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An exact non-stationary solution of simple shear flow in a Bingham fluid
Journal of Non-Newtonian Elsevier Science Publishers
107
107-113 B.V., Amsterdam
An exact non-stationary solution of simple shear flow in a Bingham fluid Ken Sekimoto Department (Received
*
of Applied Physics, Nagoya University, Nagoya 464 (Japan) August
3, 1990; in revised form October
3, 1990)
Abstract The flow of Bingham fluids can have yield surfaces that separate unyielded regions from yielded regions. The analytical solution for the Bingham flow equation that contains a moving yield surface is presented. An initially homogeneous simple shear flow in the semi-infinite bulk bounded by a planar boundary is assumed, and consideration is given to what disturbance occurs upon the reduction of the applied shear stress on the boundary after time t = 0. The solution is of the ‘similarity’ type in which all length scales grow as - t’12 and the unyielded region in particular grows from the boundary such that its thickness is proportional to t1/2. Keywords: surfaces
analytical
solution;
Bingham
fluids;
non-stationary
solution;
shear
flow;
yield
1. Introduction The flow of Bingham fluids [l] can have an interface between yielded regions and unyielded regions. Several analytical solutions to the flow equations of the Bingham fluids are known [2]. To the author’s knowledge there are, however, no such solutions that describe the moving interface between the yielded region and the nonyielded region (hereafter called the yield surface). This paper presents one such solution for the geometry of simple shear flow. I assume an initially homogeneous simple shear flow in the semi-infinite bulk of a Bingham fluid bounded by a planar boundary,
* Temporary
address:
0377-0257/91/$03.50
ESPCI,
10 rue Vauquelin,
75231 Paris (France).
0 1991 - Elsevier Science Publishers
B.V.
108 and consider what disturbance occurs upon the reduction of the applied shear stress on the boundary at time t = 0. The solution is of the similarity type in which all length scales grow as - t’/2 especially the thickness of the unyielded region that appears near the boundary. The next section defines the problem and gives the solution to it. Section 3 illustrates results in two cases of strong and weak initial shear rate. I also discuss the validity of neglecting the elasticity within the unyielded region. Finally, the present result is compared with the known similarity solution [3] for another fluid model. 2. Modeling and the solution of the problem We consider a simple shear flow with the velocity field v = (u( y, t),O,O) in the semi-infinite region y > 0. The balance of momentum along the y-axis is au “z=ay?
a?(2.1)
where p is the mass density of the fluid and r = T( y, t) is the pertinent equation of the component of the shear stress tensor. The constitutive Bingham fluid is written in this geometry as
(2.2) where q is the Newtonian viscosity (constant), stress, and sgn( a) is the generalised sign function. y>Ois u(V,O> = KY,
K>O.
?Y> 0 is the yield shear The initial condition for
(2.3)
The shear stress at the boundary ~(0, t) that has driven this flow for t -c 0 is q~ + 7v (> TV,). (We note that if v is the solution of the above equations, the velocity field when shifted by a constant v0 (v + u - vO) is also a solution because of the Galilean invariance of the system.) After t = 0 we reduce the applied stress to a constant value below the yield stress r(OJ)
= 70,
t > 0,
(24
where 17. I -C TV. (In order to apply a constant stress to the system undergoing an unsteady process we must take into account the inertia of the equipment [4].) The velocity field as y tends to infinity will not be affected by this disturbance and we assume the boundary condition I +‘,t>
- KY I + 0,
We define the position
y+
cc).
of the yield surface
(2.5) y,.(t) such that the fluid has not
109
require
17 1 < TV) for 0
&(t)
- O,t) = ++(t>
Further
we expect that the stress at the yield surface
7( y,(t)
- O,t) = 7( y,(t)
+ O,t).
(2.6) is
+ OJ) = TV’
(2.7)
which implies from (2.2) that au/ay is continuous. When the governing equations are the heat equations, as is the case for the present model, it has been shown [S] that the only possible similarity solution is of the following form: y,(t)
= e( “t)1’2,
U(YJ>=
(2.8)
“Yfi +> q.
