### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Formulation of the problem
- 3. Generalized characteristics
- 4. Propagation of a weak discontinuity
- 5. Approximate differential model
- 6. Numerical results
- 7. Conclusion
- Acknowledgments
- References

Nonlinear integrodifferential equations describing the propagation of disturbances in a thin layer of viscous liquid with free surface are studied. These equations admit solutions with weak discontinuities, which are located on the characteristics. The possibility of an unbounded increase in the amplitude of the weak discontinuity and the formation of the shock in the process of flow evolution is established. Differential balance laws approximating the integrodifferential model are proposed. These laws are used to perform numerical simulation of wave propagation in a fluid.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Formulation of the problem
- 3. Generalized characteristics
- 4. Propagation of a weak discontinuity
- 5. Approximate differential model
- 6. Numerical results
- 7. Conclusion
- Acknowledgments
- References

Since the pioneering work of Kapitsa [1], wave motions of liquid films have been a subject of numerous experimental and theoretical studies [2, 3]. Depth-averaged models, in particular, Benney's evolution equation for the film thickness [4], Shkadov's equations [5] and their modifications [6, 7] are often used to describe wave flow regimes of a thin layer of viscous fluid. If the film thickness is not sufficiently small, then the velocity distribution over the depth may differ significantly from the parabolic law. This leads to the necessity to use the equations of the long-wave approximation [8, 9]. In contrast to the theory of Prandtl's boundary layer, the pressure in the fluid is not specified in advance. It should be determined simultaneously with the velocity field and the fluid depth (this is similar to models with self-induced pressure [10, 11]). A generalized method of characteristics for equations with operator coefficients [12, 13] is suitable for theoretical study of wave propagation in a thin layer of viscous liquid in the long-wave approximation. This approach has already been used to study the wave propagation in a supersonic boundary layer within the framework of the model of viscous-inviscid interaction [14].

In this paper, we apply the theory of generalized characteristics for systems of integrodifferential equations describing the wave propagation in a thin layer of viscous liquid with free surface flowing down an inclined plane. Transport equations for the amplitudes of weak discontinuities are derived and the possibility of a nonlinear wave-breaking is established. Numerical calculations based on “multi-layer” approximation of the model confirm the possibility of shock formation.

### 2. Formulation of the problem

- Top of page
- Abstract
- 1. Introduction
- 2. Formulation of the problem
- 3. Generalized characteristics
- 4. Propagation of a weak discontinuity
- 5. Approximate differential model
- 6. Numerical results
- 7. Conclusion
- Acknowledgments
- References

We consider the two-dimensional flow of viscous fluid down an inclined plane in the gravity field (Figure 1). The Navier–Stokes equations in dimensionless variables are written as

- (1)

The boundary conditions at the plane and at the free surface are

- (2)

Here, , , , , , are dimensionless components of the velocity, pressure, Cartesian coordinates and time, respectively; , , , , , and are the corresponding dimensional variables. The parameters *L*_{0} and *H*_{0} determine the characteristic scales on the *x* and *y* axes; the quantity *U*_{0} is the characteristic velocity. The kinematic viscosity is denoted by ν_{0}, and θ refers to the angle of inclination of the plane to the horizontal. The constants ρ and *g*_{0} are the density of the fluid and the gravity acceleration. The surface tension is not taken into account. The shallowness parameter ε, the Reynolds number Re, and the Froude number Fr are defined as

In contrast to Benney's equations [15], the model (4) includes a term with the second derivative with respect to *y* and additional boundary conditions at the bottom and at the free surface. If the Reynolds number is sufficiently large, such that , then the same scaling excludes in Equations (1) terms with the second order derivatives and simplifies dynamic boundary condition in Equations (2).

The simplest stationary solution of the Equations (4) is

- (7)

where the depth *h* and the flow discharge are constants. Let us integrate the first equation of system (4) with respect to *y* under assumption that the velocity profile is given by formula (7) with the unknown functions and . Using other equations of system (4), we obtain the dispersionless Shkadov model [5]

- (8)

This system is hyperbolic and its characteristic velocities are

Further, we show that the integrodifferential model (5) is also hyperbolic (with respect to the spatial variable *x*) in the sense of [12, 13].

### 4. Propagation of a weak discontinuity

- Top of page
- Abstract
- 1. Introduction
- 2. Formulation of the problem
- 3. Generalized characteristics
- 4. Propagation of a weak discontinuity
- 5. Approximate differential model
- 6. Numerical results
- 7. Conclusion
- Acknowledgments
- References

If the hyperbolicity conditions (14) hold, the equations of motion (5) can be transformed to the form (11), which includes derivatives only in the directions tangent to the characteristic surfaces. This makes it possible to construct piecewise-smooth solutions with discontinuities in the derivatives in the direction normal to characteristics.

The eigenfunction of the operator **A**, obtained from the equations

has the form

- (16)

According to Equation (15), the jump of derivatives is proportional to the eigenfunction of the operator **A**

- (17)

where the coefficient defines the weak discontinuity amplitude.

Let us differentiate the relation on the characteristic (11) with respect to *x*

(hereinafter the superscript “2” in the notation of **F**^{2} and *k*^{2} is omitted; action of the functional **F** is determined by (13)). After calculating the difference between limiting values of this expression on Γ we have

Taking into account the identity

the previous relation can be reduced to the form

- (18)

Here, is the operator of differentiation along the characteristic of slope . The subscript “1” denotes the limiting values of the functions on one side of the surface Γ.

Taking into account the identity , and using formulae (13),(16), and (19), we represent the left-hand side of the Equation (18) in the form

where

Similarly, we transform the right-hand side of Equation (18). By using formulae (13) and (17), we determine the action of the functional

and the result of the action **F** on the function (**f** is the right side of Equations (5))

On the basis of the previous calculations, we collect the terms in (18) with respect to the powers of ξ. This yields the transport equation for the weak discontinuity amplitude

- (20)

where the coefficient *C*_{2} has the form

### 5. Approximate differential model

- Top of page
- Abstract
- 1. Introduction
- 2. Formulation of the problem
- 3. Generalized characteristics
- 4. Propagation of a weak discontinuity
- 5. Approximate differential model
- 6. Numerical results
- 7. Conclusion
- Acknowledgments
- References

As we have shown above, the evolution of the smooth solution of the integrodifferential system (5) can involve a gradient catastrophe. The further description of the solution is possible only in the class of discontinuous functions. It leads to the necessity to formulate the model in the form of balance laws. To do this, we will present (5) and (6) in conservative form.

For unknowns

the following conservative formulation of the Equations (5) and (6) is used

- (21)

Let us show that the systems (5), (6), and (21) are equivalent for smooth solutions. Obviously, system (21) is a consequence of Equations (5) and (6), obtained by simple transformations. Inversely, integrating the second equation in (21) with respect to λ gives

- (22)

Next, we multiply (22) by *H* and integrate with respect to λ from 0 to 1 by using the first and the last equations in (21). This yields the equality

Comparison of this expression with the third equation in (21) allows one to determine the function and establish the equivalence of the systems under consideration.

The hyperbolic system of differential balance laws (26), in contrast to the integrodifferential Equations (21), can be solved numerically using standard Godunov type methods. In this case, it is reasonable to use central schemes because system (26) contains a large number of unknown quantities, and exact or approximate solutions of the Riemann problem are difficult to obtain. Below, we present numerical modeling of wave motion of a viscous fluid using multilayered Equations (26) and dispersionless Shkadov model (8).