A Universal Moel for Bingham Fluis with Two Characteristic Yiel Stresses NGDomostroeva 1 an NNTrunov DIMeneleyev Institute for Metrology Russia, StPetersburg 195 Moskovsky pr 19 February, 4, 9 Abstract: We introuce a new moel for Bingham fluis with a complicate epenence of the eformation velocity on the shear stress Specific features of the flow rate for such flui are stuie This moel embraces many special an limiting cases, for many of them analytic calculations become possible 1 Approximate universal moel As it turne out recently, many important fluis incluing oil reveal a more complicate rheological behavior than it is usually postulate for a Bingham flui [1] For a uni-irectional shear flow with the velocity vx ( the following epenence of the eformation velocity on the shear stress takes place, see figure 1 = v y (1 v = ( y 1 Electronic aress: NGDomostroeva@vniimru Electronic aress: trunov@vniimru
Namely, =, if < ( an asymptotically µ = (4 Fig1 for large values ( exceeing roughly (1- Thus we have two characteristic values of the shear stresses: the yiel stress an a parameter > relate to the asymptotic viscosity µ In the intermeiate interval between ( an ( there is a flow with a very high effective viscosity If <<, we have with a very large µ >> µ µ = (5 For the quantative escription of such flui we have to fin a simple enough universal an physically groune moel function ( This function epens on four parameters (, µ,, µ or (, µ, α, β µ α = < 1, β = < 1 (6 µ
with imensionless α, β As it can be seen from fig1, this function must obey two conitions >, > (7 an have asymptotics ( an (4 Besies, for a usual Newtonian flui with = = such a function must be o: ( = ( (8 A convenient function satisfying all the above conitions is ( 1 α( 1 ( = ( tanh µ for all with α (6 In particular, for small replacing tanh(z by its (9 argument z we obtain: ( α ( = = (1 µ µ Flow in a channel Now we stuy the flow with ( (9 in a channel with infinite with an the height h, so that Obviously h y h (11 v( h = v( h = (1 In the upper half = gy, g = p x (1
with g being the pressure graient Substituting (1 into (9 an integrating we obtain ( ( z cosh( 1 α ( ln 1 α cosh( 1 α( h y y y y y z v = + (14 h y z if y y an Here we have introuce v( y v( y g =, if y y (15 y =, y =, z = y y (16 g The total flow rate ( for both halves per unit with is Hereafter we use the equality h Q= v y y+ y v y ( ( (17 y ( A ( B cosh A 1+ exp ln = A B + ln cosh B 1+ exp We suppose that A, B in (18 excee - In this case we may assume that for the logarithmic terms the upper limit of integration in (17 is equal to infinity an use (18 sk π sln( 1+ e = (19 4k The final expression looks as follows : with 1 π b Q= Q a+ b + a + a b+ 41 1 ( ( α (
Q hg = ; a = ; b = (1 µ hg hg The well known stanar expression for Q we obtain if b = ie = Then for all = the viscosity is equal to µ so that µ oes not have influence on Q so that Introucing a new parameter: = + ( hg c a b a= βc, β 1, ( we obtain another convenient formula: f Q c c f Q ( αβ, β β ( 1 β ( α β = 1 +,, (4 π = + + 41 ( 1 β ( α (5 The abovementione stanar case correspons to β = 1, when f = 1 an Q = if c= a 1 Remember that in many typical cases so that the term with a may be omitte The following features of Eq (4 are remarkable: a is much smaller than a 1 Both the first term an the secon, linear in c term o not epen on β = ; there is no term of orer c at any β ; only the smallest an often neglecte term proportional to epens on α The same calculations were fulfille with several another moel functions satisfying to the previous conitions In all these cases the above properties 1- remain vali an only the specific form of f epens on ( c
The usual way for etermining the yiel stress takes into account only the linear in c asymptotics of Q (5 It is eviently that in such a way we can only etermine an not Thus namely an not remains a genuine parameter etermining the whole rheology As to, it is in the presence of a umb parameter for all or the most of situations Thus we cannot agree with the opinion [1] that is only a formal parameter Possible applications Thus we have a rather simple bi-viscosity moel (9 incluing limiting cases: µ =, if α = an so on Moreover, a very weak epenence of the whole rheology on in the presence of allows us in most cases simply to put = Then we obtain a fully analytic approximate form (9 which is more physically an mathematically well groune than the well known power law approximation: proportional to m [1,4] for the Bingham flui Our form of may be use for analytical calculation of the flow in complex geometry [] eg in a squeeze test [] 1 VVTetelmin, VAYasev The Oil Reology, Moscow, 9 [in Russian], p4 GGLipcomb, MMDenn JNon-Newton Flui Mech, 14 (1984 PP 7-46 NRoussel, ChLanos, ZToutou JNon-Newton Flui Mech,15 (6 1-7 4 GAstarita, GMarrucci Principles of Non-Newtonian Flui Mechanics McGraw-Hill, 1974, ch