Y
>Yc(t),
(2.9)
where 19 is a dimensionless parameter to be determined. Based on dimensional analysis we may give the alternative discussion for assuming the similarity solution to be of the same form as (2.8, 2.9): The quantities Y/T,, and v = q/p have the dimensions of time, velocity and diffusion ( QGV2 constant respectively. If we distinguish between the dimensions of the lengths along the flow direction (parallel to x) and in the transverse direction (parallel to y), the first two quantities include the dimensions of the former length. Since we are considering the spatial variation only in the y direction it is natural to search for a diffusion-like flow process for the simple shear geometry and to assume the form given by (2.8, 2.9). Since the velocity u is uniform within the nonyielded region the acceleration dv/Clt is also uniform there. Thus from eqn. (2.1) the spatial gradient of the stress, a~/ay, is uniform in the nonyielded region 0 < y -C y,( t). The stress 7 is therefore interpolated using the boundary values, 70 at y = 0 and 7Y at y = y,( t). The gradient of r in this region is (rY - T,)/y,( t). Then (2.1) becomes
au f%
7y - 70
= y,(t) ’
Integrating obtain s*NV>
0
(2.10) over the time interval
[O,t] and using (2.6) and (2.9), we (2.11)
= 2{,
where we have defined { = ( 7Yfrom (2.2) (2.7) and (2.9) f( e,e)
(2.10)
+
ey(e,e)
= 0
T~)/~K.
For the yielded
region
we obtain (2.12)
Y *, lb) I
I I I
I I 1
I
51: : : :_
I I I ;
I I
-
I I I 0
0
,
5
V,T
Fig. 1. The scaled velocity V () and the scaled shear stress T (- - - -) defined in the text are shown as functions of the scaled distance Y from the boundary plane on which the constant shear stress is applied. The values of the parameters 0 = 0.2 ([ = 0.126) in Fig. l(a) and 6’= 5.0 ({ = 13.38) in Fig. l(b). The thick dots correspond to the positions on the yield surface.
and from (2.1) and (2.2) Sf”(SJI)
+
(
;
+ 2 f’(.s$) 1
= 0
(2.13)
111 where f’(s,e) = af(s,B)/a condition (2.5) is rewritten f(.G)
&e)
(2.12)
= 1+
= a2f(s,0)/as2.
The
i
5
(2.13) and (2.14) can be solved to give
1
exp( - s2/4)
- 6
erfc( s/2) (2.15)
J;; erfc( 8/2)
= -&r”
error function
defined
by
exp( -t2)dt
(2.16)
x From (2.15) and (2.11) we find that 13is related l
=
boundary (2.14)
where erfc( x) is the complementary erfc(x)
f”(s,e)
s+oo
+ 1
Equations
s and as
to 1 as follows:
0 exd-e2/4)
(2.17)
J;; erfc( 8/2)
Summarize our solution. Defining the scaled quantities as Y = y/( ~t)‘/~, V = u/[ K( N)“~] and T = (7 - T~)/( 7r- Q), the flow velocity and the shear stress are expressed as follows: ej-(e,e) ‘=
i yf(y,e)
0 G y<
8
(2.18)
~28 o<
-1+;
y
T=
(2.19) i
+
[ me)
+ v-‘(w)]
y2 8
The value of 8, which is determined implicitly by (2.17), is the monotonic function of 5 with the following asymptotic relations,
(2.20) It can be shown surface.
