

######  Example 1.7.

If
$(\mu^{\prime},\mu^{\prime\prime},\nu^{\prime},\nu^{\prime\prime})=(0,0,i,-i)$,
then the tensor product $M(\mu^{\prime})\otimes M(\mu^{\prime\prime})$ is not
isomorphic to $M(\nu^{\prime})\otimes M(\nu^{\prime\prime})$ because Theorem
1.3 shows

$\displaystyle M(0)\otimes M(0)$
$\displaystyle\cong\bigoplus_{k=0}^{\infty}M(-2k),$ $\displaystyle M(i)\otimes
M(-i)$ $\displaystyle\cong P(-2)\oplus\bigoplus_{k=2}^{\infty}M(-2k),$

but $P(-2)$ is not isomorphic to the direct sum $M(0)\oplus M(-2)$. In this
case, the assumption in Theorem 1.1 is not satisfied because $0$ is not anti-
dominant.

##  2 Proof of Theorem 1.1.

Our proof relies on the earlier work [4] and [2]. Kobayashi [4] established a
general framework of the restriction of parabolic Verma modules to reductive
subalgebras, and proved the branching rule in the Grothendieck group for the
discretely decomposable case, whereas Kåhrström [2] showed that a tensoring
with Verma modules preserves Verma flags.

###  2.1 Preliminaries of parabolic Verma modules.

We collect some basic notions of parabolic Verma modules, see [1]. Let
$\mathfrak{g}$ be a semisimple Lie algebra over $\mathbb{C}$, $\mathfrak{h}$ a
Cartan subalgebra, and W the Weyl group of the root system
$\Delta:=\Delta(\mathfrak{g},\mathfrak{h})$. We fix a positive system
$\Delta^{+}$, take the corresponding Borel subalgebra
$\mathfrak{b}:=\mathfrak{h}+\mathfrak{n}$ with nilpotent radical
$\mathfrak{n}=\oplus_{\alpha\in\Delta^{+}}\mathfrak{g}_{\alpha}$, and let
$\rho$ be half the sum of positive roots. A standard parabolic subalgebra
$\mathfrak{p}$ is a parabolic subalgebra containing the Borel subalgebra
$\mathfrak{b}$. We take a Levi decomposition
$\mathfrak{p}=\mathfrak{l}+\mathfrak{u}$ with
$\mathfrak{h}\subset\mathfrak{l}$, and set
$\Delta^{+}(\mathfrak{l}):=\Delta^{+}\cap\Delta(\mathfrak{l},\mathfrak{h})$
and
$\mathfrak{n}_{-}(\mathfrak{l}):=\oplus_{\alpha\in\Delta^{+}(\mathfrak{l})}\mathfrak{g}_{-\alpha}$.
Let $U(\mathfrak{g})$ be the universal enveloping algebra of $\mathfrak{g}$,
and $Z(\mathfrak{g})$ its center.

A $\mathfrak{g}$-module $M$ is called locally $Z(\mathfrak{g})$-finite if for
every $x\in M$ the space $Z(\mathfrak{g})x$ is finite dimensional. Let
$\mathcal{Z}_{fin}$ be the category of locally $Z(\mathfrak{g})$-finite
$\mathfrak{g}$-modules. Category $\mathcal{O}$ is the full subcategory of
$\mathcal{Z}_{fin}$ such that the objects are finitely generated and
$\mathfrak{h}$-semisimple, and locally $\mathfrak{n}$-finite. Category
$\mathcal{O}^{\mathfrak{p}}$ is the full subcategory of $\mathcal{O}$ such
that the objects are locally $\mathfrak{n}_{-}(\mathfrak{l})$-finite.

We write the set of dominant integral for
$\Lambda^{+}(\mathfrak{l}):=\\{\lambda\in\mathfrak{h}^{*}:\langle\lambda,\alpha^{\vee}\rangle\in\mathbb{N}\quad\text{for
all}\ \alpha\in\Delta^{+}(\mathfrak{l})\\}$ where $\alpha^{\vee}$ denotes the
coroot of $\alpha$. For $\lambda\in\Lambda^{+}(\mathfrak{l})$, let
$F_{\lambda}$ denote the finite-dimensional irreducible $\mathfrak{l}$-module
with highest weight $\lambda$. We also write $\mathbb{C}_{\lambda}$ for
$F_{\lambda}$ if it is one-dimensional. We regard $F_{\lambda}$ (and
$\mathbb{C}_{\lambda}$) as a $\mathfrak{p}$-module by letting $\mathfrak{u}$
act trivially. The $\mathfrak{g}$-module
$M^{\mathfrak{g}}_{\mathfrak{p}}(\lambda):=U(\mathfrak{g})\otimes_{U(\mathfrak{p})}F_{\lambda}$
is referred to as a parabolic Verma module. For $\mathfrak{p}=\mathfrak{b}$,
it reduces to the usual Verma module $M(\lambda)\equiv
M^{\mathfrak{g}}_{\mathfrak{b}}(\lambda)=U(\mathfrak{g})\otimes_{U(\mathfrak{b})}\mathbb{C}_{\lambda}$
for $\lambda\in\mathfrak{h}^{*}$. An element $\lambda\in\mathfrak{h}^{*}$ is
said to be anti-dominant if
$\langle\lambda+\rho,\alpha^{\vee}\rangle\notin\mathbb{N}_{+}=\\{1,2,\cdots\\}$
for all $\alpha\in\Delta^{+}$, equivalently, it $M(\lambda)$ is simple.

The Harish-Chandra isomorphism parametrizes the set
$\mathop{\mathrm{Hom}}\nolimits_{\mathbb{C}\text{-alg}}(Z(\mathfrak{g}),\mathbb{C})$
of $\mathbb{C}$-algebraic homomorphisms as

$\displaystyle\mathfrak{h}^{*}/W\cong\mathop{\mathrm{Hom}}\nolimits_{\mathbb{C}\text{-alg}}(Z(\mathfrak{g}),\mathbb{C}),\
\nu\mapsto\chi_{\nu}.$

In our parametrization, the infinitesimal character of the trivial module is
$\rho\in\mathfrak{h}^{*}/W$ and that of the Verma module $M(\lambda)$ is
$\lambda+\rho$. Any locally $Z(\mathfrak{g})$-finite $\mathfrak{g}$-module $M$
admits a primary decomposition

$\displaystyle M=\bigoplus_{\nu\in\mathfrak{h}*/W}M^{\chi_{\nu}},$

where $M^{\chi_{\nu}}$ is the $\nu$-primary component of $\nu$, namely,
$M^{\chi_{\nu}}$ is the $\mathfrak{g}$-submodule of elements annihilated by a
power of
$\mathop{\mathrm{Ker}}\nolimits(\chi_{\nu}:Z(\mathfrak{g})\rightarrow\mathbb{C})$.

######  Definition 2.1 ([1, Ch. 3.9]).

For $M\in\mathcal{O}$, a surjective $\mathfrak{g}$-map $\phi:P\to M$ is called
a projective covering of $M$ if $P\in\mathcal{O}$ is a projective module
without any proper submodule such that its image of $\phi$ equals $M$.

We omit $\phi$ when it is obvious. For $\lambda\in\mathfrak{h}^{*}$, let
$P(\lambda)$ denote the projective cover of the Verma module $M(\lambda)$.

######  Example 2.2.

Let $\mathfrak{g}=\mathfrak{sl}_{2}(\mathbb{C})$. For
$\lambda\notin\\{-2,-3,\cdots\\}$, one has $P(\lambda)=M(\lambda)$. If
$\lambda\in\\{-2,-3,\cdots\\}$, there is a non-split short exact sequence of
$\mathfrak{g}$-modules:

$\displaystyle 0\rightarrow M(-\lambda-2)\rightarrow P(\lambda)\rightarrow
M(\lambda)\rightarrow 0.$

###  2.2 Branching problem for parabolic Verma modules.

The tensor product $M(\mu^{\prime})\otimes M(\mu^{\prime\prime})$ may be
viewed as the restriction of the Verma module
$M(\mu^{\prime},\mu^{\prime\prime})=U(\mathfrak{g}\oplus\mathfrak{g})\otimes_{U(\mathfrak{b}\oplus\mathfrak{b})}\mathbb{C}_{(\mu^{\prime},\mu^{\prime\prime})}$
to the subalgebra $\mathop{\mathrm{diag}}\nolimits(\mathfrak{g})$ of
$\mathfrak{g}\oplus\mathfrak{g}$. We then utilize the general framework in [3]
and [4]. In particular, the concept of discrete decomposability of
$\mathfrak{g}$-modules, which was introduced in [3] as an algebraic
counterpart of the unitary representation without continuous spectrum, is
useful in our study of the tensor product.

