# US 10,502,671 B2 · Viscosity measuring method and viscosity measuring apparatus

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- Number: US 10,502,671 B2
- Document ID: US10502671B2
- Status: Granted
- Kind: B2
- Application number: 15/567,439
- Filed: 2016-04-28
- Published: 2018-04-19
- Granted: 2019-12-10
- Earliest claimed: 2015-05-07
- Assignees: Aohata Corporation
- Applicants: Aohata Corporation
- Inventors: Takayoshi Hoshino
- Web page: https://patentgrep.com/patent/US10502671B2

## CPC

- G01N 11/10: by moving a body within the material
- G01N 2011/0006: {Calibrating, controlling or cleaning viscometers}
- G01N 2011/0026: {Investigating specific flow properties of non-Newtonian fluids}
- G01N: INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES (measuring or testing apparatus or processes other than immunoassay, involving enzymes or microorganisms C12M, C12Q)

## Abstract

A viscosity measuring method includes: immersing a plunger into a sample contained in a cylindrical container by an initial depth L0; further immersing the plunger at a velocity vp1 by a distance ΔL1, and measuring a force applied to the plunger; obtaining first peak value FT1 and a first convergent value FTe1 of the force; returning the plunger to the initial depth L0; further immersing the plunger at a velocity vp2 by a distance ΔL2, and measuring a force applied to the plunger; obtaining a second peak value FT2 and a second convergent value FTe2 of the force; obtaining a flow behavior index n; and calculating an apparent viscosity μa of the sample based on Expression (1).

## Claims

### Claim 1

A method of measuring an apparent viscosity of a sample of non-Newtonian fluid comprising:
(A) a step in which a plunger having an outer radius Ri is immersed into a sample contained in a cylindrical container having an inner radius R0, coaxially with the cylindrical container by an initial depth L0 and stopped thereat, the outer radius Ri of the plunger being smaller than the inner radius R0 of the cylindrical container;
(B) a step in which the plunger is further immersed into the sample coaxially with the cylindrical container at a first relative movement velocity vp1 by a first relative movement distance ΔL1, and during the further immersing operation and after the further immersing operation, a force applied to the plunger from the sample is measured with a passage of time;
(C) a step in which a first peak value FT1 and a first convergent value FTe1 of the force applied to the plunger from the sample are obtained, based on measured values of the force caused by the relative movement of the plunger at the first relative movement velocity vp1;
(D) a step in which the plunger is returned coaxially with the cylindrical container to the initial depth L0 and stopped thereat;
(E) a step in which the plunger is further immersed into the sample coaxially with the cylindrical container at a second relative movement velocity vp2 by a second relative movement distance ΔL2, and during the further immersing operation and after the further immersing operation, a force applied to the plunger from the sample is measured with a passage of time;
(F) a step in which a second peak value FT2 and a second convergent value FTe2 of the force applied to the plunger from the sample are obtained, based on measured values of the force caused by the relative movement of the plunger at the second relative movement velocity vp2;
(G) a step in which, based on a predetermined relationship established among: the first relative movement distance ΔL1, the first peak value FT1 and the first convergent value FTe1, at the first relative movement velocity vp1; the second relative movement distance ΔL2, the second peak value FT2 and the second convergent value FTe2, at the second relative movement velocity vp2; a plunger ratio κ=Ri/R0; and a flow behavior index n of the sample; the flow behavior index n is obtained; and
(H) a step in which, when the first relative movement velocity vp1 is represented as vp, the first relative movement distance ΔL1 is represented as ΔL, the first peak value FT1 is represented as FT, and the first convergent value FTe1 is represented as FTe, or when the second relative movement velocity vp2 is represented as vp, the second relative movement distance ΔL2 is represented as ΔL, the second peak value FT2 is represented as FT, and the second convergent value FTe2 is represented as FTe, an apparent viscosity μa of the sample is calculated based on the vp, the ΔL, the FT, the FTe, the plunger ratio κ, the flow behavior index n, a density ρ of the sample, a gravitational acceleration g, and the following Expression (1):
⁢
[
NO
⁢
⁢
1
]
⁢
μ
a
=
σ
w
dy
/
dt
=
σ
0
+
K
⁡
(
dy
dt
)
n
dy
/
dt
⁢
⁢
⁢
in
⁢
⁢
which
⁢
⁢
⁢
σ
w
=
T
w
⁢
PR
0
2
,
dy
dt
=
(
λ
2
κ
-
κ
-
T
0
)
s
⁢
(
PR
0
2
⁢
K
)
s
,
⁢
⁢
K
=
PR
0
2
⁢
(
Φ
⁢
⁢
R
0
v
p
⁢
κ
2
)
n
,
P
=
2
⁢
σ
y
T
0
⁢
R
0
,
T
w
=
λ
2
κ
-
κ
,
(
1
)
Φ
=
{
∫
κ
λ
-
⁢
(
λ
2
-
x
2
-
T
0
⁢
x
)
s
⁢
x
-
s
⁢
⁢
dx
-
∫
λ
+
1
⁢
(
x
2
-
λ
2
-
T
0
⁢
x
)
s
⁢
⁢
x
-
s
⁢
dx
}
⁢
κ
2
-
∫
κ
λ
-
⁢
(
λ
2
-
x
2
-
T
0
⁢
x
)
s
⁢
x
2
-
s
⁢
dx
+
⁢
∫
λ
+
1
⁢
(
x
2
-
λ
2
-
T
0
⁢
x
)
s
⁢
x
2
-
s
⁢
⁢
dx
(
2
)
0
=
∫
κ
λ
-
⁢
(
λ
2
-
x
2
-
T
0
⁢
x
)
s
⁢
x
-
s
⁢
⁢
dx
-
∫
λ
+
1
⁢
(
x
2
-
λ
2
-
T
0
⁢
x
)
s
⁢
x
-
s
⁢
⁢
dx
,
⁢
⁢
s
=
1
n
,
(
3
)
⁢
λ
-
=
1
2
⁡
[
(
T
0
2
+
4
⁢
λ
2
)
1
2
-
T
0
]
,
(
4
)
⁢
λ
+
=
1
2
⁡
[
(
T
0
2
+
4
⁢
λ
2
)
1
2
+
T
0
]
,
⁢
⁢
T
0
=
F
y
F
cb
,
σ
y
=
F
y
2
⁢
π
⁢
⁢
R
i
⁡
(
L
0
+
L
2
)
,
F
y
=
F
Te
-
F
b
,
⁢
⁢
F
cb
=
F
T
-
F
b
,
F
b
=
π
⁢
⁢
R
i
2
⁢
L
2
⁢
ρ
⁢
⁢
g
,
L
2
=
Δ
⁢
⁢
L
1
-
κ
2
.
(
5
)

### Claim 2 (depends on 1)

The viscosity measuring method according to claim 1, wherein
the first relative movement distance ΔL1 and the second relative movement distance ΔL2 are smaller than the initial depth L0.

### Claim 3 (depends on 1)

The viscosity measuring method according to claim 1, wherein
in the step of obtaining the flow behavior index n, it is judged whether the convergent value FTe of the force applied to the plunger from the sample at the first relative movement velocity vp1 or the second relative movement velocity vp2 is larger or not than a buoyancy force Fb applied to the plunger from the sample, and when the convergent value FTe is judged to be larger than the buoyancy force Fb, the process for obtaining the flow behavior index n is performed.