that
in our solution
a~/ay
is continuous
at the yield
3. Discussion Figure 1 shows the scaled velocity V and the scaled stress T as functions of the scaled coordinate Y for 8 = 0.2 (Fig. l(a)) and 8 = 5.0 (Fig. l(b)). The former case corresponds to a large initial shear rate or to a small yield stress,
112 and from Fig. l(a) we see there is little effect of the existence of the yield stress. The thickness of the disturbed region is = (vt)‘/*. On the other hand, in the opposite case (5 > 1) we see from Fig. l(b) or from (2.16) that there is a narrow (scaled) cross-over region with the scaled width - (2/5)‘/* that joins the yield surface and the outer flow region that memorizes the initial shear rate due to the inertia effect. From (2.11), (2.18) and (2.20) the velocity of the boundary plane at t( > 0) is [u(O,t) - u(O,O)] = K(Vt) I/* for {+x 1 and [u(O,t) - u(O,O)] = [(rY i/* for 5 % 1. (Here we have explicitly written the initial velocity of %)Kt/P] the boundary plane considering the Galilean invariance of the system.) Note that in the latter case the velocity change is independent of the Newtonian viscosity. The solution we have obtained will also describe qualitatively the case of the finite thickness of the flow region as long as this thickness is much larger than the typical thickness of the disturbed region, min{(vt)“2, u,(t)}. Another limitation on the applicability of the present solution to reality, setting aside the limitation due to the approximate nature of the Bingham model, stems from the neglect of the elasticity in the unyielded region. Similar to the familiar argument on the validity of the incompressibility assumption for ordinary fluids [6], we expect that the assumption of the rigid unyielded region against shear stresses is valid only if the time for the mechanical equilibration within that region ( = y,( t)/cl) is much shorter than the typical time of the growth of the region (=v,( t)[dy,.(t)/&]-‘), where c, is the transverse sound velocity in the unyielded region. Thus from (2.8) we see that our solution is physically meaningful only in the time domain (Y/C:) +Z t. This condition could be rephrased in a more general way: in the phenomena with a given frequency w the elastic effects in the unyielded region cannot be neglected if the viscous skin depth S, = (v/w)‘/* exceeds the wavelength of transverse sound in the unyielded region, X = 2 lTc,/w. After completion of the main part of this work we became aware of work by Phan-Thien [3] and Anshus and Astarita [7]. Phan-Thien [3] considered the flow commencement induced by the application of constant stress (> TV) on a boundary plane and calculated the simple shear flow in the semi-infinite bulk. He proposed the following constitutive equation,
where p = au/ay and (Y is a dimensionless (Y= 1, the shear rate i, has a discontinuity I = TV. This
parameter. In his model, unless at the yield surface defined by
113 limit. The bi-viscosity is replaced by
model is recovered
if the above constitutive
equation
Here the limit of (Y+ cc corresponds to the Bingham fluid (see (2.2)). Also in the bi-viscosity model we can construct the similarity solution for the flow commencement as was done in [3]. There is, however, an important difference between this similarity solution and that obtained in [3]: we can demonstrate [9] that, in the bi-viscosity model (3.2), the position of the yield surface (or, more precisely, the surface at which ) T ) = TV) tends to infinity when 0 = (In (~)r/* in the limit where (Y tends to infinity, whereas this does not occur in the model of [3], which used eqn. (3.1). Anshus and Astarita [7] have already shown that the similarity solution describing the flow commencement of the Bingham fluids does not have the yield surface. The fact that the yield surface approaches infinity in the limit where CI tends to infinity implies the instantaneous propagation of the yield surface from y = 0 to y = CC for the Bingham fluids. According to the above mentioned validity condition (V/C:) K t for the Bingham fluids, the actual propagation of the yield surface must be controlled by the elastic effect. On the other hand the solution for the Bingham fluids described in the previous section really possesses a moving yield surface. Acknowledgements The author acknowledges N. Phan-Thien for informing 5 and for valuable comments on the manuscript.
the author
of Ref.
References E.C. Bit&am, Fluidity and Plasticity, McGraw-Hill, New York, 1922. R.D. Bird, R.C. Arnstrong and 0. Hassager, Dynamics of Polymeric Liquids, Vol. 1, Wiley, New York, 1977. N. Phan-Thien, J. Appl. Mech., 50 (1983) 229. I.M. Krieger, J. Rheol., 34 (1990) 471. G.W. Bluman and J.D. Cole, Similarity Methods for Differential Equations, SpringerVerlag, 1974. L.D. Landau and E.M. Lifshitz, Fluid Mechanics, Pergamon, New York, 1959. B.E. Anshus and G. Astarita, AIChE J., 20 (1974) 832. E.J. O’Donovan and R.I. Tanner, J. Non-Newtonian Fluid Mech., 15 (1984) 75. K. Sekimoto, unpublished work.