######  Definition 2.3 ([4, Def. 3.1]).

A $\mathfrak{g}$-module $X$ is called discretely decomposable if there is an
ascending sequence $\\{X_{i}\\}_{i\in\mathbb{N}}$ of submodules such that
$\bigcup X_{i}=X$ and that each $X_{i}$ has finite length.
Moreover for a full subcategory $\mathcal{A}$ of the category of
$\mathfrak{g}$-modules, we say $X\in\mathcal{A}$ is discretely decomposable in
$\mathcal{A}$ if every $X_{i}$ belongs to $\mathcal{A}$.

Consider the restriction of a generalized Verma module
$M^{\mathfrak{g}}_{\mathfrak{p}}(\mu)$ with respect to a reductive subalgebra
$\mathfrak{g}^{\prime}$ of $\mathfrak{g}$. In general, the restriction may not
contain any simple $\mathfrak{g}^{\prime}$-module. A geometric criterion on
the pair $(\mathfrak{g}^{\prime},\mathfrak{p})$ was proved in [4] about when
the restriction
$M^{\mathfrak{g}}_{\mathfrak{p}}(\mu)\rvert_{\mathfrak{g}^{\prime}}$ is
descretely deconposable. In what follows, we recall an algebraic condition
which is slightly stronger than this geometric condition.
A semisimple element $H\in\mathfrak{h}$ is said to be hyperbolic if
$\mathop{\mathrm{ad}}\nolimits(H)$ is diagonalizable with all eigenvalues
being real. We write $\mathfrak{g}=\mathfrak{u}_{-}+\mathfrak{l}+\mathfrak{u}$
for the decomposition into the sum of eigenspaces of
$\mathop{\mathrm{ad}}\nolimits(H)$ with negative, $0$ and positive
eigenvalues, respectively. Then $\mathfrak{p}(H):=\mathfrak{l}+\mathfrak{u}$
is a parabolic subalgebra of $\mathfrak{g}$.

######  Definition 2.4 ([4, Def. 3.7]).

A parabolic subalgebra $\mathfrak{p}$ is said to be
$\mathfrak{g}^{\prime}$-$compatible$ if there exists a hyperbolic element $H$
in $\mathfrak{g}^{\prime}$ such that $\mathfrak{p}=\mathfrak{p}(H)$.

If $\mathfrak{p}=\mathfrak{l}+\mathfrak{u}$ is
$\mathfrak{g}^{\prime}$-compatible, then
$\mathfrak{p}^{\prime}:=\mathfrak{p}\cap\mathfrak{g}^{\prime}$ becomes a
parabolic subalgebra of $\mathfrak{g}^{\prime}$ with the Levi decomposition:

$\mathfrak{p}^{\prime}=\mathfrak{l}^{\prime}+\mathfrak{u}^{\prime}:=(\mathfrak{l}\cap\mathfrak{g}^{\prime})+(\mathfrak{u}\cap\mathfrak{g}^{\prime}).$
(2)

We use the following fact that guarantees the discrete decomposability of the
restriction.

######  Fact 2.5 ([4, Prop. 3.8]).

If a parabolic subalgebra $\mathfrak{p}$ of $\mathfrak{g}$ is
$\mathfrak{g}^{\prime}$-$compatible$, then the restriction
$X\rvert_{\mathfrak{g}^{\prime}}$ is discretely decomposable in
$\mathcal{O}^{\mathfrak{p}^{\prime}}$ for any $X\in\mathcal{O}^{\mathfrak{p}}$

For a $\mathfrak{g}^{\prime}$-compatible parabolic subalgebra $\mathfrak{p}$,
each eigenspace of $H$ on $M^{\mathfrak{g}}_{\mathfrak{p}}(\mu)\cong
U(\mathfrak{u}_{-})\otimes F_{\mu}$ is finite-dimensional, hence one has:

######  Remark 2.6.

The weight spaces of
$M^{\mathfrak{g}}_{\mathfrak{p}}(\mu)\rvert_{\mathfrak{g}^{\prime}}$ are all
finite-dimensional.

Given a vector space V, we denote by $S(V):=\oplus_{k=0}^{\infty}S^{k}(V)$ the
symmetric tensor algebra. Retain the notation as in $\eqref{eqn:smallpara}$.
We extend the adjoint action of $\mathfrak{l}^{\prime}$ on
$\mathfrak{u}_{-}/\mathfrak{u}_{-}\cap\mathfrak{g}^{\prime}$ to
$S(\mathfrak{u}_{-}/\mathfrak{u}_{-}\cap\mathfrak{g}^{{}^{\prime}})$. As in
[4, Sect. 3.3], we take a Cartan subalgebra $\mathfrak{h}^{\prime}$ of
$\mathfrak{g}^{\prime}$ such that $H\in\mathfrak{h}^{\prime}$, and extend it
to a Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$. Clearly
$\mathfrak{h}\subset\mathfrak{l}$ and
$\mathfrak{h}^{\prime}\subset\mathfrak{l}^{\prime}$. For
$\delta\in\Lambda^{+}(\mathfrak{l}^{\prime})$, we denote by
$F_{\delta}^{\prime}$ the simple $\mathfrak{l}^{\prime}$-module of highest
weight $\delta$. We set

$\displaystyle
m(\delta;\mu):=\dim\mathop{\mathrm{Hom}}\nolimits_{\mathfrak{l}^{\prime}}(F^{\prime}_{\delta},F_{\mu}\otimes
S(\mathfrak{u}_{-}/\mathfrak{u}_{-}\cap\mathfrak{g}^{\prime})).$

######  Fact 2.7 ([4, Thm. 3.10]).

Suppose that $\mathfrak{p}=\mathfrak{l}+\mathfrak{u}$ is
$\mathfrak{g}^{\prime}$-compatible. Then for any
$\mu\in\Lambda^{+}(\mathfrak{l})$,
(1) $m(\delta;\mu)<\infty\ for\ all\
\delta\in\Lambda^{+}(\mathfrak{l^{\prime}})$.
(2) In the Grothendieck group,
$[M^{\mathfrak{g}}_{\mathfrak{p}}(\mu)\rvert_{\mathfrak{g}^{\prime}}]=\bigoplus_{\delta\in\Lambda^{+}(\mathfrak{l^{\prime}})}m(\delta;\mu)\
[M^{\mathfrak{g}^{\prime}}_{\mathfrak{p}^{\prime}}(\delta)]$.

######  Example 2.8.

We write $\Gamma$ for the semigroup generated by positive roots, and
$M(\mu)_{\nu}$ for the $\nu$-weight space of $M(\mu)$. Applying Fact 2.7 to
$(\mathfrak{g}\oplus\mathfrak{g},\mathop{\mathrm{diag}}\nolimits(\mathfrak{g}))$,
we get the following equation in the Grothendieck group:

$\displaystyle[M(\mu^{\prime})\otimes
M(\mu^{\prime\prime})]=\bigoplus_{\nu\in\mu^{\prime}+\mu^{\prime\prime}-\Gamma}\dim
M(\mu^{\prime}+\mu^{\prime\prime})_{\nu}\ [M(\nu)].$

We summarize some direct consequences of Fact 2.5, Remark 2.6, and Fact 2.7.

######  Corollary 2.9.

Retain the setting of Fact 2.7. For any $M\in\mathcal{O}^{\mathfrak{p}}$, in
particular for
$M=M^{\mathfrak{g}}_{\mathfrak{p}}(\mu)\rvert_{\mathfrak{g}^{\prime}}$, the
restriction $M\rvert_{\mathfrak{g}^{\prime}}$ admits a
$Z(\mathfrak{g}^{\prime})$-primary decomposition and each component belongs to
$\mathcal{O}^{\mathfrak{p}^{\prime}}$.

We then reduce a question to find the branching rules to some discussion in
BGG category $\mathcal{O}$.

###  2.3 Tensor products and Verma flags.

We recall from [1] and [2] the notion of Verma flags and some compatibility of
the tensor product of two modules in $\mathcal{O}$ and Verma flags.

######  Definition 2.10 ([2]).

Let $\tilde{\mathcal{O}}$ be the full subcategory of $\mathcal{Z}_{fin}$ of
which objects $M$ have the property that there are finitely many weights
$\lambda_{1},\cdots\lambda_{n}$ such that
$\Pi(M)\subset\bigcup(\lambda_{i}-\Gamma)$, where we denote by $\Pi(M)$ the
set of weights of $M$.