### Claim 4 (depends on 1)

The viscosity measuring method according to claim 1, wherein
the relationship established among: the first relative movement distance ΔL1, the first peak value FT1 and the first convergent value FTe1, at the first relative movement velocity vp1; the second relative movement distance ΔL2, the second peak value FT2 and the second convergent value FTe2, at the second relative movement velocity vp2; the plunger ratio κ; and the flow behavior index n of the sample includes the above Expressions (2) to (5) and the following Expression (6):
[
NO
⁢
⁢
2
]
s
=
1
n
=
ln
⁡
(
v
p
⁢
⁢
2
/
Φ
2
v
p
⁢
⁢
1
/
Φ
1
)
ln
⁡
(
F
cb
⁢
⁢
2
/
(
L
0
+
L
2
-
⁢
2
)
⁢
λ
2
2
F
cb
⁢
⁢
1
/
(
L
0
+
L
2
-
⁢
1
)
⁢
λ
1
2
)
.
(
6
)

### Claim 5 (depends on 4)

The viscosity measuring method according to claim 4, wherein
the step of obtaining the flow behavior index n includes:
a step in which a provisional flow behavior index na is determined;
a step in which, based on the above Expressions (3) to (5) established among:
the provisional flow behavior index na; the first relative movement distance ΔL1, the first peak value FT1 and the first convergent value FTe1, at the first relative movement velocity vp1; the plunger ratio κ; and a first dimensionless coordinate λ1; or based on a graph or a table created based on the above Expressions (3) to (5), the first dimensionless coordinate λ1 is obtained;
a step in which, based on the above Expression (2) established among the provisional flow behavior index na, the first dimensionless coordinate λ1 and a first dimensionless flow velocity Φ1, or based on a graph or a table created based on the above Expression (2), the first dimensionless flow velocity Φ1 is obtained;
a step in which, based on the above Expressions (3) to (5) established among: the provisional flow behavior index na; the second relative movement distance ΔL2, the second peak value FT2 and the second convergent value FTe2, at the second relative movement velocity vp2; the plunger ratio κ; and a second dimensionless coordinate λ2; or based on a graph or a table created based on the above Expressions (3) to (5), the second dimensionless coordinate λ2 is obtained;
a step in which, based on the above Expression (2) established among the provisional flow behavior index na, the second dimensionless coordinate λ2 and a second dimensionless flow velocity Φ2, or based on a graph or a table created based on the above Expression (2), the second dimensionless flow velocity Φ2 is obtained;
a step in which a flow behavior index n is obtained based on: the first dimensionless coordinate λ1 and the first dimensionless flow velocity Φ1 at the first relative movement velocity vp1; the second dimensionless coordinate λ2 and the second dimensionless flow velocity Φ2 at the second relative movement velocity vp2; and the above Expression (6); and
a step in which the method returns to the step of determining a provisional flow behavior index na, when the obtained flow behavior index n is compared to the provisional flow behavior index na to find out that the obtained flow behavior index n is different from the provisional flow behavior index na.

### Claim 6

A method of measuring an apparent viscosity of a sample of non-Newtonian fluid comprising:
(A) a step in which a plunger having an outer radius Ri is immersed into a sample contained in a cylindrical container having an inner radius R0, coaxially with the cylindrical container by an initial depth L0 and stopped thereat, the outer radius Ri of the plunger being smaller than the inner radius R0 of the cylindrical container;
(B) a step in which the plunger is further immersed into the sample coaxially with the cylindrical container at a first relative movement velocity vp1 by a relative movement distance ΔL, and during the further immersing operation and after the further immersing operation, a force applied to the plunger from the sample is measured with a passage of time;
(C) a step in which a first peak value FT1 and a first convergent value FTe1 of the force applied to the plunger from the sample are obtained, based on measured values of the force caused by the relative movement of the plunger at the first relative movement velocity vp1;
(D) a step in which the plunger is returned coaxially with the cylindrical container to the initial depth L0 and stopped thereat;
(E) a step in which the plunger is further immersed into the sample coaxially with the cylindrical container at a second relative movement velocity vp2 by the relative movement distance ΔL, and during the further immersing operation and after the further immersing operation, a force applied to the plunger from the sample is measured with a passage of time;
(F) a step in which a second peak value FT2 and a second convergent value FTe2 of the force applied to the plunger from the sample are obtained, based on measured values of the force caused by the relative movement of the plunger at the second relative movement velocity vp2;
(G) a step in which, based on a predetermined relationship established among: the first peak value FT1 and the first convergent value FTe1 at the first relative movement velocity vp1; the second peak value FT2 and the second convergent value FTe2 at the second relative movement velocity vp2; the relative movement distance ΔL; a plunger ratio κ=Ri/R0; and a flow behavior index n of the sample; the flow behavior index n is obtained;
(H) a step in which, when the first relative movement velocity vp1 is represented as vp, the first peak value FT1 is represented as FT, and the first convergent value FTe1 is represented as FTe, or when the second relative movement velocity vp2 is represented as vp, the second peak value FT2 is represented as FT, and the second convergent value FTe2 is represented as FTe, an apparent viscosity μa of the sample is calculated based on the vp, the FT, the FTe, the relative movement distance ΔL, the plunger ratio κ, the flow behavior index n, a density ρ of the sample, a gravitational acceleration g, and the following Expression (1):
⁢
[
NO
⁢
⁢
3
]
⁢
μ
a
=
σ
w
dy
/
dt
=
σ
0
+
K
⁡
(
dy
dt
)
n
dy
/
dt
⁢
⁢
⁢
in
⁢
⁢
which
⁢
⁢
⁢
σ
w
=
T
w
⁢
PR
0
2
,
dy
dt
=
(
λ
2
κ
-
κ
-
T
0
)
s
⁢
(
PR
0
2
⁢
K
)
s
,
⁢
⁢
K
=
PR
0
2
⁢
(
Φ
⁢
⁢
R
0
v
p
⁢
κ
2
)
n
,
P
=
2
⁢
σ
y
T
0
⁢
R
0
,
T
w
=
λ
2
κ
-
κ
,
(
1
)
Φ
=
{
∫
κ
λ
-
⁢
(
λ
2
-
x
2
-
T
0
⁢
x
)
s
⁢
x
-
s
⁢
⁢
dx
-
∫
λ
+
1
⁢
(
x
2
-
λ
2
-
T
0
⁢
x
)
s
⁢
⁢
x
-
s
⁢
dx
}
⁢
κ
2
-
∫
κ
λ
-
⁢
(
λ
2
-
x
2
-
T
0
⁢
x
)
s
⁢
x
2
-
s
⁢
dx
+
⁢
∫
λ
+
1
⁢
(
x
2
-
λ
2
-
T
0
⁢
x
)
s
⁢
x
2
-
s
⁢
⁢
dx
⁢
(
2
)
0
=
∫
κ
λ
-
⁢
(
λ
2
-
x
2
-
T
0
⁢
x
)
s
⁢
x
-
s
⁢
⁢
dx
-
∫
λ
+
1
⁢
(
x
2
-
λ
2
-
T
0
⁢
x
)
s
⁢
x
-
s
⁢
⁢
dx
,
⁢
⁢
s
=
1
n
,
(
3
)
⁢
λ
-
=
1
2
⁡
[
(
T
0
2
+
4
⁢
λ
2
)
1
2
-
T
0
]
,
(
4
)
⁢
λ
+
=
1
2
⁡
[
(
T
0
2
+
4
⁢
λ
2
)
1
2
+
T
0
]
,
⁢
⁢
T
0
=
F
y
F
cb
,
σ
y
=
F
y
2
⁢
π
⁢
⁢
R
i
⁡
(
L
0
+
L
2
)
,
F
y
=
F
Te
-
F
b
,
⁢
⁢
F
cb
=
F
T
-
F
b
,
F
b
=
π
⁢
⁢
R
i
2
⁢
L
2
⁢
ρ
⁢
⁢
g
,
L
2
=
Δ
⁢
⁢
L
1
-
κ
2
.
(
5
)

