Acoustic force of the gravitational type
Acoustic force of the gravitational type
Ion Simaciu , Gheorghe Dumitrescu and Zoltan Borsos Petroleum-
Gas University of Ploieşti, Ploiești 100680, Romania High School Toma N.
Socolescu, Ploiești, Romania
Abstract:
In this paper it was shown that, under certain restrictive conditions, Bjerknes secondary forces are attractive and proportionate to the product of the virtual masses of the two bubbles. Keyword: secondary Bjerknes force, acoustic force of gravitational type, the acoustic world Introduction
The secondary Bjerknes force is a mechanism of mutual interaction between bubbles oscillating in a sound field.
Secondary Bjerknes forces manifest between pairs of bubbles in the same acoustic field. Under certain restrictive conditions, the secondary Bjerknes force can present a gravitational relation as we will further show in this paper. Our derivation leads to a form of the force independent on the sign of the expression of the force and proportional to the product of the cube of the two bubbles radii,
Batr
F R R . This means that the secondary Bjerknes force is independent of the phase difference. We will show that this form also implies the proportionality with the product of the virtual masses (the added masses) of the oscillating bubble i i m R i π ρ= = [1]. For two bubbles with different radii, secondary Bjerknes force is both attractive and repulsive. For identical bubbles these forces are attractive for any frequency up to a rank of the radius for which the forces become repulsive, i.e. when small variations of the two radii occurs [2, 3, 4]. We adopt this case in order to use it for the electromagnetic interaction where charged particles, e.g. electrons and positrons, have the same interaction cross section and therefore the same radius.
2. Setting the conditions for an attractive acoustic force that does not depend on the phase difference
In order to find the expression of the gravitational acoustic force between two bubbles, we will target to reach the following properties: - be attractive, whereas the bubbles oscillating are in phase or in phase opposition (i.e. electrostatically, not dependent on the charges sign), - be proportional to the product of the radii of bubbles (i.e. the product of the virtual masses of the two bubbles), a [email protected], b [email protected] - be dependent on the viscosity coefficient, ti i thi υ υ β β β= + , i.e. be a force emerging from the energy absorption from a plane wave which induces the oscillations of the bubbles and which is also converted into the thermal energy of the fluid and gas / vapor within the bubbles, - have a significantly smaller value than that of the electrostatic acoustic force. In order to infer the expression of gravitational force, we start from the expression of the Bjerknes secondary force [5] ( ) ( )( )
2, cos 1 2 cos , coscos 42 cos cos 2 cos , cos ,4 B R R a a a aF r a a a a O arR R a aa a a a a a O ar πρωϕ ϕ ϕ ϕϕπρω ϕ ϕ ϕ ϕ += − − + + + = + − − + + + (1) where ( ) ( )
2, arctan , 1, 2.4 ii i ii i i
Aa iR βωϕ ω ωρ ω ω β ω= = =− − + (2) When bubbles oscillate in phase, ϕ ϕ ϕ= − = , and the expression of force (1) becomes ( ) ( )
2, 0 1 2 .4 B R R a aF r a a a a a a O ar πρωϕ + = = − − + + + (3) When bubbles oscillate in phase opposition, ϕ π= , and the expression of force (1) becomes ( ) ( )
2, 1 2 .4 B R R a aF r a a a a a a O ar πρωϕ π + = = − − − − + + (4) One can see from equations (3) and (4) that the second and the fourth terms of equation (1) are independent of the sign of the phase function cos ϕ . Then we adopt: ( ) ( ) ( ) Batr
R R R RF r a a a a a a a ar r πρω πρωϕ ϕ ϕ= − − = − − (5) Continuing by subtracting in (5) the expression of the dimensional amplitude from (2), yields ( ) ( ) ( ) ( )
Batr
AF r r R R πω ϕρ ω ω β ω ω ω β ω= − − − + − + (6) This force is attractive, regardless of the phase difference. This is a consequence of both scattering and absorption of plan wave energy by the bubble. This force is much smaller than the corresponding electroacoustic force (see (19) of [7]). The gravitational acoustic force is proportional to the product of the masses and the absorption coefficients of both bubbles. We will adopt a damping coefficient which has three components: ii aci i thi thii
Ru R υ ω µβ β β β βρ= + + = + + (7) an acoustic one ( ( ) ( ) i R u ω ), one of the liquid viscosity ( ( ) i i R υ β µ ρ= ) and thermal one associated to the fluid viscosity of the bubbles gas or vapor ( thi β ).