######  Remark 2.11.

For the symmetric pair
$(\mathfrak{g}\oplus\mathfrak{g},\mathop{\mathrm{diag}}\nolimits(\mathfrak{g}))$,
the Borel subalgebra $(\mathfrak{b}\oplus\mathfrak{b})$ is
$\mathop{\mathrm{diag}}\nolimits(\mathfrak{g})$-compatible, see Definition
2.4. Thus it follows from Fact 2.5 that for any $M,N\in\mathcal{O}$, the
tensor product $M\otimes N\in\tilde{\mathcal{O}}$ and admits a primary
decomposition.

Let us review dual modules in $\mathcal{O}$ and the Verma flags, see [1, Ch.
3].
Fix an anti-involution $\tau$ of a complex semisimple Lie algebra
$\mathfrak{g}$ such that $\tau$ switches $\mathfrak{g}_{\alpha}$ with
$\mathfrak{g}_{-\alpha}$ and is the identity on $\mathfrak{h}$. For
$M\in\mathcal{Z}_{fin}$, we define a $\mathfrak{g}$-module on

$\displaystyle
M^{\vee}:=\bigoplus_{\lambda\in\mathfrak{h}^{*}}\mathop{\mathrm{Hom}}\nolimits_{\mathbb{C}}(M_{\lambda},\mathbb{C})$

by $(X.f)(v):=f(\tau(X).v)$ for $f\in M^{\vee}$, $v\in M$, and
$X\in\mathfrak{g}$. For $M,N\in\mathcal{O}$, one has $(M\otimes N)^{\vee}\cong
M^{\vee}\otimes N^{\vee}$, see [2] for example.

We say $M\in\mathcal{Z}_{fin}$ has a Verma flag or standard filtration if
there is an ascending sequence of submodules $\\{M_{i}\\}$ such that $\bigcup
M_{i}=M$, $M_{0}=0$, and $M_{i+1}/M_{i}$ are isomorphic to some Verma module
for every $i$. If $M\in\mathcal{O}$, ${M_{i}}$ is stable for sufficiently
large $i$.

We write $M\in\bigtriangleup$ if $M$ has a Verma flag, $M\in\bigtriangledown$
if $M^{\vee}$ has a Verma flag, and $M\in\bigtriangleup\cap\bigtriangledown$
if $M$ and $M^{\vee}$ have a Verma flag.

######  Definition 2.12 ([1, Ch. 11]).

$X\in\mathcal{O}$ is called a tilting module if
$M\in\bigtriangleup\cap\bigtriangledown$.
$X\in\tilde{\mathcal{O}}$ is called a tilting module if all primary components
are tilting modules.

######  Fact 2.13 ([2, Prop. 2.6]).

Let $M,N\in\mathcal{O}$. For
$\mathcal{F}\in\\{\bigtriangleup,\bigtriangledown,\bigtriangleup\cap\bigtriangledown\\}$,
one has

$\displaystyle M\in\mathcal{F}\Rightarrow M\otimes N\in\mathcal{F}.$

It is known, see [1, Ch. 11], that the $\mathfrak{g}$-module structure of a
tilting module is determined by its Grothendieck group.

###  2.4 Proof of Theorem 1.1.

We are ready to complete the proof of Theorem 1.1.

######  Proposition 2.14.

If $\mu\in\mathfrak{h}^{*}$ and $N\in\mathcal{O}$, then one has the following
isomorphism in the Grothendieck group:

$\displaystyle[M(\mu)\otimes N]=\bigoplus_{\nu\in\Pi(N)}\dim N_{\nu}\
[M(\mu+\nu)].$

###### Proof.

Since $N$ is a $\mathfrak{g}$-module, one has the following isomorphism of
$\mathfrak{g}$-modules.

$\displaystyle M(\mu)\otimes
N=(U(\mathfrak{g})\otimes_{U(\mathfrak{b})}\mathbb{C}_{\mathfrak{\mu}})\otimes
N\cong U(\mathfrak{g})\otimes_{U(\mathfrak{b})}(\mathbb{C}_{\mu}\otimes N).$

We take a filtration $0=B_{0}\subset B_{1}\subset B_{2}\subset\cdots$ of
$\mathbb{C}_{\mu}\otimes N$ as a $\mathfrak{b}$-module such that
$B_{i+1}/B_{i}$ is isomorphic to $\mathbb{C}_{\mu_{i}}$ for some
$\mu_{i}\in\mu+\Pi(N)$. In turn,
$\\{U(\mathfrak{g})\otimes_{U(\mathfrak{b})}B_{i}\\}$ is a Verma flag of
$M(\mu)\otimes N$. Then the proposition follows because
$(U(\mathfrak{g})\otimes_{U(\mathfrak{b})}B_{i+1})/(U(\mathfrak{g})\otimes_{U(\mathfrak{b})}B_{i})\cong
M(\mu_{i})$. ∎

######  Remark 2.15.

If $N$ is a Verma module, this is a special case of [4, Thm. 3.10].

As a consequence of Proposition 2.14, we get the following corollary.

######  Corollary 2.16.

Suppose that
$\mu^{\prime},\mu^{\prime\prime},\nu^{\prime},\nu^{\prime\prime}\in\mathfrak{h}^{*}$
satisfy $\mu^{\prime}+\mu^{\prime\prime}=\nu^{\prime}+\nu^{\prime\prime}$,
then one has an isomorphism in the Grothendieck group:

$\displaystyle[M(\mu^{\prime})\otimes
M(\mu^{\prime\prime})]=[M(\nu^{\prime})\otimes M(\nu^{\prime\prime})].$

Under the assumption of Theorem 1.1, at least one of $M(\mu^{\prime})$ and
$M(\mu^{\prime\prime})$ is simple. Similarly for $M(\nu^{\prime})$ and
$M(\nu^{\prime\prime})$. Therefore we obtain Theorem 1.1 since simple Verma
modules belong to $\bigtriangleup\cap\bigtriangledown$.

##  3 Indecomposable components of $M(\mu^{\prime})\otimes
M(\mu^{\prime\prime})$ in the $\mathfrak{sl}_{2}$ case.

Throughout this section, we suppose $\mathfrak{g}$ is
$\mathfrak{sl}_{2}(\mathbb{C})$. We prove Theorem 1.3 and Corollary 1.4.

Since each weight space of Verma modules is one-dimensional for
$\mathfrak{sl}_{2}(\mathbb{C})$, the following equation holds in the
Grothendieck group from Proposition 2.14.

$\displaystyle[M(\mu^{\prime})\otimes
M(\mu^{\prime\prime})]=\bigoplus_{k\in\mathbb{N}}[M(\mu^{\prime}+\mu^{\prime\prime}-2k)].$
(3)

Let $M^{\chi_{\nu+1}}$ denote the $Z(\mathfrak{g})$-primary component of the
tensor product $M(\mu^{\prime})\otimes M(\mu^{\prime\prime})$ for the
generalized eigenvalue
$\nu+1\in\mathfrak{h}^{*}/W\cong\mathop{\mathrm{Hom}}\nolimits_{\mathbb{C}\text{-alg}}(Z(\mathfrak{g}),\mathbb{C})$.
Then one has a primary decomposition:

$\displaystyle M(\mu^{\prime})\otimes
M(\mu^{\prime\prime})\cong\bigoplus_{\nu+1\in
W\cdot(\mu^{\prime}+\mu^{\prime\prime}+1-2\mathbb{N})}M^{\chi_{\nu+1}}.$

We know that the $\mathfrak{g}$-module $M^{\chi_{\nu+1}}$ is of finite length,
and want to determine the structure of each $M^{\chi_{\nu+1}}$ for the proof
of Theorem 1.3.