### Claim 7

A method of measuring an apparent viscosity of a sample of non-Newtonian fluid comprising:
(A) a step in which a plunger having an outer radius Ri is immersed into a sample contained in a cylindrical container having an inner radius R0, coaxially with the cylindrical container by an initial depth L0 and stopped thereat, the outer radius Ri of the plunger being smaller than the inner radius R0 of the cylindrical container;
(B) a step in which the plunger is further immersed into the sample coaxially with the cylindrical container at a first relative movement velocity vp1 by a first relative movement distance ΔL1, and during the further immersing operation and after the further immersing operation, a force applied to the plunger from the sample is measured with a passage of time;
(C) a step in which a first peak value FT1 and a first convergent value FTe1 of the force applied to the plunger from the sample are obtained, based on measured values of the force caused by the relative movement of the plunger at the first relative movement velocity vp1;
(D) a step in which the plunger is returned coaxially with the cylindrical container to the initial depth L0 and stopped thereat;
(E) a step in which the plunger is further immersed into the sample coaxially with the cylindrical container at a second relative movement velocity vp2 by a second relative movement distance ΔL2, and during the further immersing operation and after the further immersing operation, a force applied to the plunger from the sample is measured with a passage of time;
(F) a step in which a second peak value FT2 and a second convergent value FTe2 of the force applied to the plunger from the sample are obtained, based on measured values of the force caused by the relative movement of the plunger at the second relative movement velocity vp2;
(G) a step in which, based on a predetermined relationship established among: the first relative movement distance ΔL1, the first peak value FT1 and the first convergent value FTe1, at the first relative movement velocity vp1; the second relative movement distance ΔL2, the second peak value FT2 and the second convergent value FTe2, at the second relative movement velocity vp2; a plunger ratio κ=Ri/R0; and a flow behavior index n of the sample; the flow behavior index n is obtained;
(H) a step in which the plunger is returned coaxially with the cylindrical container to the initial depth L0 and stopped thereat;
(I) a step in which the plunger is further immersed into the sample coaxially with the cylindrical container at a third relative movement velocity vp3 by a third relative movement distance ΔL3, and during the further immersing operation and after the further immersing operation, a force applied to the plunger from the sample is measured with a passage of time;
(J) a step in which a third peak value FT3 and a third convergent value FTe3 of the force applied to the plunger from the sample are obtained, based on measured values of the force caused by the relative movement of the plunger at the third relative movement velocity vp3;
(K) a step in which, when the third relative movement velocity vp3 is represented as vp, the third relative movement distance ΔL3 is represented as ΔL, the third peak value FT3 is represented as FT and the third convergent value FTe3 is represented as FTe, an apparent viscosity μa of the sample is calculated based on the vp, the ΔL, the FT and the FTe, the plunger ratio κ, the flow behavior index n, a density ρ of the sample, a gravitational acceleration g, and the following Expression (1):
⁢
[
NO
⁢
⁢
4
]
⁢
μ
a
=
σ
w
dy
/
dt
=
σ
0
+
K
⁡
(
dy
dt
)
n
dy
/
dt
⁢
⁢
⁢
in
⁢
⁢
which
⁢
⁢
⁢
σ
w
=
T
w
⁢
PR
0
2
,
dy
dt
=
(
λ
2
κ
-
κ
-
T
0
)
s
⁢
(
PR
0
2
⁢
K
)
s
,
⁢
⁢
K
=
PR
0
2
⁢
(
Φ
⁢
⁢
R
0
v
p
⁢
κ
2
)
n
,
P
=
2
⁢
σ
y
T
0
⁢
R
0
,
T
w
=
λ
2
κ
-
κ
,
(
1
)
Φ
=
{
∫
κ
λ
-
⁢
(
λ
2
-
x
2
-
T
0
⁢
x
)
s
⁢
x
-
s
⁢
⁢
dx
-
∫
λ
+
1
⁢
(
x
2
-
λ
2
-
T
0
⁢
x
)
s
⁢
⁢
x
-
s
⁢
dx
}
⁢
κ
2
-
∫
κ
λ
-
⁢
(
λ
2
-
x
2
-
T
0
⁢
x
)
s
⁢
x
2
-
s
⁢
dx
+
⁢
∫
λ
+
1
⁢
(
x
2
-
λ
2
-
T
0
⁢
x
)
s
⁢
x
2
-
s
⁢
⁢
dx
⁢
(
2
)
0
=
∫
κ
λ
-
⁢
(
λ
2
-
x
2
-
T
0
⁢
x
)
s
⁢
x
-
s
⁢
⁢
dx
-
∫
λ
+
1
⁢
(
x
2
-
λ
2
-
T
0
⁢
x
)
s
⁢
x
-
s
⁢
⁢
dx
,
⁢
⁢
s
=
1
n
,
(
3
)
⁢
λ
-
=
1
2
⁡
[
(
T
0
2
+
4
⁢
λ
2
)
1
2
-
T
0
]
,
(
4
)
⁢
λ
+
=
1
2
⁡
[
(
T
0
2
+
4
⁢
λ
2
)
1
2
+
T
0
]
,
⁢
⁢
T
0
=
F
y
F
cb
,
σ
y
=
F
y
2
⁢
π
⁢
⁢
R
i
⁡
(
L
0
+
L
2
)
,
F
y
=
F
Te
-
F
b
,
⁢
⁢
F
cb
=
F
T
-
F
b
,
F
b
=
π
⁢
⁢
R
i
2
⁢
L
2
⁢
ρ
⁢
⁢
g
,
L
2
=
Δ
⁢
⁢
L
1
-
κ
2
.
(
5
)