3. Thermal damping coefficient thi β According to the paper [6], the thermal damping constant ( thi thi β ωδ= ) is ( ) ( ) ( )( ) ( )( ) i i i i i ithi i i i i i i X X X X XX X X X X X γ ωβ γ ω− + − − = − + − − (8) with ( ) i i X R ω χ= , γ the ratio of specific heats, ( ) ( ) g P g g K c χ ρ µ ργ= ≅ the thermal diffusivity of gas in bubbles and K the thermal conductivity. For ( ) i i X R ω χ= << , the parentheses in (8) can be approached by sinh sin 2 1 , sinh sin 1 , cosh cos 1 ,120 3 840 360 i i i ii i i i i i i i
X X X XX X X X X X X X + = + − = + − = + (9) since sinh .., cosh 1 ..,2 6 120 5040 2 2 24 720sin .., cos 1 ...6 120 5040 2 24 720 i i i i
X X X Xi i i i i ii i ii i i i i ii i i
X X X X X Xe e e eX X XX X X X X XX X X − − − += ≅ + + + + = ≅ + + + +≅ − + − + ≅ − + − + (10) Subtracting (9) in (8), it follows ( ) ( ) i i i ithi
X R γ ω γ ωβ γω γχ− −≅ = (11) From the same paper [6], it is shown that the natural angular frequency (the resonance frequency), with eff p p γ= , is ( )( )( ) eff effi ii i i i i i p pX XR X X X R γω ρ ργ − − − = + ≅ − (12) Considering the pressure caused by the surface tension [5], the natural angular frequency becomes ( )2 2 13 . effi i i i i ppR R R R σσ σω γ ρ ρ ρ γ ργ ≅ + − = (13) Subtracting (13) into (11), one can approach ( ) ( ) effi ithi pR γ σγ ωβ γχ γ ρχ−−≅ = (14a) If ( ) i p R γ γ σ>> − , it results eff p p γ≅ and the thermal damping coefficient is independent of the bubble radius ( ) ( ) i ithi R p γ ω γβ γχ γρχ− −≅ = (14b) If ( ) i R p γ σ γ− >> , it results ( ) eff i p R γ σ≅ − and the thermal damping coefficient is dependent of the bubble radius ( )( ) thi i R γ γ σβ γ ρχ− −≅ (14c) For ( ) i i X R ω χ= >> , the parentheses in (8) can be approached by sinh sin , sinh sin , cosh cos ,2 2 2 i i i
X X Xi i i i i i e e eX X X X X X + ≅ − ≅ − ≅ (15) since sinh , cosh , 1 sin 1, 1 cos 1.2 2 2 2 i i i i i i
X X X X X Xi i i i e e e e e eX X X X − − − += ≅ = ≅ − ≤ ≤ − ≤ ≤ (16) Hence, in this case, subtracting (15) into (8), the thermal damping coefficient becomes ( ) ( ) ( ) effi ithi i i i pX R R γ χγ ω γ ω χβ ω ω ω γω ωρ−− −≅ = = (17a) If ( ) i p R γ γ σ>> − , it results eff p p γ≅ and the thermal damping coefficient is dependent of the bubble radius ( ) ( ) effthi i i p pR R γ χ γ χβ γω ωρ ω ωρ− −= = (17b) If ( ) i R p γ σ γ− >> , it results ( ) eff i p R γ σ≅ − and the thermal damping coefficient ( ) ( )( ) effthi i i p R R γ χ γ γ χ σβ γω ωρ γω ωρ− − −= = (17c) is dependent of the bubble radius. Requirements for the force of the gravitational attractive type 4.1. Attractive force at resonance
When we address the case of identical bubbles, the force (6) can be rewritten using (7) as ( ) ( )
Batr th
AF r Rr R u R πωω µρ ω ω β ωρ−= − + + + (18) Depending on the order of the scale adopted for ω , i ω , aci β , i υ β and thi β , the interaction can be influenced by the scattering of the plane wave or by the absorption of the energy of the plane wave and conversion to thermal energy. At resonance [5], ω ω≅ , the relationship (18) becomes ( ) .28 2 Batrr th
AF r Rr R u R πω µρ β ωρ−= + + (19) For ac t υ β β>> , the expression of the attractive force becomes ( )