######  Lemma 3.1.

For every $\nu\in X$, $M^{\chi_{\nu+1}}$ has a Verma flag. Moreover, in the
Grothendieck group, one has

$\displaystyle[M^{\chi_{\nu+1}}]=\begin{cases}[M(\nu)+M(-\nu-2)]&\nu\in
A^{\prime}\cup A^{\prime*},\\\ [M(\nu)]&\nu\notin A^{\prime}\cup
A^{\prime*}.\end{cases}$

###### Proof.

Since $M(\mu^{\prime})\otimes M(\mu^{\prime\prime})$ has a Verma flag, so does
every component $M^{\chi_{\nu+1}}$, whence the first assertion. The second
assertion is derived from $\eqref{eqn:C}$ because

$\displaystyle X\cap X^{*}\setminus\\{-1\\}=A^{\prime}\cup A^{\prime*}$

is the set of $\nu\in X$ such that $\nu\neq\nu^{*}$ and that $[M(\nu^{*})]$
occurs in $\eqref{eqn:C}$. ∎

We also use the following lemma [6, Lem. 9.4]:

######  Lemma 3.2.

For all $k\in\mathbb{N}$ and for all
$\mu^{\prime},\mu^{\prime\prime}\in\mathbb{C}$,

$\displaystyle\mathop{\mathrm{Hom}}\nolimits_{\mathfrak{g}}(M(\mu^{\prime}+\mu^{\prime\prime}-2k),M(\mu^{\prime})\otimes
M(\mu^{\prime\prime}))\neq\\{0\\}.$

For the sake of completeness, we give a direct proof. The dimension of weight
spaces is given as

$\displaystyle\dim(M(\mu^{\prime})\otimes
M(\mu^{\prime\prime}))_{\mu^{\prime}+\mu^{\prime\prime}-2k}=k+1.$

Since $\mathop{\mathrm{diag}}\nolimits(\mathfrak{n})$ maps
$(M(\mu^{\prime})\otimes
M(\mu^{\prime\prime}))_{\mu^{\prime}+\mu^{\prime\prime}-2k}$ to
$(M(\mu^{\prime})\otimes
M(\mu^{\prime\prime}))_{\mu^{\prime}+\mu^{\prime\prime}-2k+\nu}$, it has a
non-trivial kernel, namely, there exists a non-zero singular vector in
$(M(\mu^{\prime})\otimes
M(\mu^{\prime\prime}))_{\mu^{\prime}+\mu^{\prime\prime}-2k}$.

By Lemma 3.1 and 3.2, one has an isomorphism $M^{\chi_{\nu+1}}\cong M(\nu)$
for $\nu\in X\setminus(A^{\prime}\cup A^{\prime*})$. We note that
$\chi_{\nu+1}=\chi_{\nu^{*}+1}$ where $\nu^{*}=-\nu-2$. What remains is to
determine the structure of $M^{\chi_{\nu+1}}$ for $\nu\in A^{\prime}$. We
compute $M^{\chi_{\nu+1}}$ for $\nu\in A^{\prime}$ in by case-by-case argument
as below.

###  3.1 Generic case.

We begin with the case $\mu^{\prime}+\mu^{\prime\prime}\notin\mathbb{N}$.
Every Verma module $M(\mu^{\prime}+\mu^{\prime\prime}-2k)$ is irreducible, and
the set of (generalized) eigenvalues
$\\{\mu^{\prime}+\mu^{\prime\prime}-2k+1:k\in\mathbb{N}\\}$ are distinct from
each other. Hence the equation of the Grothendieck group (3) is the
isomorphism of modules.

###  3.2 Singular case 1.

We assume $\mu^{\prime}+\mu^{\prime\prime}\in\mathbb{N}$ but at least one of
$\mu^{\prime}$ or $\mu^{\prime\prime}$ does not belong to $\mathbb{N}$. We
observe that $A^{\prime}$ coincides with $A(\mu^{\prime},\mu^{\prime\prime})$
in Theorem 1.3. On the other hand, $M(\mu^{\prime})\otimes
M(\mu^{\prime\prime})$ is a tilting module by Fact 2.13. Suppose $\nu\in
A^{\prime}$. Since $[M^{\chi_{\nu+1}}]=[M(\nu)]+[M(-\nu-2)]=[P(\nu)]$ by
Example 2.2 and Lemma 3.1, we conclude $M^{\chi_{\nu+1}}\cong P(\nu)$ because
both the primary component $M^{\chi_{\nu+1}}$ and the projective covering
$P(\nu)$ are tilting modules. Hence Theorem 1.3 is shown in this case.

###  3.3 Singular case 2.

Finally, we assume $\mu^{\prime},\mu^{\prime\prime}\in\mathbb{N}$. Without
loss of generality, we may assume $\mu^{\prime}\geq\mu^{\prime\prime}$. One
has the following short exact sequence:

$\displaystyle 0\rightarrow M(\mu^{\prime})\otimes
M(-\mu^{\prime\prime}-2)\rightarrow M(\mu^{\prime})\otimes
M(\mu^{\prime\prime})\rightarrow M(\mu^{\prime})\otimes
L(\mu^{\prime\prime})\rightarrow 0.$

We have already determined the structure of $M(\mu^{\prime})\otimes
M(-\mu^{\prime\prime}-2)$ in the previous cases. We note that the set
$A(\mu^{\prime},\mu^{\prime\prime})$ in Theorem 1.3 is equal to
$A^{\prime}(\mu^{\prime},-\mu^{\prime\prime}-2)$ since
$-\mu^{\prime}-(-\mu^{\prime\prime}-2)-2=-|\mu^{\prime}-\mu^{\prime\prime}|$.
Moreover, since $M(\mu^{\prime})\otimes L(\mu^{\prime\prime})$ is a projective
module, the above short exact sequence splits. In view of a Verma flag of
$M(\mu^{\prime})\otimes L(\mu^{\prime\prime})$ one has the following direct
sum decomposition

$\displaystyle M(\mu^{\prime})\otimes
L(\mu^{\prime\prime})\cong\bigoplus_{k=0}^{\mu^{\prime\prime}}M(\mu^{\prime}+\mu^{\prime\prime}-2k)$

because the $Z(\mathfrak{g})$-infinitesimal characters in the summands are all
distinct. Thus the proof of Theorem 1.3 is completed.

###  3.4 Proof of the equivalence (ii)$\Leftrightarrow$(iv) in Theorem 1.2.

Suppose $\mathop{\mathrm{Re}}\nolimits\nu\leq 0$. By Theorem 1.3 and its
proof,

$\displaystyle(M(\nu’)\otimes
M(\nu^{\prime\prime}))^{\chi_{\nu+1}}=M(\nu)\oplus M(-\nu-2)$

If and only if $\nu\in A’\setminus A$. Moreover,

$\displaystyle A’\setminus A=\\{-|\mu’-\mu^{\prime\prime}|-2j-2:0\leq
j\leq\min(\mu’.\mu^{\prime\prime})\\}$

If $\mu’,\mu^{\prime\prime}\in\mathbb{N}$, and $A’\setminus A=\emptyset$
otherwise. Hence the equivalence (ii)$\Leftrightarrow$(iv) in Theorem 1.2 is
shown by putting $\lambda^{\prime\prime\prime}=-\nu$.

######  Example 3.1.

We cannot drop the condition $2\leq\lambda^{\prime\prime\prime}$ in (iv) in
Theorem 1.2. For instance, take
$(\lambda^{\prime},\lambda^{\prime\prime},\lambda^{\prime\prime\prime})$ to
$(-1,-3,-2)$. From Theorem 1.3, we have

$\displaystyle(M(1)\otimes M(3))^{\chi_{2+1}}=M(2)\oplus M(-4),$

but $-2<|-1-(-3)|+2$. It does not satisfy (ii).

###  3.5 Proof of Corollary 1.4

###### Proof.

The implication (i)$\Rightarrow$(ii) is obvious. We recall $B\equiv
B(\mu^{\prime},\mu^{\prime\prime})$ from Theorem 1.3, which tells us that the
condition (iii) is equivalent to $B\cap\mathbb{N}=\emptyset$ because the
projective covering $P(a)$ and irreducible $M(b)$ are self-dual. On the other
hand, $B\cap\mathbb{N}\neq\emptyset$ if and only if
$\mu^{\prime},\mu^{\prime\prime}\in\mathbb{N}$, and in this case both $\nu$
and $\nu^{*}=-\nu-2$ belong to

$\displaystyle B\cap\mathbb{N}$
$\displaystyle=\\{|\mu^{\prime}-\mu^{\prime\prime}|+2j:0\leq
j\leq\min(\mu^{\prime},\mu^{\prime\prime})\\}$ $\displaystyle=\\{-\nu-2:\nu\in
A^{\prime}\setminus A\\}.$

By Theorem 1.3,
$\dim\mathop{\mathrm{Hom}}\nolimits(M(-\nu-2),M(\mu^{\prime})\otimes
M(\mu^{\prime\prime}))=\dim\mathop{\mathrm{Hom}}\nolimits(M(-\nu-2),M(-\nu-2)\oplus
M(\nu))=2$ for any $\nu\in B\cap\mathbb{N}$. Hence the implication
(ii)$\Rightarrow$(iii) is shown. On the other hand, it follows from Theorem
1.3 that indecomposable summand of $M(\mu^{\prime})\otimes
M(\mu^{\prime\prime})$ are given as