### Claim 8

A viscosity measuring apparatus comprising:
a cylindrical container having a predetermined inner radius R0, in which a sample is contained;
a plunger having an outer radius Ri that is smaller than the inner radius R0 of the cylindrical container, the plunger being arranged inside the cylindrical container coaxially therewith so as to be relatively movable;
a driving unit configured to relatively move the plunger coaxially with the cylindrical container;
a measuring unit disposed on the plunger and configured to measure a force applied to the plunger from the sample; and
a control unit configured to control the driving unit;
wherein:
the control unit is configured to control the driving unit such that the plunger is immersed into the sample contained in the cylindrical container coaxially with the cylindrical container by a predetermined initial depth L0 and stopped thereat, and that the plunger is then further immersed into the sample coaxially with the cylindrical container at a first relative movement velocity vp1 by a first relative movement distance ΔL1;
the measuring unit is configured to measure a force applied to the plunger from the sample with a passage of time, during the further immersing operation of the plunger at the first relative movement velocity vp1 and after the further immersing operation thereof;
the control unit is configured to obtain a first peak value FT1 and a first convergent value FTe1 of the force applied to the plunger from the sample, based on measured values of the force caused by the relative movement of the plunger at the first relative movement velocity vp1;
the control unit is further configured to control the driving unit such that the plunger is returned coaxially with the cylindrical container to the initial depth L0 and stopped thereat, and that the plunger is then further immersed into the sample coaxially with the cylindrical container at a second relative movement velocity vp2 by a second relative movement distance ΔL2;
the measuring unit is configured to measure a force applied to the plunger from the sample with a passage of time, during the further immersing operation of the plunger at the second relative movement velocity vp2 and after the further immersing operation thereof;
the control unit is configured to obtain a second peak value FT2 and a second convergent value FTe2 of the force applied to the plunger from the sample, based on measured values of the force caused by the relative movement of the plunger at the second relative movement velocity vp2;
the control unit is further configured to obtain, based on a predetermined relationship established among: the first relative movement distance ΔL1, the first peak value FT1 and the first convergent value FTe1, at the first relative movement velocity vp1; the second relative movement distance ΔL2, the second peak value FT2 and the second convergent value FTe2, at the second relative movement velocity vp2; a plunger ratio is κ=Ri/R0; and a flow behavior index n of the sample; the flow behavior index n; and
the control unit is further configured to calculate, when the first relative movement velocity vp1 is represented as vp, the first relative movement distance ΔL1 is represented as ΔL, the first peak value FT1 is represented as FT, and the first convergent value FTe1 is represented as FTe, or when the second relative movement velocity vp2 is represented as vp, the second relative movement distance ΔL2 is represented as ΔL, the second peak value FT2 is represented as FT, and the second convergent value FTe2 is represented as FTe, an apparent viscosity μa of the sample based on the vp, the ΔL, the FT, the FTe, the plunger ratio κ, the flow behavior index n, a density ρ of the sample, a gravitational acceleration g, and the following Expression (1):
⁢
[
NO
⁢
⁢
5
]
⁢
μ
a
=
σ
w
dy
/
dt
=
σ
0
+
K
⁡
(
dy
dt
)
n
dy
/
dt
⁢
⁢
⁢
in
⁢
⁢
which
⁢
⁢
⁢
σ
w
=
T
w
⁢
PR
0
2
,
dy
dt
=
(
λ
2
κ
-
κ
-
T
0
)
s
⁢
(
PR
0
2
⁢
K
)
s
,
⁢
⁢
K
=
PR
0
2
⁢
(
Φ
⁢
⁢
R
0
v
p
⁢
κ
2
)
n
,
P
=
2
⁢
σ
y
T
0
⁢
R
0
,
T
w
=
λ
2
κ
-
κ
,
(
1
)
Φ
=
{
∫
κ
λ
-
⁢
(
λ
2
-
x
2
-
T
0
⁢
x
)
s
⁢
x
-
s
⁢
⁢
dx
-
∫
λ
+
1
⁢
(
x
2
-
λ
2
-
T
0
⁢
x
)
s
⁢
⁢
x
-
s
⁢
dx
}
⁢
κ
2
-
∫
κ
λ
-
⁢
(
λ
2
-
x
2
-
T
0
⁢
x
)
s
⁢
x
2
-
s
⁢
dx
+
⁢
∫
λ
+
1
⁢
(
x
2
-
λ
2
-
T
0
⁢
x
)
s
⁢
x
2
-
s
⁢
⁢
dx
⁢
(
2
)
0
=
∫
κ
λ
-
⁢
(
λ
2
-
x
2
-
T
0
⁢
x
)
s
⁢
x
-
s
⁢
⁢
dx
-
∫
λ
+
1
⁢
(
x
2
-
λ
2
-
T
0
⁢
x
)
s
⁢
x
-
s
⁢
⁢
dx
,
⁢
⁢
s
=
1
n
,
(
3
)
⁢
λ
-
=
1
2
⁡
[
(
T
0
2
+
4
⁢
λ
2
)
1
2
-
T
0
]
,
(
4
)
⁢
λ
+
=
1
2
⁡
[
(
T
0
2
+
4
⁢
λ
2
)
1
2
+
T
0
]
,
⁢
⁢
T
0
=
F
y
F
cb
,
σ
y
=
F
y
2
⁢
π
⁢
⁢
R
i
⁡
(
L
0
+
L
2
)
,
F
y
=
F
Te
-
F
b
,
⁢
⁢
F
cb
=
F
T
-
F
b
,
F
b
=
π
⁢
⁢
R
i
2
⁢
L
2
⁢
ρ
⁢
⁢
g
,
L
2
=
Δ
⁢
⁢
L
1
-
κ
2
.
(
5
)

### Claim 9 (depends on 8)

The viscosity measuring apparatus according to claim 8, wherein
the first relative movement distance ΔL1 and the second relative movement distance ΔL2 are smaller than the initial depth L0.

### Claim 10 (depends on 8)

The viscosity measuring apparatus according to claim 8, wherein
before the control unit obtains the flow behavior index n, the control unit is configured to judge whether the convergent value FTe of the force applied to the plunger from the sample at the first relative movement velocity vp1 or the second relative movement velocity vp2 is larger or not than a buoyancy force Fb applied to the plunger from the sample, and when the control unit judges that the convergent value FTe is larger than the buoyancy force Fb, the control unit is configured to perform the process for obtaining the flow behavior index n.

### Claim 11 (depends on 8)

The viscosity measuring apparatus according to claim 8, wherein
the relationship established among: the first relative movement distance ΔL1, the first peak value FT1 and the first convergent value FTe1, at the first relative movement velocity vp1; the second relative movement distance ΔL2, the second peak value FT2 and the second convergent value FTe2, at the second relative movement velocity vp2; the plunger ratio κ; and the flow behavior index n of the sample includes the above Expressions (2) to (5) and the following Expression (6):
[
NO
⁢
⁢
6
]
s
=
1
n
=
ln
⁡
(
v
p
⁢
⁢
2
/
Φ
2
v
p
⁢
⁢
1
/
Φ
1
)
ln
⁡
(
F
cb
⁢
⁢
2
/
(
L
0
+
L
2
-
⁢
2
)
⁢
λ
2
2
F
cb
⁢
⁢
1
/
(
L
0
+
L
2
-
⁢
1
)
⁢
λ
1
2
)
.
(
6
)

### Claim 12 (depends on 11)

The viscosity measuring apparatus according to claim 11, wherein
when the control unit obtains the flow behavior index n, the control unit is configured:
to determine a provisional flow behavior index na;
to obtain, based on the above Expressions (3) to (5) established among: the provisional flow behavior index na; the first relative movement distance ΔL1, the first peak value FT1 and the first convergent value FTe1, at the first relative movement velocity vp1; the plunger ratio κ; and a first dimensionless coordinate λ1; or based on a graph or a table created based on the above Expressions (3) to (5), the first dimensionless coordinate λ1;
to obtain, based on the above Expression (2) established among the provisional flow behavior index na, the first dimensionless coordinate λ1 and a first dimensionless flow velocity Φ1, or based on a graph or a table created based on the above Expression (2), the first dimensionless flow velocity Φ1;
to obtain, based on the above Expressions (3) to (5) established among: the provisional flow behavior index na; the second relative movement distance ΔL2, the second peak value FT2 and the second convergent value FTe2, at the second relative movement velocity vp2; the plunger ratio κ; and a second dimensionless coordinate λ2; or based on a graph or a table created based on the above Expressions (3) to (5), the second dimensionless coordinate λ2;
to obtain, based on the above Expression (2) established among the provisional flow behavior index na, the second dimensionless coordinate λ2 and a second dimensionless flow velocity Φ2, or based on a graph or a table created based on the above Expression (2), the second dimensionless flow velocity Φ2;
to obtain a flow behavior index n based on: the first dimensionless coordinate) λi and the first dimensionless flow velocity Φ1 at the first relative movement velocity vp1; the second dimensionless coordinate λ2 and the second dimensionless flow velocity Φ2 at the second relative movement velocity vp2; and the above Expression (6); and
to return to the step of determining a provisional flow behavior index na, when the control unit compares the obtained flow behavior index n to the provisional flow behavior index na to find out that the obtained flow behavior index n is different from the provisional flow behavior index na.