44 5 4 4 2 44 4 0 0 0 02 3 6 10 2 3 2 2 50 0 0 0 0 0 05 4 4 2 4 2 2 2 2 20 0 0 02 5 2 2 20 0 th thBatrr eff eff effth theff eff eff eff eff u R A u u Ru A u uF r r R R R r p p R pR A u u R uu ur p p R p p p R β πγ ρ γ ρ βπ µ γ µρ ω ρω ωπγ ρ γ ρ β γ ρµβγ µ γ µ −−≅ − − = − − = − − − + + +
22 2 2 2 20 025 4 4 2 4 3 2 2 2 2 2 7 4 2 4 2 25 2 2 2 8 4 4 40 0 0 02 5 2 2 2 4 4 40 04 4 4 4 4 4 5 2 2 9 4 4 30 0 0 04 2 4 20
42 2 3 2 32 3 212 2 3 2 theffth theff eff eff eff effth th theff eff eff u RpR A u u R uu ur p p R p p p Ru R u up p p R γ ρ βπγ ρ γ ρ β γ ρ µ βγ µ γ µγ ρ β γ ρµβ γ ρ µ β = − + + + + ++ + + theff u Rp γ ρ µ β (20a) If the thermal damping coefficient is in accordance with the relationship (14a), the expression (20a) of the attractive force becomes ( ) th thBatrr eff eff eff eff effth th th theff eff eff eff
R A u u R uu uF r r p p R p p p Ru R u u u Rp p p R p πγ ρ γ ρ β γ ρ µ βγ µ γ µγ ρ β γ ρµβ γ ρ µ β γ ρ µ β−≅ + + + + ++ + + ( ) ( )( ) ( ) ( ) ( ) .2 1 2 12 2 3 21 5 3 5 32 1 2 1 2 1 2 1 .3 5 5 15 3 5 eff eff eff effeff eff eff u R uR A u u ur p p R p p Ru R u u u Rp p R p γ γ µπγ ρ γ µ γ µγ χ χγ γ µ γ γ µ γ µγ χ χ χ γ χ = − −− + + + + + − − − − + + + (20b)
If the thermal damping coefficient is in accordance with the relationship (17a), ( ) ( ) ( ) th eff i eff i p R p R β γ χ γω ω ρ γ χ γρ = − = − , the expression (20a) of the attractive force becomes ( ) th thBatrr eff eff eff eff effth th th theff eff eff eff
R A u u R uu uF r r p p R p p p Ru R u u u Rp p p R p πγ ρ γ ρ β γ ρ µ βγ µ γ µγ ρ β γ ρµβ γ ρ µ β γ ρ µ β−≅ + + + + ++ + + ( ) ( )( ) ( )( )
15 2 3 3 4 4 3 1 215 4 eff eff eff effeff eff eff eff u uR A u ur p p R p R p Ru uup R p R p Rup R γ χρ γ γ ρ µ χπγ ρ γ µγ γ ρ χ γ γ ρ µχγ µ γγ γ ρ µ χ = − −− + + + + − −+ + +− ( )
35 2 3 9 4 4 3 27 2 13 4 5 20 0 eff up R γ γ ρ µχ − + (20c) The third and ninth terms in (20b), for eff p p γ≅ , are attractive forces which is proportional to the radius having exponent and hence is related to the virtual mass m R = π ρ , as the square of it ( ) ( ) ( )( ) ( )( ) ( ) Batrr u R u RR A uF R r r p pR A u ur p pA u umr p p γ γ µπ ρ γ χ γ χπ γ ρ γ µγ χ γ χγ γ µπγ χ γ χ − −− ≅ + = − − − + = − − − + (21) At resonance, the first term of the gravitational acoustic force (21) depends on the thermal diffusivity of gas/ vapor, ( ) g g χ µ ργ≅ . It follows that this force is the effect of the wave energy absorption by the gas/vapor in the bubble. The second term also depends on the viscosity coefficient, µ , of the liquid and so is the effect of the energy absorption by the liquid. For ( ) eff p R γ σ≅ − , in the expression (20b) of the attractive force, there is no term proportional to the radius that has exponent 6. In the expression (20c) of the attractive force, for eff p p γ≅ and ( ) eff p R γ σ≅ − , there is no term proportional to the radius that has exponent 6. The attractive force (21) is proportional to the radius having the exponent 6, i.e. the square of the virtual masses of the bubbles in the oscillation motion [1]. The condition ( ) X R ω χ= << must also be fulfilled so that th β , in accordance with the relationship (14b), to be independent of the radius. When ac υ β β>> , the expression of force (19) may be written as ( )
44 6 4 2 8 300 0 0 04 7 2 4 22 3 20 0 0 07 2 2 40 2 eff thBatrr effth