$\displaystyle\bigoplus_{a\in A}P(a)\oplus\bigoplus_{b\in
B\cap\mathbb{N}}M(b)\oplus\bigoplus_{b\in B\setminus\mathbb{N}}M(b).$

Since $\\{b^{*}:b\in B\cap\mathbb{N}\\}\subset B\setminus\mathbb{N}$, this
direct sum is multiplicity-free if and only if $B\cap\mathbb{N}=\emptyset$,
hence one has the equivalence (i)$\Leftrightarrow$(iii). By Theorem 1.3,
$M(\mu^{\prime})\otimes M(\mu^{\prime\prime})$ contains a reducible Verma
module as a direct summand if and only if $B\cap\mathbb{N}=\emptyset$, hence
(iii)$\Leftrightarrow$(iv) is shown. ∎

## References

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  * [2] Johan Kåhrström, _Tensoring with infinite-dimensional modules in $\mathscr{O}_{0}$_, Algebr. Represent. Theory 13 (2010), no. 5, 561–587. MR 2684221
  * [3] Toshiyuki Kobayashi, _Discrete decomposability of the restriction of $A_{\mathfrak{q}}(\lambda)$ with respect to reductive subgroups. III. Restriction of Harish-Chandra modules and associated varieties_, Invent. Math. 131 (1998), no. 2, 229–256. MR 1608642
  * [4] , _Restrictions of generalized Verma modules to symmetric pairs_ , Transform. Groups 17 (2012), no. 2, 523–546. MR 2921076
  * [5] Toshiyuki Kobayashi, Bent Ørsted, Petr Somberg, and Vladimír Souček, _Branching laws for Verma modules and applications in parabolic geometry. I_ , Adv. Math. 285 (2015), 1796–1852. MR 3406542
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# Success and breakdown of Tanner’s law for drops of dense granular
suspensions

Alice Pelosse<EMAIL_ADDRESS>Élisabeth Guazzelli  Matthieu Roché
Université Paris Cité, CNRS, Matière et Systèmes Complexes UMR 7057, F-75013,
Paris

###### Abstract

The spreading of viscous drops of density-matched suspensions on a solid
surface is experimentally investigated at the global drop scale. The spreading
dynamics still obeys Tanner’s law provided one uses an effective viscosity
which happens to be smaller than the bulk viscosity of the suspension. When
the height of the drop is of the order of the particle size, Tanner’s law
breaks down as the particles start to freeze and the pure fluid drains out of
the solid matrix.

###### keywords:

##  1 Introduction

The spreading of a liquid on a solid substrate is a common phenomenon, e.g. on
plant leaves after the impact of raindrops or insecticide droplets, or during
the coating of surfaces in industry. In these examples, the spreading fluid is
not always a pure liquid as it may carry rigid bodies, such as fine dust
particles in raindrops or solid particles added to the coating fluid. It is
therefore of great importance to examine whether the spreading laws
established for a pure fluid are still valid for such a complex fluid.

In this work, we focus on granular suspensions, a type of complex fluid
extensively studied in bulk-flow situations, but quite unexplored when
confined by a free interface such as for a spreading droplet. Granular
suspensions are made of large particles (i.e. having diameter $d>$
$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$) such that Brownian motion and
colloidal forces are negligible. Despite their high degree of heterogeneity at
the microscopic scale, these particulate systems can be seen as continuous
effective fluids at the macroscopic scale, with a bulk viscosity that solely
depends on the particle volume fraction, $\phi$, and is notably independent of
particle diameter $d$ for monodisperse rigid spheres (see e.g. Guazzelli &
Pouliquen, 2018).

Such a macroscopic description fails when the size of the system approaches
that of the particle as seen in thin films (Palma & Lhuissier, 2019; Gans et
al., 2019) or near an advancing contact line (Zhao et al., 2020; Pelosse et
al., 2023). In that latter case, the relation between the dynamic contact
angle and the spreading velocity is similar to the classic Cox-Voinov law
observed in the case of a pure liquid (Voinov, 1976; Cox, 1986). However, the
wetting viscosity involved in this law differs from that of the bulk as it
depends not only on $\phi$ but also on $d$. This observation is linked to the
ability of the particles to approach the contact line close enough to affect
dissipation (Zhao et al., 2020; Pelosse et al., 2023).

In the present work, the spreading of viscous drops of density-matched
suspensions (having a volume $V_{0}$, a surface tension with air $\gamma$, and
a density $\rho$) on a solid surface is experimentally investigated at the
global drop scale, i.e. by recording the time evolution of their radius,
$R(t)$. For pure fluids, the radius growth is expected to follow Tanner’s law
(Tanner, 1979) such that $R(t)=A\,t^{n}$, where both the factor $A$ and the
exponent $n$ depend on $V_{0}$. For drop radii smaller than the capillary
length, i.e. $R_{0}\sim V_{0}^{1/3}<\ell_{c}=\sqrt{\gamma/\rho g}$ (with
$\ell_{c}=$ $1.8\text{\,}\mathrm{mm}$ in the present work), the spreading is
driven by the balance of capillary and viscous forces, leading to
$A\propto(V_{0}^{3}\gamma/\eta)^{1/10}$ and $n=1/10$ (De Gennes, 1985).
Conversely, for large drops, i.e. $R_{0}>\ell_{c}$, gravity prevails over
capillarity and the spreading behaviour is established through the balance
between gravity and viscous dissipation, resulting in $A\propto(V_{0}^{3}\rho
g/\eta)^{1/8}$ and $n=1/8$ (Lopez et al., 1976; Hocking, 1983). This paper
discusses the validity of Tanner’s law for large drops of granular suspensions
and compares the present dynamics at the global drop scale to that previously
obtained at a local scale in the vicinity of the contact angle.

##  2 Experimental methods

Two different types of granular suspensions have been used in the experiments.
The first combination of particles and fluid consists of spherical polystyrene
beads (Dynoseeds TS, Microbeads, Norway) suspended in a density-matched
Newtonian PEG copolymer [Poly(ethylene glycol-ran-propylene glycol) monobutyl
ether] (Sigma) widely used in previous experimental work (see e.g. Guazzelli &
Pouliquen, 2018). The fluid density $\rho=1056$ kg/m3 is close to that of
polystyrene and its dynamic viscosity is measured to be $\eta_{f}=2.4\pm 0.1$
Pa.s at $22\text{\,}\mathrm{\SIUnitSymbolCelsius}$. Different batches of
particles are used with varying mean diameters $d$ = 20, 40, 80, 140, 250 and
$550\text{\,}\mathrm{\SIUnitSymbolMicro m}$ (with dispersion in size of 10%).
The suspension mixture is made by weighting a mass of suspending fluid and
adding the amount of solid needed to reach the desired particle volume
fraction $\phi=40\,\%$ in most of the situations presented in §3. Larger
loading in particles are also investigated as described in §3.4. Mixing is
achieved by (i) first slowly stirring with a spatula and (ii) then using a
rolling device overnight. The particles are found to be completely wet by the
fluid and to experience no aggregation. The surface tension of the suspensions
is that of the PEG copolymer, $\gamma=\gamma_{f}\simeq$
$35\text{\,}\mathrm{mN}\text{\,}{\mathrm{m}}^{-1}$, as confirmed by pendant
drop experiments. The second type of suspension consists of
60-$\mathrm{\SIUnitSymbolMicro m}$ PMMA spheres (Spheromers CA, Microbeads,
Norway) immersed in a fluid chosen to match the density and index of the
particles. This fluid is a mixture of Triton X-100 (73 wt%), zinc chloride (16
wt%), and water (11 wt%) having a viscosity $\eta_{f}=3.3\pm 0.1$ Pa.s at
$22\text{\,}\mathrm{\SIUnitSymbolCelsius}$. A fluorescent dye, Rhodamine 6G,
is added to the fluid to aid the visualisation with a laser sheet as described
in §3.3. Only a single solid volume fraction of $\phi=40\%$ has been studied
for this suspension mixture.

Drops comprising the smallest particles (from
$20\text{\,}\mathrm{\SIUnitSymbolMicro m}$ to
$250\text{\,}\mathrm{\SIUnitSymbolMicro m}$) are made using a syringe pump
pushing the target volume of fluid at a flow rate
$1\text{\,}\mathrm{m}\mathrm{L}\mathrm{/}\min$ out of a
$3\text{\,}\mathrm{mm}$ needle (inner diameter). To avoid transient
heterogeneity in the suspension microstructure, at least
$3\text{\,}\mathrm{mL}$ of suspension is discarded before any experiments. For
the largest $550\text{\,}\mathrm{\SIUnitSymbolMicro m}$ particles, due to
concerns about confinement effects in the tubing, the drops are made manually
with a spatula ensuring a better control of $\phi$. In all cases, the drops
are deposited onto (Neyco Fused Silica) Quartz substrates carefully enough to
produce axisymmetric spreading. To this extent, before each experiment, the
substrate is cleaned thoroughly with deionised water and ethanol and treated
with a plasma cleaner to avoid pining of the advancing contact line. Under
these conditions, the continuous phase of the suspensions wets completely the
surface of the wafer.