### Claim 13

A viscosity measuring apparatus comprising:
a cylindrical container having a predetermined inner radius R0, in which a sample is contained;
a plunger having an outer radius Ri that is smaller than the inner radius R0 of the cylindrical container, the plunger being arranged inside the cylindrical container coaxially therewith so as to be relatively movable;
a driving unit configured to relatively move the plunger coaxially with the cylindrical container;
a measuring unit disposed on the plunger and configured to measure a force applied to the plunger from the sample; and
a control unit configured to control the driving unit;
wherein:
the control unit is configured to control the driving unit such that the plunger is immersed into the sample contained in the cylindrical container coaxially with the cylindrical container by a predetermined initial depth L0 and stopped thereat, and that the plunger is then further immersed into the sample coaxially with the cylindrical container at a first relative movement velocity vp1 by a relative movement distance ΔL;
the measuring unit is configured to measure a force applied to the plunger from the sample with a passage of time, during the further immersing operation of the plunger at the first relative movement velocity vp1 and after the further immersing operation thereof;
the control unit is configured to obtain a first peak value FT1 and a first convergent value FTe1 of the force applied to the plunger from the sample, based on measured values of the force caused by the relative movement of the plunger at the first relative movement velocity vp1;
the control unit is further configured to control the driving unit such that the plunger is returned coaxially with the cylindrical container to the initial depth L0 and stopped thereat, and that the plunger is then further immersed into the sample coaxially with the cylindrical container at a second relative movement velocity vp2 by the relative movement distance ΔL;
the measuring unit is configured to measure a force applied to the plunger from the sample with a passage of time, during the further immersing operation of the plunger at the second relative movement velocity vp2 and after the further immersing operation thereof;
the control unit is configured to obtain a second peak value FT2 and a second convergent value FTe2 of the force applied to the plunger from the sample, based on measured values of the force caused by the relative movement of the plunger at the second relative movement velocity vp2;
the control unit is further configured to obtain, based on a predetermined relationship established among: the first peak value FT1 and the first convergent value FTe1, at the first relative movement velocity vp1; the second peak value FT2 and the second convergent value FTe2, at the second relative movement velocity vp2; the relative movement distance ΔL; a plunger ratio κ=Ri/R0; and a flow behavior index n of the sample; the flow behavior index n; and
the control unit is further configured, when the first relative movement velocity vp1 is represented as vp, the first peak value FT1 is represented as FT, and the first convergent value FTe1 is represented as FTe, or when the second relative movement velocity vp2 is represented as vp, the second peak value FT2 is represented as FT, and the second convergent value FTe2 is represented as FTe, to calculate an apparent viscosity μa of the sample based on the vp, the FT, the FTe, the relative movement distance ΔL, the plunger ratio κ, the flow behavior index n, a density ρ of the sample, a gravitational acceleration g, and the following Expression (1):
⁢
[
NO
⁢
⁢
7
]
⁢
μ
a
=
σ
w
dy
/
dt
=
σ
0
+
K
⁡
(
dy
dt
)
n
dy
/
dt
⁢
⁢
⁢
in
⁢
⁢
which
⁢
⁢
⁢
σ
w
=
T
w
⁢
PR
0
2
,
dy
dt
=
(
λ
2
κ
-
κ
-
T
0
)
s
⁢
(
PR
0
2
⁢
K
)
s
,
⁢
⁢
K
=
PR
0
2
⁢
(
Φ
⁢
⁢
R
0
v
p
⁢
κ
2
)
n
,
P
=
2
⁢
σ
y
T
0
⁢
R
0
,
T
w
=
λ
2
κ
-
κ
,
(
1
)
Φ
=
{
∫
κ
λ
-
⁢
(
λ
2
-
x
2
-
T
0
⁢
x
)
s
⁢
x
-
s
⁢
⁢
dx
-
∫
λ
+
1
⁢
(
x
2
-
λ
2
-
T
0
⁢
x
)
s
⁢
⁢
x
-
s
⁢
dx
}
⁢
κ
2
-
∫
κ
λ
-
⁢
(
λ
2
-
x
2
-
T
0
⁢
x
)
s
⁢
x
2
-
s
⁢
dx
+
⁢
∫
λ
+
1
⁢
(
x
2
-
λ
2
-
T
0
⁢
x
)
s
⁢
x
2
-
s
⁢
⁢
dx
⁢
(
2
)
0
=
∫
κ
λ
-
⁢
(
λ
2
-
x
2
-
T
0
⁢
x
)
s
⁢
x
-
s
⁢
⁢
dx
-
∫
λ
+
1
⁢
(
x
2
-
λ
2
-
T
0
⁢
x
)
s
⁢
x
-
s
⁢
⁢
dx
,
⁢
⁢
s
=
1
n
,
(
3
)
⁢
λ
-
=
1
2
⁡
[
(
T
0
2
+
4
⁢
λ
2
)
1
2
-
T
0
]
,
(
4
)
⁢
λ
+
=
1
2
⁡
[
(
T
0
2
+
4
⁢
λ
2
)
1
2
+
T
0
]
,
⁢
⁢
T
0
=
F
y
F
cb
,
σ
y
=
F
y
2
⁢
π
⁢
⁢
R
i
⁡
(
L
0
+
L
2
)
,
F
y
=
F
Te
-
F
b
,
⁢
⁢
F
cb
=
F
T
-
F
b
,
F
b
=
π
⁢
⁢
R
i
2
⁢
L
2
⁢
ρ
⁢
⁢
g
,
L
2
=
Δ
⁢
⁢
L
1
-
κ
2
.
(
5
)

### Claim 14

A viscosity measuring apparatus comprising:
a cylindrical container having a predetermined inner radius R0, in which a sample is contained;
a plunger having an outer radius Ri that is smaller than the inner radius R0 of the cylindrical container, the plunger being arranged inside the cylindrical container coaxially therewith so as to be relatively movable;
a driving unit configured to relatively move the plunger coaxially with the cylindrical container;
a measuring unit disposed on the plunger and configured to measure a force applied to the plunger from the sample; and
a control unit configured to control the driving unit;
wherein:
the control unit is configured to control the driving unit such that the plunger is immersed into the sample contained in the cylindrical container coaxially with the cylindrical container by a predetermined initial depth L0 and stopped thereat, and that the plunger is then further immersed into the sample coaxially with the cylindrical container at a first relative movement velocity vp1 by a first relative movement distance ΔL1;
the measuring unit is configured to measure a force applied to the plunger from the sample with a passage of time, during the further immersing operation of the plunger at the first relative movement velocity vp1 and after the further immersing operation thereof;
the control unit is configured to obtain a first peak value FT1 and a first convergent value FTe1 of the force applied to the plunger from the sample, based on measured values of the force caused by the relative movement of the plunger at the first relative movement velocity vp1;
the control unit is further configured to control the driving unit such that the plunger is returned coaxially with the cylindrical container to the initial depth L0 and stopped thereat, and that the plunger is then further immersed into the sample coaxially with the cylindrical container at a second relative movement velocity vp2 by a second relative movement distance ΔL2;
the measuring unit is configured to measure a force applied to the plunger from the sample with a passage of time, during the further immersing operation of the plunger at the second relative movement velocity vp2 and after the further immersing operation thereof;
the control unit is configured to obtain a second peak value FT2 and a second convergent value FTe2 of the force applied to the plunger from the sample, based on measured values of the force caused by the relative movement of the plunger at the second relative movement velocity vp2;
the control unit is further configured to obtain, based on a predetermined relationship established among: the first relative movement distance ΔL1, the first peak value FT1 and the first convergent value FTe1, at the first relative movement velocity vp1; the second relative movement distance ΔL2, the second peak value FT2 and the second convergent value FTe2, at the second relative movement velocity vp2; a plunger ratio κ=Ri/R0; and a flow behavior index n of the sample; the flow behavior index n;
the control unit is further configured to control the driving unit such that the plunger is returned coaxially with the cylindrical container to the initial depth L0 and stopped thereat, and that the plunger is then further immersed into the sample coaxially with the cylindrical container at a third relative movement velocity vp3 by a third relative movement distance ΔL3;
the measuring unit is configured to measure a force applied to the plunger from the sample with a passage of time, during the further immersing operation of the plunger at the third relative movement velocity vp3 and after the further immersing operation thereof;
the control unit is configured to obtain a third peak value FT3 and a third convergent value FTe3 of the force applied to the plunger from the sample, based on measured values of the force caused by the relative movement of the plunger at the third relative movement velocity vp3; and
the control unit is further configured to calculate, when the third relative movement velocity vp3 is represented as vp, the third relative movement distance ΔL3 is represented as ΔL, the third peak value FT3 is represented as FT and the third convergent value FTe3 is represented as FTe, an apparent viscosity μa of the sample based on the vp, the ΔL, the FT and the FTe, the plunger ratio κ, the flow behavior index n, a density ρ of the sample, a gravitational acceleration g, and the following Expression (1):
⁢
[
NO
⁢
⁢
8
]
⁢
μ
a
=
σ
w
dy
/
dt
=
σ
0
+
K
⁡
(
dy
dt
)
n
dy
/
dt
⁢
⁢
⁢
in
⁢
⁢
which
⁢
⁢
⁢
σ
w
=
T
w
⁢
PR
0
2
,
dy
dt
=
(
λ
2
κ
-
κ
-
T
0
)
s
⁢
(
PR
0
2
⁢
K
)
s
,
⁢
⁢
K
=
PR
0
2
⁢
(
Φ
⁢
⁢
R
0
v
p
⁢
κ
2
)
n
,
P
=
2
⁢
σ
y
T
0
⁢
R
0
,
T
w
=
λ
2
κ
-
κ
,
(
1
)
Φ
=
{
∫
κ
λ
-
⁢
(
λ
2
-
x
2
-
T
0
⁢
x
)
s
⁢
x
-
s
⁢
⁢
dx
-
∫
λ
+
1
⁢
(
x
2
-
λ
2
-
T
0
⁢
x
)
s
⁢
⁢
x
-
s
⁢
dx
}
⁢
κ
2
-
∫
κ
λ
-
⁢
(
λ
2
-
x
2
-
T
0
⁢
x
)
s
⁢
x
2
-
s
⁢
dx
+
⁢
∫
λ
+
1
⁢
(
x
2
-
λ
2
-
T
0
⁢
x
)
s
⁢
x
2
-
s
⁢
⁢
dx
⁢
(
2
)
0
=
∫
κ
λ
-
⁢
(
λ
2
-
x
2
-
T
0
⁢
x
)
s
⁢
x
-
s
⁢
⁢
dx
-
∫
λ
+
1
⁢
(
x
2
-
λ
2
-
T
0
⁢
x
)
s
⁢
x
-
s
⁢
⁢
dx
,
⁢
⁢
s
=
1
n
,
(
3
)
⁢
λ
-
=
1
2
⁡
[
(
T
0
2
+
4
⁢
λ
2
)
1
2
-
T
0
]
,
(
4
)
⁢
λ
+
=
1
2
⁡
[
(
T
0
2
+
4
⁢
λ
2
)
1
2
+
T
0
]
,
⁢
⁢
T
0
=
F
y
F
cb
,
σ
y
=
F
y
2
⁢
π
⁢
⁢
R
i
⁡
(
L
0
+
L
2
)
,
F
y
=
F
Te
-
F
b
,
⁢
⁢
F
cb
=
F
T
-
F
b
,
F
b
=
π
⁢
⁢
R
i
2
⁢
L
2
⁢
ρ
⁢
⁢
g
,
L
2
=
Δ
⁢
⁢
L
1
-
κ
2
.
(
5
)