107
107-113 B.V., Amsterdam
An exact non-stationary solution of simple shear flow in a Bingham fluid Ken Sekimoto Department (Received
*
of Applied Physics, Nagoya University, Nagoya 464 (Japan) August
3, 1990; in revised form October
3, 1990)
Abstract The flow of Bingham fluids can have yield surfaces that separate unyielded regions from yielded regions. The analytical solution for the Bingham flow equation that contains a moving yield surface is presented. An initially homogeneous simple shear flow in the semi-infinite bulk bounded by a planar boundary is assumed, and consideration is given to what disturbance occurs upon the reduction of the applied shear stress on the boundary after time t = 0. The solution is of the ‘similarity’ type in which all length scales grow as - t’12 and the unyielded region in particular grows from the boundary such that its thickness is proportional to t1/2. Keywords: surfaces
analytical
solution;
Bingham
fluids;
non-stationary
solution;
shear
flow;
yield
1. Introduction The flow of Bingham fluids [l] can have an interface between yielded regions and unyielded regions. Several analytical solutions to the flow equations of the Bingham fluids are known [2]. To the author’s knowledge there are, however, no such solutions that describe the moving interface between the yielded region and the nonyielded region (hereafter called the yield surface). This paper presents one such solution for the geometry of simple shear flow. I assume an initially homogeneous simple shear flow in the semi-infinite bulk of a Bingham fluid bounded by a planar boundary,
* Temporary
address:
0377-0257/91/$03.50
ESPCI,
10 rue Vauquelin,
75231 Paris (France).
0 1991 - Elsevier Science Publishers
B.V.
108 and consider what disturbance occurs upon the reduction of the applied shear stress on the boundary at time t = 0. The solution is of the similarity type in which all length scales grow as - t’/2 especially the thickness of the unyielded region that appears near the boundary. The next section defines the problem and gives the solution to it. Section 3 illustrates results in two cases of strong and weak initial shear rate. I also discuss the validity of neglecting the elasticity within the unyielded region. Finally, the present result is compared with the known similarity solution [3] for another fluid model. 2. Modeling and the solution of the problem We consider a simple shear flow with the velocity field v = (u( y, t),O,O) in the semi-infinite region y > 0. The balance of momentum along the y-axis is au “z=ay?
a?(2.1)
where p is the mass density of the fluid and r = T( y, t) is the pertinent equation of the component of the shear stress tensor. The constitutive Bingham fluid is written in this geometry as
(2.2) where q is the Newtonian viscosity (constant), stress, and sgn( a) is the generalised sign function. y>Ois u(V,O> = KY,
K>O.
?Y> 0 is the yield shear The initial condition for
(2.3)
The shear stress at the boundary ~(0, t) that has driven this flow for t -c 0 is q~ + 7v (> TV,). (We note that if v is the solution of the above equations, the velocity field when shifted by a constant v0 (v + u - vO) is also a solution because of the Galilean invariance of the system.) After t = 0 we reduce the applied stress to a constant value below the yield stress r(OJ)
= 70,
t > 0,
(24
where 17. I -C TV. (In order to apply a constant stress to the system undergoing an unsteady process we must take into account the inertia of the equipment [4].) The velocity field as y tends to infinity will not be affected by this disturbance and we assume the boundary condition I +‘,t>
- KY I + 0,
We define the position
y+
cc).
of the yield surface
(2.5) y,.(t) such that the fluid has not
109
require
17 1 < TV) for 0
&(t)
- O,t) = ++(t>
Further
we expect that the stress at the yield surface
7( y,(t)
- O,t) = 7( y,(t)
+ O,t).
(2.6) is
+ OJ) = TV’
(2.7)
which implies from (2.2) that au/ay is continuous. When the governing equations are the heat equations, as is the case for the present model, it has been shown [S] that the only possible similarity solution is of the following form: y,(t)
= e( “t)1’2,
U(YJ>=
(2.8)
“Yfi +> q.