The spreading of the drop is recorded by a (Imaging Source) monochrome digital
camera (with a spatial resolution of 23 pixels/mm) located
$50\text{\,}\mathrm{cm}$ above the solid substrate and operated at a frame
rate of $1\text{\,}{\mathrm{s}}^{-1}$ which provides a capture of both early
and long-time dynamics. Automatic measurement of the radius is performed by a
Python script that (i) first blurs locally the raw picture over roughly 5
pixels, (ii) then detects the edges of the drop using the (scikit-image
toolbox) Canny edge detector with a Gaussian width of 1, and (iii) finally
returns the best circle fitting the outer radius using the (scikit-image
toolbox) Hough transform function. Additional side-view recording is performed
by a camera positioned at a few centimetres from the drop (with spatial
resolution of $0.6\text{\,}\mathrm{\SIUnitSymbolMicro m}$/pixel) which is
synchronised with the top-view camera. Post-treatment of the recording uses
the Sobel filter to detect drop contour and extract the drop profile as a
function of time (Pelosse et al., 2023).

In our experiments, the initial time is ill-defined, see figure 1($a$).
Indeed, as the drop touches the solid surface and starts spreading, it is
attached to the needle for a certain amount of time which goes from a few
seconds to one minute depending on the drop composition. The growth of the
radius according to Tanner’s law is therefore only investigated once the drop
is fully detached from the needle, i.e. at a delayed time $t_{0}$. The radius
data are therefore fitted with the parameterised function
$R(t,t_{0},A)=A(t+t_{0})^{1/8}$, where $A=k(\rho gV_{0}^{3}/\eta_{f})^{1/8}$
with $k$ a constant, $V_{0}$ the drop volume inferred from its mass, and
$\eta_{f}$ the viscosity of the PEG copolymer measured by a capillary
viscometer for each experiment. The fitted parameters $t_{0}$ and $A$ are
estimated numerically. An alternative method is to use an initial radius
instead of an initial time, i.e.
$R(t,R_{0},A^{\prime})=R_{0}+A^{\prime}t^{1/8}$ (Saiseau et al., 2022). This
is equally efficient but we have chosen the time-offset correction in the
present data analysis. For smaller drops, a transition from a (short-time)
capillary to a (long-time) gravity regime has been reported (Cazabat & Cohen
Stuart, 1986; Levinson et al., 1988). Such changes in power law are not
observed with the drop sizes investigated in the present work. It should be
stressed that the time offset correction only affect the behaviour at short
time and does not modify that at long time which is undoubtedly driven by
gravity with a power law exponent of 1/8 as evidenced in figure 1($a$)
discussed in § 3.1.

##  3 Results

###  3.1 Tanner’s law

Figure 1: ($a$) Normalised radius $R^{*}=R(\eta_{f}/V_{0}^{3})^{1/8}$ versus
time (with the $t_{0}$ offset) for different drop volume $V_{0}=$100, 300, and
$1000\text{\,}\mathrm{\SIUnitSymbolMicro L}$ and different fluids: suspensions
with $d=$ 40 $\mathrm{\SIUnitSymbolMicro m}$ (purple symbols) and 550
$\mathrm{\SIUnitSymbolMicro m}$ (gold symbols) at $\phi=$ 40% and pure PEG
(grey and black symbols). Inset: raw data of the radius versus time. ($b$)
Normalised factor $A^{*}=A(\eta_{f}/V^{3})^{1/8}$versus particle diameter $d$
(from $20\text{\,}\mathrm{\SIUnitSymbolMicro m}$ to
$550\text{\,}\mathrm{\SIUnitSymbolMicro m}$). The open symbols correspond to
experiments with pure PEG. Inset: ratio of the relative effective viscosity
obtained from Tanner’s law, $\eta_{T}$, to the relative bulk viscosity,
$\eta_{s}$, versus particle diameter $d$. The red horizontal line corresponds
to this ratio for a polydisperse suspension made of particles of
$20\text{\,}\mathrm{\SIUnitSymbolMicro m}40\text{\,}\mathrm{\SIUnitSymbolMicro
m}80\text{\,}\mathrm{\SIUnitSymbolMicro
m}140\text{\,}\mathrm{\SIUnitSymbolMicro m}$, each size representing 10 % of
the suspension volume. The star symbols corresponds to the relative effective
wetting viscosity of the same suspensions obtained from the Cox-Voinov law
(Zhao et al., 2020).

Figure 1($a$) shows the typical time-evolution of the radius of drops
consisting of pure fluid and granular suspensions. The colour coding indicates
the different fluids while the symbol coding the varying drop volume. The raw
data are plotted versus time $t$ in the inset whereas the main graph displays
the normalised radius $R^{*}=R(\eta_{f}/V_{0}^{3})^{1/8}$ versus $t+t_{0}$.
This radius normalisation removes volume and viscosity effects, as evidenced
by the tight collapse of the curves for a given fluid. The time offset
corrects the initial bending seen in the inset as already explained at the end
of § 2. The spreading clearly obeys a gravity-driven dynamic, i.e. $R\sim
t^{1/8}$, as expected for the relatively large Bond numbers considered,
$2.6\leq Bo=(3V_{0}/4\pi)^{2/3}/\ell_{c}^{2}\leq 11.9$. Drops of suspensions
of any volume and concentration seem to obey Tanner’s law established for
continuous Newtonian fluids. However, there are two main differences. First,
the curves of the main log-log plot for suspension drops exhibit a small
vertical shift compared to those for the pure fluid which indicates that the
factor $A$ of Tanner’s law differs when adding particles to the fluid. Second,
for the largest particles ($d=$ $550\text{\,}\mathrm{\SIUnitSymbolMicro m}$),
the dynamic slows down for $t\gtrsim$ $200\text{\,}\mathrm{s}$. Deviations
from Tanner’s law will be successively discussed in the following § 3.2 and
3.4.

###  3.2 Tanner’s effective viscosity

We start by discussing the impact of adding particles on the factor $A$. In
the main graph of figure 1($b$), the normalised factor,
$A^{*}=A(\eta_{f}/V_{0}^{3})^{1/8}$, is plotted for all the experiments. This
normalisation eliminates any $V_{0}$-dependence for a given fluid. We see that
the value of $A^{*}$ is smaller for suspensions than for the pure continuous
phase and its value is roughly independent of $d$. The viscosity of the
continuous phase $\eta_{f}$ used to compute $A^{*}$ underestimates the
viscosity of the suspensions. Seeking to compare the effective viscosity
inferred from Tanner’s law with its bulk counterpart, we write the Tanner
suspension viscosity as $\eta_{T}\eta_{f}$. To shift the suspension data onto
the solid line corresponding to the PEG data in the figure 1($a$), the
relative viscosity $\eta_{T}$ should be computed as $\eta_{T}=(\rho
g/\eta_{f}V_{0}^{3})(k/A)^{8}$, where $k=0.61\pm 0.02$ has been inferred from
the pure PEG data (open symbols) using $A^{*}_{PEG}=k(\rho g)^{1/8}=1.94$.
This relative viscosity extracted from Tanner’s law is reported in the inset
of figure 1($b$) for the different suspensions and drop volumes. It is found
to be independent of $d$ and $V_{0}$ with a value $\eta_{T}=2.2\pm 0.5$.

This relative Tanner viscosity $\eta_{T}$ which describes the global dynamics
of the drop differs from the relative wetting viscosity $\eta_{w}$ extracted
from the Cox-Voinov law found in the viscous–capillary region corresponding to
the very close vicinity of the contact line, see the star symbols on the inset
of figure 1($b$). In contrast with the constant behaviour of $\eta_{T}$,
$\eta_{w}$ decreases with increasing $d$ and reaches a value of one (that
corresponding to the pure fluid) for a cut-off size ($\approx$
$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$) above which particles are too
large to affect dissipation in this region close to the contact line (Zhao et
al., 2020; Pelosse et al., 2023). This difference between $\eta_{w}$ and
$\eta_{T}$ may not come as a surprise as it derives from two different energy
balances at the local and global scales, i.e. from a balance of capillary and
viscous forces at the local scale of the contact line and of gravity and
viscous forces at the global scale of the drop.