### Claim 15

A control apparatus for controlling a viscosity measuring apparatus including:
a cylindrical container having a predetermined inner radius R0, in which a sample is contained;
a plunger having an outer radius Ri that is smaller than the inner radius R0 of the cylindrical container, the plunger being arranged inside the cylindrical container coaxially therewith so as to be relatively movable;
a driving unit configured to relatively move the plunger coaxially with the cylindrical container; and
a measuring unit disposed on the plunger and configured to measure a force applied to the plunger from the sample;
wherein:
the control apparatus is configured to control the driving unit such that the plunger is immersed into the sample contained in the cylindrical container coaxially with the cylindrical container by a predetermined initial depth L0 and stopped thereat, and that the plunger is then further immersed into the sample coaxially with the cylindrical container at a first relative movement velocity vp1 by a first relative movement distance ΔL1;
the measuring unit is configured to measure a force applied to the plunger from the sample with a passage of time, during the further immersing operation of the plunger at the first relative movement velocity vp1 and after the further immersing operation thereof;
the control apparatus is configured to obtain a first peak value FT1 and a first convergent value FTe1 of the force applied to the plunger from the sample, based on measured values of the force caused by the relative movement of the plunger at the first relative movement velocity vp1;
the control apparatus is further configured to control the driving unit such that the plunger is returned coaxially with the cylindrical container to the initial depth L0 and stopped thereat, and that the plunger is then further immersed into the sample coaxially with the cylindrical container at a second relative movement velocity vp2 by a second relative movement distance ΔL2;
the measuring unit is configured to measure a force applied to the plunger from the sample with a passage of time, during the further immersing operation of the plunger at the second relative movement velocity vp2 and after the further immersing operation thereof;
the control apparatus is configured to obtain a second peak value FT2 and a second convergent value FTe2 of the force applied to the plunger from the sample, based on measured values of the force caused by the relative movement of the plunger at the second relative movement velocity vp2;
the control apparatus is further configured to obtain, based on a predetermined relationship established among: the first relative movement distance ΔL1, the first peak value FT1 and the first convergent value FTe1, at the first relative movement velocity vp1; the second relative movement distance ΔL2, the second peak value FT2 and the second convergent value FTe2, at the second relative movement velocity vp2; a plunger ratio κ=Ri/R0; and a flow behavior index n of the sample; the flow behavior index n;
the control apparatus is further configured to calculate, when the first relative movement velocity vp1 is represented as vp, the first relative movement distance ΔL1 is represented as ΔL, the first peak value FT1 is represented as FT, and the first convergent value FTe1 is represented as FTe, or when the second relative movement velocity vp2 is represented as vp, the second relative movement distance ΔL2 is represented as ΔL, the second peak value FT2 is represented as FT, and the second convergent value FTe2 is represented as FTe, an apparent viscosity μa of the sample based on the vp, the ΔL, the FT, the FTe, the plunger ratio κ, the flow behavior index n, a density ρ of the sample, a gravitational acceleration g, and the following Expression (1):
⁢
[
NO
⁢
⁢
9
]
⁢
μ
a
=
σ
w
dy
/
dt
=
σ
0
+
K
⁡
(
dy
dt
)
n
dy
/
dt
⁢
⁢
⁢
in
⁢
⁢
which
⁢
⁢
⁢
σ
w
=
T
w
⁢
PR
0
2
,
dy
dt
=
(
λ
2
κ
-
κ
-
T
0
)
s
⁢
(
PR
0
2
⁢
K
)
s
,
⁢
⁢
K
=
PR
0
2
⁢
(
Φ
⁢
⁢
R
0
v
p
⁢
κ
2
)
n
,
P
=
2
⁢
σ
y
T
0
⁢
R
0
,
T
w
=
λ
2
κ
-
κ
,
(
1
)
Φ
=
{
∫
κ
λ
-
⁢
(
λ
2
-
x
2
-
T
0
⁢
x
)
s
⁢
x
-
s
⁢
⁢
dx
-
∫
λ
+
1
⁢
(
x
2
-
λ
2
-
T
0
⁢
x
)
s
⁢
⁢
x
-
s
⁢
dx
}
⁢
κ
2
-
∫
κ
λ
-
⁢
(
λ
2
-
x
2
-
T
0
⁢
x
)
s
⁢
x
2
-
s
⁢
dx
+
⁢
∫
λ
+
1
⁢
(
x
2
-
λ
2
-
T
0
⁢
x
)
s
⁢
x
2
-
s
⁢
⁢
dx
⁢
(
2
)
0
=
∫
κ
λ
-
⁢
(
λ
2
-
x
2
-
T
0
⁢
x
)
s
⁢
x
-
s
⁢
⁢
dx
-
∫
λ
+
1
⁢
(
x
2
-
λ
2
-
T
0
⁢
x
)
s
⁢
x
-
s
⁢
⁢
dx
,
⁢
⁢
s
=
1
n
,
(
3
)
⁢
λ
-
=
1
2
⁡
[
(
T
0
2
+
4
⁢
λ
2
)
1
2
-
T
0
]
,
(
4
)
⁢
λ
+
=
1
2
⁡
[
(
T
0
2
+
4
⁢
λ
2
)
1
2
+
T
0
]
,
⁢
⁢
T
0
=
F
y
F
cb
,
σ
y
=
F
y
2
⁢
π
⁢
⁢
R
i
⁡
(
L
0
+
L
2
)
,
F
y
=
F
Te
-
F
b
,
⁢
⁢
F
cb
=
F
T
-
F
b
,
F
b
=
π
⁢
⁢
R
i
2
⁢
L
2
⁢
ρ
⁢
⁢
g
,
L
2
=
Δ
⁢
⁢
L
1
-
κ
2
.
(
5
)

### Claim 16 (depends on 15)

The control apparatus according to claim 15, wherein
the first relative movement distance ΔL1 and the second relative movement distance ΔL2 are smaller than the initial depth L0.

