Curved-pipe flows have been the subject of many theoretical investigations due to their importance in various applications. The goal of this paper is to study the flow of incompressible fluid with a pressure-dependent viscosity through a curved pipe with an arbitrary central curve and constant circular cross section. The viscosity-pressure dependence is described by the well-known Barus law extensively used by the engineers. We introduce the small parameter
Curved-pipe flows have gained much attention over past years due to their importance in numerous industrial and engineering applications. Air conditioners, refrigeration systems, central heating radiators, and chemical reactors are only few examples of devices where we can find different types of curved pipes. From the theoretical point of view, curved-pipe flows are interesting due to the appearance of secondary flows caused by the effects of the pipe’s distortion. Therefore, when analyzing such problems, the main attempt is to detect the effects of the pipe’s geometry on the velocity and pressure distribution through the pipe. The engineering approach to the curved-pipe flows is often based on the
In his celebrated work, Stokes [
Motivated by the above discussion, the aim of this paper is to study the incompressible fluid with a pressure-dependent viscosity obeying Barus law and flowing through a curved pipe with constant circular cross section. Introducing the viscosity-pressure dependence ( rewriting the governing system by replacing the original pressure with a new, finding the solution of the reconstructing the effective pressure by applying the inverse transformation.
Naturally, it is not reasonable to expect that we will succeed to find the exact solution of the governing
Along with the viscosity-pressure dependence, our aim is to treat as general domain as possible. Thus, we assume that the pipe’s central curve, denoted by
We conclude the introduction by providing more bibliographic remarks on the subject. Curved-pipe flow in case of constant viscosity (
In this section we formally describe the complex pipe’s geometry. As emphasized in Introduction, we want to address the case of a general curved pipe with circular cross section. In view of that, we introduce a generic curve in
The normal
Next, we introduce the small parameter
The reference system.
Observe that by putting
We are now in position to define our three-dimensional domain representing thin (or long) curved pipe with an arbitrary central curve
Finally, we denote by
From the strictly mathematical point of view, we have to ensure the local injectivity of the parametrization
We suppose that the pipe
Substituting the Barus law into the momentum equation (
We divide it by
The form of the above equation suggests introducing a new function, denoted by
Here and in the sequel we will call it the
By a simple calculation we deduce
The obtained transformed system is in the form of a nonlinear Stokes-like system (with nonlinearity appearing on the right-hand side in (
It is essential to be aware that, throughout the whole process,
In order to write the transformed equations (
Using the fact that
To make the complex notation more compact, here and in the sequel we introduce
Observe that
We employed this fact to assure the local injectivity of the parametrization
The covariant basis is complemented with the
In our case it reads
Finally,
Its fundamental property is that they are symmetric in lower indices; that is,
Now we are going to establish the asymptotic behavior of the above quantities needed in the sequel. Having in mind that
In [
Here
Rather complicated but straightforward calculations based on (
Now we apply the two-scale asymptotic technique on the transformed problem (
Plugging the above expansions in (
First, let us note that substituting (
Observe that, at this stage, the angle
Seeking for the effects of the pipe’s distortion, we continue the computation and try to construct the velocity corrector. The
Taking into account (
Since
By a simple integration (passing to polar coordinates), one can easily check that
Finally, from (
To conclude this section we write the asymptotic solution of the transformed problem (
For the transformed pressure we obtain
Now we apply the inverse transformation, that is, we reconstruct the original pressure from (
In view of (
This represents our asymptotic approximation for the effective pressure. Obviously, it is well-defined since
For the velocity approximation, we have
Finally, to emphasize the importance of the obtained result, let us compare it with the one from [
In the previous section, we formally derived an asymptotic model describing the flow of incompressible fluid with a pressure-dependent viscosity through a thin (or long) curved pipe. We assumed that the pressure-dependent viscosity obeyed Barus formula (most commonly used in the engineering community) and worked with physically relevant Dirichlet boundary conditions. Since the pipe’s central curve is assumed to be a general (generic) curve, the applicability of the obtained result is broad. Moreover, by obtaining the explicit expressions (
From the rigorous mathematical point of view, we should link our formally obtained solution with the original solution by proving some kind of convergence result. It can be accomplished by evaluating the difference between those two solutions in the appropriate rescaled norm. Though rigorous justification is out of scope of the present paper, let us comment on this as well. Since
To conclude, it is important to emphasize that the presented approach can be extended to a case of general viscosity-pressure relation
The author declares that there is no conflict of interests regarding the publication of this paper.
The author of this work has been supported by the