R pA R A R RF r r p uR Rr u π ρ πγ ρ ρ βµ γ µ µω ρ βω µ µ µ − −= ≅ − − + + (22) Regardless of the expressions (14) and (17) of the thermal damping coefficient at resonance, the force expression (22) has no terms proportional to the radius with the exponent 6. If th ac β β>> and th υ β β>> , the expression (19) of force becomes ( )
44 44 2 2 4 22 0 0 0 0 00 02 3 2 2 40 0 0 20 0 0
21 .8 228 1 2 effBatrr eff th th thth th th pA AF r r p uR RRr R u R π πγ µρ β γρ β ρ βω µρ ω β β ρ β − −= ≅ − − + + (23) Regardless of the expressions (14) and (17) of the thermal damping coefficient at resonance, the force expression (23) has no terms proportional to the radius with the exponent 6. It follows that, at resonance, the attractive force fulfills the criteria listed in the second section for the case ac t υ β β>> , according to the expression (21). ω ω<< and ω ω>> When ω ω<< , the square bracket from (18) can be approached as ( )
22 2 2 2 2 222 2 2 2 4 40 0 02 4 2 40 0 0 0 0 0 ω β ω ω ω β β ωω ω β ω ω ωω ω ω ω ω ω − + = − + ≅ − + > (24) Subtracting (24) in (18), one can express the attractive force as ( )
22 4 41 12 4 21 3 .....2
Batr eff eff effeff eff effeff
A RAF r R Rr R r p p pA R R Rr p p pA Rr p πγ ω ρπωω ω β γρ ω γ ρ ω βρ ω ω ωπγ ω ρ γρ ω γρ ωβπγ ω ρ −−≅ = ≅ − + − + − + + + ≅ − eff eff eff
R A R Rp r p p γρ ωβ πγ ω ρ γρ βω − + + = + + (25a) If eff p p γ≅ , the expression (25a) of the attractive force becomes ( ) Batr
A R R A R RF r p r p p r p πω ρ ρ ωβ πω ρ ρ βω − − ≅ + + = + + (25b) If ( ) eff p R γ σ≅ − , the expression (25a) of the attractive force becomes ( ) ( ) ( )
44 2 4 10 30 0443 4 2
Batr
A R RF r r πγ ω ρ γρ βω γ σγ σ − ≅ + + − − (25c) The first term in (25b) is an attractive force which is proportional to the radius having exponent and hence is related to the virtual mass m R = π ρ , as the square of it. The second is attractive and independent of the radius, unless the terms ( ) ( )
420 0 R p ρ β ( )
23 3 3 22 2 2 24 6 4 4 ...3 223 ac th thth th th th th
RR R Rp p p u RR Rp R R R R uRp υ ρ β β βρ β ρ ω µ βρρ ωµ µ µ µβ β β β βρ ρ ρ ρρµ + + = = + + ≅ + + + + + + = + th th th th th R R R Rp p p p up µ ρ µ ρ µρ ρωβ β β β β+ + + + + (26a) is independent of the radius. In the expression (26a), only the first term and the sixth term, with th β having the expression (17b), are independent of the radius
42 4 2 94 4 4 4 2 3 3 20 0 34 4 4 4 4 4 7 2 5 20 0 0 0 0
22 2 3 ( 1)... ...3 3 3 3 2 th R Rp p up p up ρ β ρωµ µ γ ρχβ ω −≅ + + = + + (26b) Under the above conditions, the expression of the attractive force (25b) becomes ( )
42 4 6 20 06 40 4 4 20 02 4 6 8 4 4 1 2 3 3 3 2 3 204 4 2 3 40 0 02 2 4 8 4 4 1 2 3 3 3 2 3 22 3 4 4 3 40 0 0
2, 1 48 ...3 32 2 2 3 ( 1)1 ...3 3 2 2 3 ( 1)1 ... .2 3 3
Batr
A R RF r R p r pA Rp r p upm Ar p p up πω ρ ρ βωπω ρ µω γ ρω χω µω γ ρω χπ ρ − ≅ + + = − −+ + + = − −+ + + (27) We have demonstrated that, for ω ω<< , the gravitational acoustic forces are the effect of the induction wave scattering (first term) and the absorption of the energy of the wave by liquid and gas / vapor (the second and third terms). The first term in (25c) is an attractive force which is proportional to the radius having exponent . The expression (25c) is an attractive force of gravitational type, only the terms ( ) R ρ β γ σ − ( ) ( )( ) ( )( )