### Claim 17 (depends on 15)

The control apparatus according to claim 15, configured to judge, before the control apparatus obtains the flow behavior index n, whether the convergent value FTe of the force applied to the plunger from the sample at the first relative movement velocity vp1 or the second relative movement velocity vp2 is larger or not than a buoyancy force Fb applied to the plunger from the sample, and configured to perform the process for obtaining the flow behavior index n, when the control apparatus judges that the convergent value FTe is larger than the buoyancy force Fb.

### Claim 18 (depends on 15)

The control apparatus according to claim 15, wherein
the relationship established among: the first relative movement distance ΔL1, the first peak value FT1 and the first convergent value FTe1, at the first relative movement velocity vp1; the second relative movement distance ΔL2, the second peak value FT2 and the second convergent value FTe2, at the second relative movement velocity vp2; the plunger ratio κ; and the flow behavior index n of the sample includes the above Expressions (2) to (5) and the following Expression (6):
[
NO
⁢
⁢
10
]
s
=
1
n
=
ln
⁡
(
v
p
⁢
⁢
2
/
Φ
2
v
p
⁢
⁢
1
/
Φ
1
)
ln
⁡
(
F
cb
⁢
⁢
2
/
(
L
0
+
L
2
-
⁢
2
)
⁢
λ
2
2
F
cb
⁢
⁢
1
/
(
L
0
+
L
2
-
⁢
1
)
⁢
λ
1
2
)
.
(
6
)

### Claim 19 (depends on 18)

The control apparatus according to claim 18,
when the control apparatus obtains the flow behavior index n, the control apparatus is configured:
to determine a provisional flow behavior index na;
to obtain, based on the above Expressions (3) to (5) established among: the provisional flow behavior index na; the first relative movement distance ΔL1, the first peak value FT1 and the first convergent value FTe1, at the first relative movement velocity vp1; the plunger ratio κ; and a first dimensionless coordinate λ1; or based on a graph or a table created based on the above Expressions (3) to (5), the first dimensionless coordinate λ1;
to obtain, based on the above Expression (2) established among the provisional flow behavior index na, the first dimensionless coordinate λ1 and a first dimensionless flow velocity Φ1, or based on a graph or a table created based on the above Expression (2), the first dimensionless flow velocity Φ1;
to obtain, based on the above Expressions (3) to (5) established among: the provisional flow behavior index na; the second relative movement distance ΔL2, the second peak value FT2 and the second convergent value FTe2, at the second relative movement velocity vp2; the plunger ratio κ; and a second dimensionless coordinate λ2;
or based on a graph or a table created based on the above Expressions (3) to (5), the second dimensionless coordinate λ2;
to obtain, based on the above Expression (2) established among the provisional flow behavior index na, the second dimensionless coordinate λ2 and a second dimensionless flow velocity Φ2, or based on a graph or a table created based on the above Expression (2), the second dimensionless flow velocity Φ2;
to obtain a flow behavior index n based on: the first dimensionless coordinate λ1 and the first dimensionless flow velocity Φ1 at the first relative movement velocity vp1; the second dimensionless coordinate λ2 and the second dimensionless flow velocity Φ2 at the second relative movement velocity vp2; and the above Expression (6); and
to return to the step of determining a provisional flow behavior index na, when the control apparatus compares the obtained flow behavior index n to the provisional flow behavior index na to find out that the obtained flow behavior index n is different from the provisional flow behavior index na.

### Claim 20

A control apparatus for controlling a viscosity measuring apparatus including:
a cylindrical container having a predetermined inner radius R0, in which a sample is contained;
a plunger having an outer radius Ri that is smaller than the inner radius R0 of the cylindrical container, the plunger being arranged inside the cylindrical container coaxially therewith so as to be relatively movable;
a driving unit configured to relatively move the plunger coaxially with the cylindrical container; and
a measuring unit disposed on the plunger and configured to measure a force applied to the plunger from the sample;
wherein:
the control apparatus is configured to control the driving unit such that the plunger is immersed into the sample contained in the cylindrical container coaxially with the cylindrical container by a predetermined initial depth L0 and stopped thereat, and that the plunger is then further immersed into the sample coaxially with the cylindrical container at a first relative movement velocity vp1 by a relative movement distance ΔL;
the measuring unit is configured to measure a force applied to the plunger from the sample with a passage of time, during the further immersing operation of the plunger at the first relative movement velocity vp1 and after the further immersing operation thereof;
the control apparatus is configured to obtain a first peak value FT1 and a first convergent value FTe1 of the force applied to the plunger from the sample, based on measured values of the force caused by the relative movement of the plunger at the first relative movement velocity vp1;
the control apparatus is further configured to control the driving unit such that the plunger is returned coaxially with the cylindrical container to the initial depth L0 and stopped thereat, and that the plunger is then further immersed into the sample coaxially with the cylindrical container at a second relative movement velocity vp2 by the relative movement distance ΔL;
the measuring unit is configured to measure a force applied to the plunger from the sample with a passage of time, during the further immersing operation of the plunger at the second relative movement velocity vp2 and after the further immersing operation thereof;
the control apparatus is configured to obtain a second peak value FT2 and a second convergent value FTe2 of the force applied to the plunger from the sample, based on measured values of the force caused by the relative movement of the plunger at the second relative movement velocity vp2;
the control apparatus is further configured to obtain, based on a predetermined relationship established among: the first peak value FT1 and the first convergent value FTe1, at the first relative movement velocity vp1; the second peak value FT2 and the second convergent value FTe2, at the second relative movement velocity vp2; the relative movement distance ΔL; a plunger ratio κ=Ri/R0; and a flow behavior index n of the sample; the flow behavior index n; and
the control apparatus is further configured to calculate, when the first relative movement velocity vp1 is represented as vp, the first peak value FT1 is represented as FT, and the first convergent value FTe1 is represented as FTe, or when the second relative movement velocity vp2 is represented as vp, the second peak value FT2 is represented as FT, and the second convergent value FTe2 is represented as FTe, an apparent viscosity μa of the sample based on the vp, the FT, the FTe, the relative movement distance ΔL, the plunger ratio κ, the flow behavior index n, a density ρ of the sample, a gravitational acceleration g, and the following Expression (1):
⁢
[
NO
⁢
⁢
11
]
⁢
μ
a
=
σ
w
dy
/
dt
=
σ
0
+
K
⁡
(
dy
dt
)
n
dy
/
dt
⁢
⁢
⁢
in
⁢
⁢
which
⁢
⁢
⁢
σ
w
=
T
w
⁢
PR
0
2
,
dy
dt
=
(
λ
2
κ
-
κ
-
T
0
)
s
⁢
(
PR
0
2
⁢
K
)
s
,
⁢
⁢
K
=
PR
0
2
⁢
(
Φ
⁢
⁢
R
0
v
p
⁢
κ
2
)
n
,
P
=
2
⁢
σ
y
T
0
⁢
R
0
,
T
w
=
λ
2
κ
-
κ
,
(
1
)
Φ
=
{
∫
κ
λ
-
⁢
(
λ
2
-
x
2
-
T
0
⁢
x
)
s
⁢
x
-
s
⁢
⁢
dx
-
∫
λ
+
1
⁢
(
x
2
-
λ
2
-
T
0
⁢
x
)
s
⁢
⁢
x
-
s
⁢
dx
}
⁢
κ
2
-
∫
κ
λ
-
⁢
(
λ
2
-
x
2
-
T
0
⁢
x
)
s
⁢
x
2
-
s
⁢
dx
+
⁢
∫
λ
+
1
⁢
(
x
2
-
λ
2
-
T
0
⁢
x
)
s
⁢
x
2
-
s
⁢
⁢
dx
⁢
(
2
)
0
=
∫
κ
λ
-
⁢
(
λ
2
-
x
2
-
T
0
⁢
x
)
s
⁢
x
-
s
⁢
⁢
dx
-
∫
λ
+
1
⁢
(
x
2
-
λ
2
-
T
0
⁢
x
)
s
⁢
x
-
s
⁢
⁢
dx
,
⁢
⁢
s
=
1
n
,
(
3
)
⁢
λ
-
=
1
2
⁡
[
(
T
0
2
+
4
⁢
λ
2
)
1
2
-
T
0
]
,
(
4
)
⁢
λ
+
=
1
2
⁡
[
(
T
0
2
+
4
⁢
λ
2
)
1
2
+
T
0
]
,
⁢
⁢
T
0
=
F
y
F
cb
,
σ
y
=
F
y
2
⁢
π
⁢
⁢
R
i
⁡
(
L
0
+
L
2
)
,
F
y
=
F
Te
-
F
b
,
⁢
⁢
F
cb
=
F
T
-
F
b
,
F
b
=
π
⁢
⁢
R
i
2
⁢
L
2
⁢
ρ
⁢
⁢
g
,
L
2
=
Δ
⁢
⁢
L
1
-
κ
2
.
(
5
)