The Tanner viscosity happens also to be much smaller than the bulk viscosity
of the suspension as the relative bulk viscosity is $\eta_{s}\simeq 10$ at the
same $\phi=40\%$. This may be due to particle slip along the very smooth
substrate. Another difference may come from the suspension microstructure
which certainly is dissimilar in the present non-viscometric flow and in a
pure shearing flow. Besides, confinement by both the substrate and the mobile
interface may create layering that could span over 10 particle diameters at
$\phi=40\%$ (Gallier et al., 2016) and end up in a strong dip in dissipation
(Ramaswamy et al., 2017). However, experiments with polydisperse suspensions
do not support this explanation as suspensions consisting of a mixture of 20,
40, 80, and 140 $\mathrm{\SIUnitSymbolMicro m}$ particles (meant to hinder
crystallisation) do not exhibit a significant increase in dissipation, see red
line in the inset in figure 1($b$). To discriminate between possible causes of
this smaller value of $\eta_{T}$, we examine in detail the particulate flow
during spreading in the following § 3.3.

###  3.3 Visualisation of the particulate flow in the rivulet configuration

Figure 2: ($a$) Sketch of the apparatus used to visualize the particle flow in
a spreading rivulet with a laser sheet. A fluorescent dye suspended in the
fluid is excited by the green light and fluoresces red light collected by a
side view camera. ($b$) Imaging for a suspension made of
60-$\mathrm{\SIUnitSymbolMicro m}$ particles at $\phi=40\%$ (scale bar:
$1\text{\,}\mathrm{mm}$). See accompanying movie 1 in the supplementary
material. ($c$) Horizontal speed $v_{x}$ from PIV analysis of the particle
flow in ($b$). The speed is normalised by the contact line velocity $U$. ($d$)
Sum of raw pictures between $t$ and $t+\Delta t$ with $\Delta t=2d/U$ in the
corresponding red (left) and blue (right) squares in subplot $(b)$.

To gain further insight into the particle motion during spreading, particle
imaging is performed in a more convenient rivulet configuration, i.e. for a
non-isotropic spreading along only one direction, using a density and index-
matched suspension described in § 2. As sketched in figure 2($a$), imaging
without optical distortion is achieved using two orthogonal microscope slides.
A green laser sheet (Coherent lasers, $\lambda=$ $532\text{\,}\mathrm{nm}$,
$50\text{\,}\mathrm{mW}$) illuminates the fluid rivulet through the bottom
slide and a side view camera images the flow. The laser sheet is not deflected
across the flat walls nor in the index-matched suspension and induces
fluorescence of Rhodamine 6G in the illuminated slice (about
$30\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thick) which is excited by the
green light and emits red wavelengths. This light is filtered by a high-pass
red filter (Thorlabs, $\lambda>$ $550\text{\,}\mathrm{nm}$) and collected by
the monochrome camera (Imaging Source, DMK33UX174, 2.3 MP) mounted with a zero
distortion lens (Opto Engineering, MC100X), and orthogonal to the laser sheet.
The spatial resolution is of $3\text{\,}\mathrm{\SIUnitSymbolMicro m}$/pixel
and the acquisition frame-rate of $1\text{\,}{\mathrm{s}}^{-1}$. With this
apparatus, the non-fluorescing particles appear dark in the illuminated slice
as seen in figure 2($b$) and the accompanying movie. The horizontal speed
coming from the PIV analysis of a frame is displayed in figure 2($c$). The
analysis is performed with the trackpy module of Python on two consecutive
frames. The algorithm uses an interrogation window size of 16 pixels, a search
area size of 18 pixels, and an overlap of 9 pixels. The velocity is normalised
by the speed of the advancing contact line (red arrow). In figure 2($d$), the
sum of the pictures between $t$ and $t+\Delta t$ is shown for $\Delta t=2d/U$
in two specific regions. With this time interval, particles moving at a speed
close to that of the contact line should not appear as fixed dark dots but
should be blurred.

Visualisation of the particulate flow during the spreading of the suspension
rivulet unveils several significant elements. Importantly, figure 2($b$) and
its accompanying movie do not exhibit any clear development of layering or
ordering during spreading, meaning that particle self-organisation cannot
explain the strong decrease in viscosity as that reported by e.g. Ramaswamy et
al. (2017). However, PIV measurements presented in figure 2($c$) and the sum
of consecutive pictures in figure 2($d$) reveal two crucial features of the
particle flow. First, a strong particle slip develops along the wall near the
advancing contact line as shown by the velocity signal in figure 2($c$) or by
the smearing of dark dots near the rivulet edge in figure 2($d$) (right).
Second, in the centre of the rivulet, particles barely move as illustrated by
the weak velocity signal from the PIV in figure 2($c$) or by the sum of the
pictures in figure 2($d$) (left). In this later figure, sharp dots at the
centre of the rivulet (i.e. on the left) indicate that the particles have not
moved within the time interval $\Delta t=2d/U$, in contrast with the
observations near the contact line (i.e. on the right) where the particulate
phase undergoes a strong flow.

These observations of the structure of the particulate flow can explain the
low value of the effective spreading viscosity displayed in the inset of
figure 1($b$). In rheometers, slip velocity of the particles leads to a
smaller stress on the particles and results in an effective smaller viscosity
(Jana et al., 1995; Yoshimura & Prud’homme, 1988; Ahuja & Singh, 2009). In
spreading experiments, particles close to the tip of the rivulet exhibit a
strong slip at roughly the contact line speed. This slip is not surprising
under such strong confinement and given the very smooth surface of the solid
substrate. Yet, slip by itself cannot account for such a large decrease in
effective viscosity. This decrease could also be the result of the almost
frozen particulate phase at the centre of the rivulet such that particles act
more like a porous medium than like a flowing suspension in this region.
Interestingly, the effective viscosity of a granular bed is found to be
independent of particle size and to possess a value close to the Einstein
viscosity ($\eta_{s}=1+2.5\phi=2$ at $\phi=0.4$), i.e. a value much smaller
than that of a sheared dense suspension due to the screening of the
hydrodynamic interactions (Vowinckel et al., 2021). This bears a striking
resemblance to the behaviour of $\eta_{T}$ in figure 1($b$). In summary, slip
at the tip of the drop and porous-like behaviour at the centre of the
drop/rivulet could both account for the observed $\eta_{T}$.

###  3.4 Long time behaviour: validity of Tanner’s law

Figure 3: Top view pictures of the spreading of different suspensions. From
top to bottom: ($a$) $d=$ $20\text{\,}\mathrm{\SIUnitSymbolMicro m}$,
$\phi=40\%$; ($b$) $d=$ $250\text{\,}\mathrm{\SIUnitSymbolMicro m}$,
$\phi=40\%$; ($c$) $d=$ $550\text{\,}\mathrm{\SIUnitSymbolMicro m}$,
$\phi=40\%$; ($d$) $d=$ $550\text{\,}\mathrm{\SIUnitSymbolMicro m}$,
$\phi=47\%$ (scale bar: $5\text{\,}\mathrm{mm}$). See accompanying movie 2 in
the supplementary material.

We now turn to the slowdown of the spreading dynamics observed above
$200\text{\,}\mathrm{s}$ for suspensions made of the largest
$550$-$\mathrm{\SIUnitSymbolMicro m}$ particles. In figure 3, top view
pictures of drop spreading evidence different patterns for suspensions having
different particle size of $40\text{\,}\mathrm{\SIUnitSymbolMicro
m}250\text{\,}\mathrm{\SIUnitSymbolMicro
m}550\text{\,}\mathrm{\SIUnitSymbolMicro m}$ at $\phi=40\%$. For the largest
particles, top-views at a higher volume fraction, namely $\phi=47\%$, are also
presented on the last row ($d$). Clearly, the particle phase initially spreads
with the fluid but then does not expand further beyond a critical radius which
decreases with increasing particle size, as depicted on the first three rows
($a$), ($b$), ($c$), and with increasing particle volume fraction, as shown on
the two last rows ($c$), ($d$). There is a striking difference between the
behaviour for the smallest particles where the particle arrest is barely seen
(top row) and that for the largest particles (bottom row) where particle
freeze happens at very early stage. These observations suggest that the
departure from Tanner’s law can be associated with the freezing of the
particle matrix while the fluid continues to drain outward.