Y
>Yc(t),
(2.9)
where 19 is a dimensionless parameter to be determined. Based on dimensional analysis we may give the alternative discussion for assuming the similarity solution to be of the same form as (2.8, 2.9): The quantities Y/T,, and v = q/p have the dimensions of time, velocity and diffusion ( QGV2 constant respectively. If we distinguish between the dimensions of the lengths along the flow direction (parallel to x) and in the transverse direction (parallel to y), the first two quantities include the dimensions of the former length. Since we are considering the spatial variation only in the y direction it is natural to search for a diffusion-like flow process for the simple shear geometry and to assume the form given by (2.8, 2.9). Since the velocity u is uniform within the nonyielded region the acceleration dv/Clt is also uniform there. Thus from eqn. (2.1) the spatial gradient of the stress, a~/ay, is uniform in the nonyielded region 0 < y -C y,( t). The stress 7 is therefore interpolated using the boundary values, 70 at y = 0 and 7Y at y = y,( t). The gradient of r in this region is (rY - T,)/y,( t). Then (2.1) becomes
au f%
7y - 70
= y,(t) ’
Integrating obtain s*NV>
0
(2.10) over the time interval
[O,t] and using (2.6) and (2.9), we (2.11)
= 2{,
where we have defined { = ( 7Yfrom (2.2) (2.7) and (2.9) f( e,e)
(2.10)
+
ey(e,e)
= 0
T~)/~K.
For the yielded
region
we obtain (2.12)
Y *, lb) I
I I I
I I 1
I
51: : : :_
I I I ;
I I
-
I I I 0
0
,
5
V,T
Fig. 1. The scaled velocity V () and the scaled shear stress T (- - - -) defined in the text are shown as functions of the scaled distance Y from the boundary plane on which the constant shear stress is applied. The values of the parameters 0 = 0.2 ([ = 0.126) in Fig. l(a) and 6’= 5.0 ({ = 13.38) in Fig. l(b). The thick dots correspond to the positions on the yield surface.
and from (2.1) and (2.2) Sf”(SJI)
+
(
;
+ 2 f’(.s$) 1
= 0
(2.13)
111 where f’(s,e) = af(s,B)/a condition (2.5) is rewritten f(.G)
&e)
(2.12)
= 1+
= a2f(s,0)/as2.
The
i
5
(2.13) and (2.14) can be solved to give
1
exp( - s2/4)
- 6
erfc( s/2) (2.15)
J;; erfc( 8/2)
= -&r”
error function
defined
by
exp( -t2)dt
(2.16)
x From (2.15) and (2.11) we find that 13is related l
=
boundary (2.14)
where erfc( x) is the complementary erfc(x)
f”(s,e)
s+oo
+ 1
Equations
s and as
to 1 as follows:
0 exd-e2/4)
(2.17)
J;; erfc( 8/2)
Summarize our solution. Defining the scaled quantities as Y = y/( ~t)‘/~, V = u/[ K( N)“~] and T = (7 - T~)/( 7r- Q), the flow velocity and the shear stress are expressed as follows: ej-(e,e) ‘=
i yf(y,e)
0 G y<
8
(2.18)
~28 o<
-1+;
y
T=
(2.19) i
+
[ me)
+ v-‘(w)]
y2 8
The value of 8, which is determined implicitly by (2.17), is the monotonic function of 5 with the following asymptotic relations,
(2.20) It can be shown surface.