23 1 23 1 3 12 2 24 63 124 4 .2 ac th thth th thth th
RR R Ru RR R R RRR u υ γ ρ β β βγρ β γ ρ ω µ βγ σ ργ σ γ σγ ρ µ µ µβ β βρ ρ ργ σ ωµ β βρ + + = = + + ≅ − − − + + + + − + + ( ) ( ) ( )( ) ( ) ( ) th thth th th R R RR R R u γ µ γ ρ β γ µ ρ βγ σ γ σ γ σγ µ ρ β γ µρ β γ ρω βγ σ γ σ γ σ = + + + − − −+ + +− − − (28a) are proportional to the radius having exponent − .The only term that fulfills this condition is the second term in the expression (28a), for the thermal damping coefficient given by the expression (17c), ( ) ( ) ( ) th R R R γ χγρ β γ ρ βγ σ ωγ σ −≅ + = + − − (28b) By replacing (28b) in the expression of force (25c), it results ( ) ( ) ( ) ( )( )( )( )
Batr
A RA RF r R Rr rAmr γ χ γ γ π χ ρπγ ω ρ ωγ σ γ σγ γ χγ πρσ − − −−≅ + = + = − − −− + − (29) We note that this attractive force is independent of angular frequency and is the effect of the absorption of wave energy by the gas / vapor from the bubbles. When ω ω>> , the square bracket from the expression of the attractive force (18) can be approached ( )
22 22 02 2 2 2 40 2 2
44 1 ω βω ω β ω ω ω ω − + = − + (30) Subtracting this new approach in (18), yields ( )
Batr eff
A AF r pr R r R R π πω β βρ ω ρ ωω ω γρ ω ω− −= = − + − + (31) Replacing the expression (13) for ω and the expression (7) for β , in (31), results ( ) Batr eff th
AF r p Rr R R u R π ω µρ ω βγρ ω ω ρ−= − + + + (32)
This force, for any approximation of the effective pressure eff p and the thermal damping coefficient th β , according to relations (14) and (17), does not have terms that meet the requirements of the second section. We have shown that in the interaction between two bubbles induced by an acoustic wave there is also a gravitational acoustic force. The gravitational acoustic force exists at resonance according to formula (21) and for the low angular frequency ( ω ω<< )according to the formulas (27) and (29).
5. Conclusions
We have shown that under certain restrictive conditions, Bjerknes secondary forces are attractive, according to formulas (21), (27) and (29). These gravitational acoustic forces depend on the virtual mass of the oscillating bubble as the square of it. We point out that the expression of gravitational force derived in this paper is not the final one. We can make the same statement for the electrostatic force. We believe that one can obtain the final expressions of these forces if one investigates the interaction of N identical bubbles which form a cluster. A previous analysis of this phenomenon, studied in the literature as a phenomenon specific to a cluster [7, 8], points out that the coupled oscillations of the N bubbles induce a mass (as a measure of mechanical inertia and gravitational mass) for each bubble, much higher than the virtual mass. The induced mass mentioned above depends on the number of bubbles and, on the size of the cluster and of the equivalent mass of the bubble. The dependencies mentioned above are equivalent to the Mach principle in the electromagnetic universe [9, 10]. In a further paper we will try to prove the above assumption. References H. N. Oguz and A. Prosperetti, The natural frequency of oscillation of gas bubbles in tubes, J. Acoust. Soc. Am. 103 (6), 3301–3308, 1998. 2.
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