Figure 4: ($a$) Drop of suspension consisting of
550-$\mathrm{\SIUnitSymbolMicro m}$ particles with $V_{0}=$
$100\text{\,}\mathrm{\SIUnitSymbolMicro L}$. Main graph: fluid radius $R$
(black) and particle radius $R_{p}$ (yellow) as a function of time. The dashed
line correspond to Tanner’s law. Thumbnails (i) to (iv): side-view pictures
corresponding to the black dots in the main graph. Top inset: top view picture
with fluid ($R$) and particle ($R_{p}$) radii. ($b$) Particle radius $R_{p}$
as a function of time for different suspensions (see legend) with a drop
volume $V_{0}\simeq$ $100\text{\,}\mathrm{\SIUnitSymbolMicro L}$. Error bars
are computed from standard deviations over different experiments.

In figure 4($a$), the fluid radius $R$ of a
$100\text{\,}\mathrm{\SIUnitSymbolMicro L}$ drop of a suspension consisting of
$550\text{\,}\mathrm{\SIUnitSymbolMicro m}$ particles is plotted as a function
of time (black solid line). The particle spreading radius, $R_{p}$, defined in
the top left inset, is also plotted with a yellow solid line. Side-view
snapshots numbered from (i) to (iv) taken during this spreading illustrate two
different regimes. First, for pictures (i) and (ii), the drop profile is
concave and $R_{p}$ increases, meaning that particles move with the fluid, in
agreement with the early time dynamics in figure 3. In this regime, the drop
radius growth obeys Tanner’s law and the suspension behaves like a continuous
fluid. Second, while the contact line still progresses from (ii) to (iv),
$R_{p}$ saturates, i.e. the particles do not move and remain at the centre of
the drop, also in agreement with observations from figure 3. This translates
into a significant slowdown of the drop radius growth $R$ and departure from
Tanner’s law (black solid line). In this second regime, the continuous
approach fails and the fluid drains out of the porous matrix consisting of the
large particles. The drop profile then takes a convex shape near the edge to
connect the contact line with the centre of the drop where the particles
remain, as seen in the thumbnails (iii) and (iv). In these pictures, the
particle protrusions become more significant and one can clearly identify a
single monolayer of particles in the picture (iv). Time evolution of
normalised particle radii for drop consisting of different particle sizes and
volume fractions are plotted in figure 4($b$). The same normalisation to
remove volume and viscosity effects is used and the drop volumes are kept
similar at approximately $100\text{\,}\mathrm{\SIUnitSymbolMicro L}$. This
graph confirms that particle freezing happens earlier with large particles and
large volume fractions. Conversely, for particle diameters
$\lesssim$$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$, $R_{p}$ follows a power
law growth with an exponent $1/8$, i.e. the same dynamics as that predicted
for a continuous fluid by Tanner’s law.

To gain insight on the transition between these two regimes, the shape and
thickness of the drop have been investigated. In figure 5($a,b$), plots of the
time-evolution of the profile $h(r,t)$, of the radius $R$, and of the drop-
centre thickness $h_{0}$, for a drop of pure fluid ($r$ is the radial distance
from the drop centre) indicate that experimental profiles are well captured by
the predictions of Hocking (1983) for Bond numbers $Bo=\mathcal{O}(1)$,

$h(r,t)=\frac{{V}}{\pi
R(t)^{2}}\frac{I_{0}\left(\frac{R(t)}{\ell_{c}}\right)-I_{0}\left(\frac{r}{\ell_{c}}\right)}{I_{2}\left(\frac{R(t)}{\ell_{c}}\right)},$
(1)

where $I_{n}(x)$ is the $n$-th modified Bessel function of the first kind. In
figure 5($b$), the radius data (computed from top views) are fitted with
Tanner’s law (red dashed line) and used to compute the drop central thickness,
$h_{0}$, according to (1) (blue dashed line). Excellent agreement is found
with drop thickness data extracted from side views. Figure 5($a$) shows the
drop profiles extracted from these side views and the comparison with (1) at
the times indicated by the large dots in figure 5($b$). While not perfect, the
agreement between the experimental and theoretical profiles is satisfactory,
especially in predicting the central drop thickness. Side-views are therefore
not required and drop profiles can be inferred using the drop radius $R(t)$
from top-view pictures.

Figure 5: Spreading of a drop of pure fluid with $V_{0}=$
$295\text{\,}\mathrm{\SIUnitSymbolMicro L}$. ($a$) Experimental profiles from
side views (solid lines) and predicted shapes (dashed lines) from (1). The
colour gradation indicates the time progression. The large dots in ($b$)
correspond to the moments of the profiles plotted in ($a$). ($b$) Right axis:
experimental radius $R$ as a function of time and Tanner’s law (red dashed
line) with the fitted parameters $A=$ 0.0059, $t_{0}=$
$10.4\text{\,}\mathrm{s}$. Left axis: central thickness $h_{0}$ as a function
of time with the prediction of (1) (blue dashed line). ($c$) Radius as a
function of time, normalised by $R_{c}$ and $t_{c}$, respectively, with
$R_{c}=At_{c}^{1/8}$ and $h_{0}(R_{c})=d$ (see legend in figure 1 for the
symbols).

The simplest criterion that can be proposed is that the breakdown of Tanner’s
law for granular suspensions happens when the drop thickness reaches
approximately a particle diameter, as suggested by the side-view thumbnails in
figure 4($a$) where drop thickness is roughly 2-diameter thick at the
transition for the 550-$\mathrm{\SIUnitSymbolMicro m}$ particles. This can be
estimated by evaluating the time $t_{c}$ (or equivalently the drop radius
$R_{c}$) for which the central thickness $h_{0}(R_{c})\simeq d$ with
$R_{c}=At_{c}^{1/8}$. Figure 5($c$) show that a nice collapse of drop radius
versus time can be obtained for different particle sizes and drop volumes by
normalising the radius by the critical radius $R_{c}$ and the time by the
critical time $t_{c}$. Spreading deviates from Tanner’s law (solid line) for
$t\simeq t_{c}$ for the 250 and $550\text{\,}\mathrm{\SIUnitSymbolMicro m}$
particles when $R\simeq R_{c}$. For the smallest particle diameter, the
experiments are too short to observe the deviation as the critical thickness
$h_{0}=d$ is far from being reached.

##  4 Concluding remarks

A global-scale investigation of the spreading of a drop of granular suspension
has been performed in the dense regime ($\phi\geq 40\%$) in order to exhibit
the strongest effects of the particle phase. The first major output, developed
in § 3.2, is that Tanner’s law which describes the growth of the drop radius
is still valid provided one uses an effective viscosity independent of the
particle size. This so-called Tanner viscosity is much smaller than the bulk
viscosity of the suspension probably due to the combined effect of particle
slip near the tip of the drop and porous-like behaviour at the centre of the
drop as shown by flow visualisation and PIV analysis in § 3.3. The second
important finding is that, when the height of the drop becomes of the order of
the particle spacing, the spreading slows down. This departure from Tanner’s
law comes from particle freezing and drainage of the pure fluid out of the
porous particle matrix. In § 3.4, simply stating that this transition happens
when the drop thickness reaches a particle diameter, $h_{0}(t_{c})=d$,
provides a decent prediction of the breakdown of Tanner’s law at a solid
volume fraction $\phi=40\%$. However, a $\phi$-dependent criterion (which is
difficult to measure) is expected in view of the earlier particle freezing
seen at larger $\phi$ in figure 4($b$), i.e. as $\phi$ becomes closer to a
jamming volume fraction upon spreading, $\phi_{c}^{s}$, which is certainly
lower than the usual bulk jamming volume fraction because of confinement
(Scott & Kilgour, 1969; Desmond & Weeks, 2009). We anticipate a weak algebraic
divergence with the distance to the jamming point similar to that found for
the transition neck diameter in the pinch-off of a viscous suspension thread
(Château et al., 2018). A tentative criterion could then be
$h_{0}(t_{c})=d/(\phi_{c}^{s}-\phi)^{\alpha}$ with $\alpha\simeq 1/3$ and
could help to rationalise the earlier freezing when $\phi$ is close to
$\phi_{c}^{s}$. The spreading of granular drops and its transition from a
continuous effective fluid to a discrete system is therefore the signature of
a confinement-dependent jamming.

[Supplementary data]Supplementary material and movies are available at
https://doi.org/10.1017/jfm.2023…

[Acknowledgements]We would like to thank two interns, Maël Lebon and Siham
Mekel, for their precious help at the beginning of this project.

[Declaration of interests]The authors report no conflict of interest.

[Author ORCIDs]
Alice Pelosse http://orcid.org/0000-0002-4554-0604;
Élisabeth Guazzelli http://orcid.org/0000-0003-3019-462X;
Matthieu Roché http://orcid.org/0000-0002-8293-7029.

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