that
in our solution
a~/ay
is continuous
at the yield
3. Discussion Figure 1 shows the scaled velocity V and the scaled stress T as functions of the scaled coordinate Y for 8 = 0.2 (Fig. l(a)) and 8 = 5.0 (Fig. l(b)). The former case corresponds to a large initial shear rate or to a small yield stress,
112 and from Fig. l(a) we see there is little effect of the existence of the yield stress. The thickness of the disturbed region is = (vt)‘/*. On the other hand, in the opposite case (5 > 1) we see from Fig. l(b) or from (2.16) that there is a narrow (scaled) cross-over region with the scaled width - (2/5)‘/* that joins the yield surface and the outer flow region that memorizes the initial shear rate due to the inertia effect. From (2.11), (2.18) and (2.20) the velocity of the boundary plane at t( > 0) is [u(O,t) - u(O,O)] = K(Vt) I/* for {+x 1 and [u(O,t) - u(O,O)] = [(rY i/* for 5 % 1. (Here we have explicitly written the initial velocity of %)Kt/P] the boundary plane considering the Galilean invariance of the system.) Note that in the latter case the velocity change is independent of the Newtonian viscosity. The solution we have obtained will also describe qualitatively the case of the finite thickness of the flow region as long as this thickness is much larger than the typical thickness of the disturbed region, min{(vt)“2, u,(t)}. Another limitation on the applicability of the present solution to reality, setting aside the limitation due to the approximate nature of the Bingham model, stems from the neglect of the elasticity in the unyielded region. Similar to the familiar argument on the validity of the incompressibility assumption for ordinary fluids [6], we expect that the assumption of the rigid unyielded region against shear stresses is valid only if the time for the mechanical equilibration within that region ( = y,( t)/cl) is much shorter than the typical time of the growth of the region (=v,( t)[dy,.(t)/&]-‘), where c, is the transverse sound velocity in the unyielded region. Thus from (2.8) we see that our solution is physically meaningful only in the time domain (Y/C:) +Z t. This condition could be rephrased in a more general way: in the phenomena with a given frequency w the elastic effects in the unyielded region cannot be neglected if the viscous skin depth S, = (v/w)‘/* exceeds the wavelength of transverse sound in the unyielded region, X = 2 lTc,/w. After completion of the main part of this work we became aware of work by Phan-Thien [3] and Anshus and Astarita [7]. Phan-Thien [3] considered the flow commencement induced by the application of constant stress (> TV) on a boundary plane and calculated the simple shear flow in the semi-infinite bulk. He proposed the following constitutive equation,
where p = au/ay and (Y is a dimensionless (Y= 1, the shear rate i, has a discontinuity I = TV. This
parameter. In his model, unless at the yield surface defined by
113 limit. The bi-viscosity is replaced by
model is recovered
if the above constitutive
equation
Here the limit of (Y+ cc corresponds to the Bingham fluid (see (2.2)). Also in the bi-viscosity model we can construct the similarity solution for the flow commencement as was done in [3]. There is, however, an important difference between this similarity solution and that obtained in [3]: we can demonstrate [9] that, in the bi-viscosity model (3.2), the position of the yield surface (or, more precisely, the surface at which ) T ) = TV) tends to infinity when 0 = (In (~)r/* in the limit where (Y tends to infinity, whereas this does not occur in the model of [3], which used eqn. (3.1). Anshus and Astarita [7] have already shown that the similarity solution describing the flow commencement of the Bingham fluids does not have the yield surface. The fact that the yield surface approaches infinity in the limit where CI tends to infinity implies the instantaneous propagation of the yield surface from y = 0 to y = CC for the Bingham fluids. According to the above mentioned validity condition (V/C:) K t for the Bingham fluids, the actual propagation of the yield surface must be controlled by the elastic effect. On the other hand the solution for the Bingham fluids described in the previous section really possesses a moving yield surface. Acknowledgements The author acknowledges N. Phan-Thien for informing 5 and for valuable comments on the manuscript.
the author
of Ref.
References E.C. Bit&am, Fluidity and Plasticity, McGraw-Hill, New York, 1922. R.D. Bird, R.C. Arnstrong and 0. Hassager, Dynamics of Polymeric Liquids, Vol. 1, Wiley, New York, 1977. N. Phan-Thien, J. Appl. Mech., 50 (1983) 229. I.M. Krieger, J. Rheol., 34 (1990) 471. G.W. Bluman and J.D. Cole, Similarity Methods for Differential Equations, SpringerVerlag, 1974. L.D. Landau and E.M. Lifshitz, Fluid Mechanics, Pergamon, New York, 1959. B.E. Anshus and G. Astarita, AIChE J., 20 (1974) 832. E.J. O’Donovan and R.I. Tanner, J. Non-Newtonian Fluid Mech., 15 (1984) 75. K. Sekimoto, unpublished work.