### Claim 21

A control apparatus for controlling a viscosity measuring apparatus including:
a cylindrical container having a predetermined inner radius R0, in which a sample is contained;
a plunger having an outer radius Ri that is smaller than the inner radius R0 of the cylindrical container, the plunger being arranged inside the cylindrical container coaxially therewith so as to be relatively movable;
a driving unit configured to relatively move the plunger coaxially with the cylindrical container; and
a measuring unit disposed on the plunger and configured to measure a force applied to the plunger from the sample;
wherein:
the control apparatus is configured to control the driving unit such that the plunger is immersed into the sample contained in the cylindrical container coaxially with the cylindrical container by a predetermined initial depth L0 and stopped thereat, and that the plunger is then further immersed into the sample coaxially with the cylindrical container at a first relative movement velocity vp1 by a first relative movement distance ΔL1;
the measuring unit is configured to measure a force applied to the plunger from the sample with a passage of time, during the further immersing operation of the plunger at the first relative movement velocity vp1 and after the further immersing operation thereof;
the control apparatus is configured to obtain a first peak value FT1 and a first convergent value FTe1 of the force applied to the plunger from the sample, based on measured values of the force caused by the relative movement of the plunger at the first relative movement velocity vp1;
the control apparatus is further configured to control the driving unit such that the plunger is returned coaxially with the cylindrical container to the initial depth L0 and stopped thereat, and that the plunger is then further immersed into the sample coaxially with the cylindrical container at a second relative movement velocity vp2 by a second relative movement distance ΔL2;
the measuring unit is configured to measure a force applied to the plunger from the sample with a passage of time, during the further immersing operation of the plunger at the second relative movement velocity vp2 and after the further immersing operation thereof;
the control apparatus is configured to obtain a second peak value FT2 and a second convergent value FTe2 of the force applied to the plunger from the sample, based on measured values of the force caused by the relative movement of the plunger at the second relative movement velocity vp2;
the control apparatus is further configured to obtain, based on a predetermined relationship established among: the first relative movement distance ΔL1, the first peak value FT1 and the first convergent value FTe1, at the first relative movement velocity vp1; the second relative movement distance ΔL2, the second peak value FT2 and the second convergent value FTe2, at the second relative movement velocity vp2; a plunger ratio κ=Ri/R0; and a flow behavior index n of the sample; the flow behavior index n;
the control apparatus is further configured to control the driving unit such that the plunger is returned coaxially with the cylindrical container to the initial depth L0 and stopped thereat, and that the plunger is then further immersed into the sample coaxially with the cylindrical container at a third relative movement velocity vp3 by a third relative movement distance ΔL3;
the measuring unit is configured to measure a force applied to the plunger from the sample with a passage of time, during the further immersing operation of the plunger at the third relative movement velocity vp3 and after the further immersing operation thereof;
the control apparatus is configured to obtain a third peak value FT3 and a third convergent value FTe3 of the force applied to the plunger from the sample, based on measured values of the force caused by the relative movement of the plunger at the third relative movement velocity vp3; and
the control apparatus is further configured to calculate, when the third relative movement velocity vp3 is represented as vp, the third relative movement distance ΔL3 is represented as ΔL, the third peak value FT3 is represented as FT and the third convergent value FTe3 is represented as FTe, an apparent viscosity μa of the sample based on the vp, the ΔL, the FT and the FTe, the plunger ratio κ, the flow behavior index n, a density ρ of the sample, a gravitational acceleration g, and the following Expression (1):
⁢
[
NO
⁢
⁢
12
]
⁢
μ
a
=
σ
w
dy
/
dt
=
σ
0
+
K
⁡
(
dy
dt
)
n
dy
/
dt
⁢
⁢
⁢
in
⁢
⁢
which
⁢
⁢
⁢
σ
w
=
T
w
⁢
PR
0
2
,
dy
dt
=
(
λ
2
κ
-
κ
-
T
0
)
s
⁢
(
PR
0
2
⁢
K
)
s
,
⁢
⁢
K
=
PR
0
2
⁢
(
Φ
⁢
⁢
R
0
v
p
⁢
κ
2
)
n
,
P
=
2
⁢
σ
y
T
0
⁢
R
0
,
T
w
=
λ
2
κ
-
κ
,
(
1
)
Φ
=
{
∫
κ
λ
-
⁢
(
λ
2
-
x
2
-
T
0
⁢
x
)
s
⁢
x
-
s
⁢
⁢
dx
-
∫
λ
+
1
⁢
(
x
2
-
λ
2
-
T
0
⁢
x
)
s
⁢
⁢
x
-
s
⁢
dx
}
⁢
κ
2
-
∫
κ
λ
-
⁢
(
λ
2
-
x
2
-
T
0
⁢
x
)
s
⁢
x
2
-
s
⁢
dx
+
⁢
∫
λ
+
1
⁢
(
x
2
-
λ
2
-
T
0
⁢
x
)
s
⁢
x
2
-
s
⁢
⁢
dx
⁢
(
2
)
0
=
∫
κ
λ
-
⁢
(
λ
2
-
x
2
-
T
0
⁢
x
)
s
⁢
x
-
s
⁢
⁢
dx
-
∫
λ
+
1
⁢
(
x
2
-
λ
2
-
T
0
⁢
x
)
s
⁢
x
-
s
⁢
⁢
dx
,
⁢
⁢
s
=
1
n
,
(
3
)
⁢
λ
-
=
1
2
⁡
[
(
T
0
2
+
4
⁢
λ
2
)
1
2
-
T
0
]
,
(
4
)
⁢
λ
+
=
1
2
⁡
[
(
T
0
2
+
4
⁢
λ
2
)
1
2
+
T
0
]
,
⁢
⁢
T
0
=
F
y
F
cb
,
σ
y
=
F
y
2
⁢
π
⁢
⁢
R
i
⁡
(
L
0
+
L
2
)
,
F
y
=
F
Te
-
F
b
,
⁢
⁢
F
cb
=
F
T
-
F
b
,
F
b
=
π
⁢
⁢
R
i
2
⁢
L
2
⁢
ρ
⁢
⁢
g
,
L
2
=
Δ
⁢
⁢
L
1
-
κ
2
.
(
5
)