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It is shown that when the compressibility of a fluid is taken into account, the nonlinear term disappears in the Euler equation. The validity of this approach is proved by the example of capillary waves.

In the monography [

In this work, we apply Euler’s refined equation towards capillary waves and show that despite the use of the boundary condition at the bottom of the liquid, the depth of the reservoir disappears from the dispersion equation, and thus the phase velocity of the capillary wave depends only on its length. In addition, it will be shown that there is no stability condition for a capillary wave, i.e. it is always stable in the first equation of system (72) in [

In existing gas and hydrodynamics theory, Euler equation is applied in the form:

ρ d V d t = ρ [ ∂ V ∂ t + ( V ∇ ) V ] = − ∇ P + ρ g (1)

where: ρ -density, V ―speed fluid particles, P ―pressure, g ―gravitational acceleration. Here supposed to that liquid is incompressible and consequently ρ = c o n s t . In several works (see e.g. [

∂ ρ ∂ t + ( V ∇ ) ρ = − ρ ∇ V − V ∇ P C p 2 (2)

where C p is isobaric sound speed in the liquid which can be considered as infinitely large, and we have:

d ρ d t = ∂ ρ ∂ t + ( V ∇ ) ρ = − ρ ∇ V (3)

From (3) this it follows that liquid is compressible medium ( ∇ V ≠ 0 ) end d ρ / d t ≠ 0 . Consequently, Euler equation should be written in the form of:

d ( ρ V ) d t = V d ρ d t + ρ d V d t = − ∇ P + ρ g (4)

Then if we substitute d ρ / d t from (3) in (4), we get:

− V ρ ∇ V + ρ d V d t = − ∇ P + ρ g ⇒ ρ [ d V d t − ( V ∇ ) V ] = − ∇ P + ρ g ⇒ ρ ∂ V ∂ t = − ∇ P + ρ g (5)

So, we have a system of two equations:

{ ρ ∂ V ∂ t = − ∇ P + ρ g ∂ ρ ∂ t + ( V ∇ ) ρ = − ρ ∇ V (6)

We see that the nonlinear term from the Euler equation has disappeared. It remains only in the mass conservation equation.

Let’s presenting all the variables from the system (6) in the form of the sum of their stationary and perturbed values f = f 0 + f ′ and supposing that f ′ / f 0 < 1 , after linearization it will take the following form:

{ ρ 0 ∂ V ∂ t = − ∇ P + g C 2 P 1 C 2 ( ∂ P ∂ t + V 0 ∇ P ) = − ρ 0 ∇ V (7)

where we used: state equation of medium ρ ′ = P ′ / C 2 (C―adiabatic speed of sound in the liquid), liquid equilibrium equation ∇ P 0 = ρ 0 g and perturbed values don’t have stokes.

If V ¯ 0 there is a stationary velocity of fluid flow, then the linearized Equation (1) will be

ρ 0 [ ∂ V ∂ t + ( V 0 ∇ ) V ] = − ∇ P + g C 2 P

As we see, this equation, which determines the acceleration of a liquid particle, contains a stationary velocity of motion, on which the acceleration should not depend. This fact irrefutably proves the validity of formula (4).

Appling to the first equation of system (7) operator ∇ and to the second operator ∂ / ∂ t , they easily reduced to differential equation for perturbed pressure

Δ P − 1 C 2 ( g → + V → 0 ∂ ∂ t ) ∇ P − 1 C 2 ∂ 2 P ∂ t 2 = 0 (8)

Presenting now the perturbed pressure in the form of periodic function

P ( x , z , t ) = P a ( z ) exp [ i ( k x − ω t ) ] (9)

and supposing that V 0 = e x V 0 and g = − e z g , we will get ordinary differential equation of second order for the amplitude of pressure disturbance in the following form:

d 2 P a ( z ) d z 2 + g C 2 d P a ( z ) d z + [ ω C 2 ( ω − k V 0 ) − k 2 ] P a ( z ) = 0 (10)

The Equation (10) describes pressure disturbance on both sides of surface of tangential discontinuity z = 0 . We solve this equation for air ( z > 0 ) in the form:

P a 1 ( z ) = A exp ( γ z ) , (11)

where upon, taking into account the attenuation of the disturbance when z → ∞ , for γ we will get:

γ = − k θ 1 { 1 + 1 + θ 1 2 [ 1 − U p ( U p − V 0 ) C 1 2 ] } < 0 (12)

where C 1 is sound speed in the air on the sea level, U p = ω / k is phase speed of surface wave and θ 1 is dimensionless value and it is equal to

θ 1 = 2 k C 1 2 g (13)

For the liquid ( z < 0 ) , because of its depth limitation, from (10) analogically we will have:

P a 2 ( z ) = B 1 exp ( δ 1 z ) + B 2 exp ( δ 2 z ) (14)

where

δ 1 = − k θ 2 [ 1 + 1 + θ 2 2 ( 1 − U p 2 C 2 2 ) ] < 0 (15)

δ 2 = − k θ 2 [ 1 − 1 + θ 2 2 ( 1 − U p 2 C 2 2 ) ] > 0 (16)

θ 2 = 2 k C 2 2 g (17)

Let us denote liquid surface displacement along the axis z through

ξ ( x , t ) = a exp [ i ( k x − ω t ) ] (18)

and in this case, the boundary conditions on the surface ( z = 0 ) and on the bottom ( z = − h ) of the liquid will have the form of:

{ ( P 2 − P 1 ) | z = 0 = − α ∂ 2 ξ ∂ x 2 V z 1 | z = 0 = ∂ ξ ∂ t + V 0 ∂ ξ ∂ x V z 2 | z = 0 = ∂ ξ ∂ t V z 2 | z = − h = 0 (19)

where α is the coefficient of surface tension.

Let’s present perturbation velocity in the form of periodical function:

V ( x , z , t ) = V a ( z ) exp [ i ( k x − ω t ) ] (20)

and denote its z component through disturbance pressure from the first equation of the system (7):

V z ( x , z , t ) = − i ρ 0 ω [ ∂ P a ( z ) ∂ z + g C 2 P a ( z ) ] exp [ i ( k x − ω t ) ] (21)

Substituting (9), (11), (14) and (21) in the boundary conditions (19), we will get the system of homogenous equations referred to unknown coefficients A , B 1 , B 2 and a:

{ A − B 1 − B 2 + α k 2 a = 0 γ + g / C 1 2 ρ 01 ω A − ( ω − k V 0 ) a = 0 δ 1 + g / C 2 2 ρ 02 ω B 1 + δ 2 + g / C 2 2 ρ 02 ω B 2 − ω a = 0 δ 1 + g / C 2 2 ρ 02 ω exp ( − δ 1 h ) B 1 + δ 2 + g / C 2 2 ρ 02 ω exp ( − δ 2 h ) B 2 = 0 (22)

Equating the determinant of the system (22) to zero, we will get dispersion relation for the wave on the liquid surface taking into account surface tension force in the form of:

δ 1 δ 2 ρ 02 ω ( ω − k V 0 − γ ρ 01 ω α k 2 ) [ exp ( − δ 1 h ) − exp ( − δ 2 h ) ] − γ ρ 01 [ δ 2 exp ( − δ 1 h ) + δ 1 exp ( − δ 2 h ) ] = 0 (23)

Taking into account that on the sea level C 1 ≅ 340 m / sec , let’s consider the inequality

θ 1 = 2 k C 1 2 g > 1 ⇒ k > g 2 C 1 2 ⇒ λ < 4 π C 1 2 g = 1.45 × 10 5 m

We can see, that to this inequality satisfies with the entire range of lengths of surface waves on the water, from capillary to tsunami. It is apparent that for the capillary waves length of which does not exceed a few centimeters, we have: θ 2 ≫ θ 1 ≫ 1 . Considering also that U p 2 / C 2 2 ≪ U p 2 / C 1 2 ≪ 1 , from (12), (15) and (16) we find γ = δ 1 = − k , δ 2 = k and then neglecting ρ 01 with respect to ρ 02 , the dispersion Equation (23) takes the form:

ρ 02 U p 2 + ρ 01 V 0 U p − α k = 0 (24)

the solution of which is

U p = − ρ 01 V 0 ± ρ 01 2 V 0 2 + 4 α k ρ 02 2 ρ 02 (25)

In order to show the truthfulness of our results, let’s consider earlier results and show their drawbacks. As it was said in the introduction, in the monography [

U p 2 = ( ω k ) 2 = ( g k + α k ρ 02 ) t h ( k h ) (26)

from which it follows that when

g k > α k ρ 02 ⇒ k < ρ 02 g α ⇒ λ > 2 π α ρ 02 g = 1.72 sm (27)

the influence of the surface tension force is negligible, and the wave becomes purely gravitational. This conclusion contradicts to the classical experiment, in which a steel needle does not sink in a glass filled with water to the brim. This is because although the diameter of glass greatly exceeds the above specified length, the force of surface tension acts which balances the pressure produced by the needle. Thus, the dependence of phase speed on gravitational acceleration is excluded and consequently there is no existing condition that limits the length of capillary wave. Such a conclusion is quite understandable from the point of view of physics, because surface tension arises due to the interaction forces between molecules on the surface of a liquid that significantly exceed the gravitational force.

The contradiction associated with the influence of the gravitational field is eliminated by taking into account the compressibility of the fluid in the mass continuity equation. The solution of the problem for such a case is given in [

U p = ω k = ρ 01 V 0 t h ( k h ) ± { t h ( k h ) [ ρ 02 α k − ρ 01 ρ 02 V 0 2 ] } 1 / 2 ρ 02 (28)

from which follows the condition of stability of capillary wave:

V 0 ≤ α k ρ 01 (29)

From (29) it is easy to calculate that the wind with the speed of V 0 = 5 m/s, will blow off capillary waves whose length λ > 1.6 cm. However, simple observations show that capillary waves exist at quite stronger winds. In addition, since the capillary wave is a purely surface phenomenon, its phase speed must not depend on the depth of the fluid.

As it is apparent in the Equation (25), this contradiction is eliminated if in Euler equation, we consider liquid as compressible. Capillary wave is stable in any wind, if only the wind force does not exceed intermolecular interaction force and in this case, setting of the problem becomes meaningless. We can also see that phase speed of capillary wave does not depend on the depth of fluid.

Contradictions that are present in the theory of surface waves are described in detail in the works [

The author declares no conflicts of interest regarding the publication of this paper.

Kirtskhalia, V. (2019) The Linearity of the Euler Equation as a Result of the Compressibility of a Fluid. Journal of Modern Physics, 10, 452-458. https://doi.org/10.4236/jmp.2019.104030