Advances in Mathematical Physics

Volume 2017 (2017), Article ID 7042686, 9 pages

https://doi.org/10.1155/2017/7042686

## Solutions of Navier-Stokes Equation with Coriolis Force

^{1}Department of Advanced Materials and Chemical Engineering, Catholic University of Daegu, Gyeongsan, Gyeongbuk 38430, Republic of Korea^{2}Korea Institute of Energy Research, Daejeon 305-343, Republic of Korea

Correspondence should be addressed to Shin-Kun Ryi and Hankwon Lim

Received 15 April 2017; Accepted 3 July 2017; Published 14 August 2017

Academic Editor: Eugen Radu

Copyright © 2017 Sunggeun Lee et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We investigate the Navier-Stokes equation in the presence of Coriolis force in this article. First, the vortex equation with the Coriolis effect is discussed. It turns out that the vorticity can be generated due to a rotation coming from the Coriolis effect, . In both steady state and two-dimensional flow, the vorticity vector gets shifted by the amount of . Second, we consider the specific expression of the velocity vector of the Navier-Stokes equation in two dimensions. For the two-dimensional potential flow , the equation satisfied by is independent of . The remaining Navier-Stokes equation reduces to the nonlinear partial differential equations with respect to the velocity and the corresponding exact solution is obtained. Finally, the steady convective diffusion equation is considered for the concentration and can be solved with the help of Navier-Stokes equation for two-dimensional potential flow. The convective diffusion equation can be solved in three dimensions with a simple choice of .

#### 1. Introduction

Our Earth, even though it is hard to feel in daily life, is rotating. However, if we look around to see what is happening, we realize that this rotation leads to many interesting phenomena. These phenomena are unavoidable as long as we live on Earth. On a macroscopic scale, this effect is substantial: the climate is affected and complicated because the rotation changes the flows of the air heated by the sun. The air has a very complicated motion compared to the one without rotation. Typhoons and their motion are another phenomenon resulting from the effect of the rotation of Earth. Furthermore, their direction of rotation is opposite to each other in the northern and southern hemispheres. The flows of the ocean can be affected by the rotation of Earth as well. Changing the motion of moving objects or fluids such as air and ocean illustrated above is caused by a force. This force on all moving bodies is called Coriolis force. The rotating effect may be seen in outer space such as accretion disks of fast rotating black hole [1] as well. In this case, the relativity and the curved space-time should be counted to understand the effect better. It has been applied commercially in the areas of, for instance, Coriolis flow meter, and the solutions to the flow equations of Navier-Stokes equation are very useful. Hence, the Coriolis effect in the Navier-Stokes equation should be included in these cases. In addition, when the long cylinder carrying flows of water or oil is concerned, the Coriolis effect is not negligible and should be considered in order to construct the cylinder safely. In wide aspects, many chemical reactions of the flowing fluids or gases should be understood with the inclusion of this effect.

In summary, in fact, almost all of the fluids flowing on Earth can be affected, so knowing the exact motion of the fluids is important for applications and problem solving. The flow even involves heat flows of the fluids as well. Our work is just a starting point to find such solutions. As a basic exercise, we investigate the vortex equation and the Navier-Stokes equations under the Coriolis force. Specifically, we assume that the solutions are time independent. In [2], the Coriolis effect on the three-dimensional Navier-Stokes equation was considered. That was the case of compressible fluid while our concern here is the case of an incompressible one.

The present work has the following structure. In Section 2, we construct the vortex equation with a term of Coriolis force. The Coriolis effect can generate the vorticity. Especially for two-dimensional flow, the interesting point of view is the fact that the angular velocity coming from Coriolis force comes into play just as a shift of vorticity vector. In Section 3, we try to find the solution to the Navier-Stokes equations with Coriolis force as well. We apply the solutions to convective diffusion for two-dimensional potential flow. The effect of rotation does not appear on the velocity potential and the concentration is independent of its effect. In three dimensions, we take simplified choice of and solve the convective diffusion equation. Finally, Section 4 finishes with conclusions and discussions.

#### 2. Vortex Equation for Two-Dimensional Flow

In this section, we construct the equation satisfied by the vortex [3] arising from the Navier-Stokes equation. When the fluid motion is viscous, the Euler equation becomes replaced by the Navier-Stokes equation. For incompressible flow, the Navier-Stokes equation in the presence of Coriolis force is given by where is Reynolds number. The last term of the first equation comes from the Coriolis effect with the angular velocity . The vorticity vector is given byLet us apply curl on both sides of the Navier-Stokes equation and use the vector identityWe will focus on the case of constant . Then, we see that the vorticity vector satisfies the following differential equations for incompressible fluid :where we changed the notation of Reynolds number as .

Note that even if the vorticity vector in (4) is only time dependent, that is, without spatial dependence, the Coriolis effect gives rise to the vorticity. In this case, the vorticity is given by Moreover, we see that, by solving the equationthe vorticity can be generated even for a constant (for independent of both time and space in (4)). This means that the phenomenon of vortex can arise everywhere on Earth even though the effect of Coriolis on Earth is too small to be seen.

If we define such thatand consider stationary solution such that , then we have the following equation:Thus, the angular velocity coming from Coriolis effect only appears as a shift of vorticity vector. Furthermore, if we consider two-dimensional flows such thatwhere and are unit vectors on and , respectively, and the angular velocity is a two-dimensional one, then the above equations are reduced to the following: Note that if has one component, say , affecting the vorticity vector, say , in two-dimensional flow (we express the velocity vector as ), hence we can discard vector notation and the equation becomes Considering the constant pressure case, the Navier-Stokes equation becomes Even though the above equation is reduced to a simplified form, it is still hard to be solved. By restricting two dimensions, we may simplify the equation and have a chance to get a solution. Alternatively, changing the above second-order differential equations into a first-order differential equation like BPS solution [4, 5] in field theory is an open and challenging problem.

Solving the Navier-Stokes equation directly is a straightforward way to get a vorticity though the exact solutions are quite restricted. In the next section, we try to solve the steady two-dimensional Navier-Stokes equation.

#### 3. Solution of Navier-Stokes Equations and Its Applications

Let us find the solution of Navier-Stokes equations of which velocities have the following expression [6]:where is the velocity potential for the flow in the plane and is out-of-plane velocity component. The vorticity vector field is given byand from the expression in (13) we have the vorticity vector The contours of are known as the vortex lines of the flow. The zero vorticity comes out if the third component of the velocity is equal to zero in which it turns out to be a two-dimensional flow.

In components, after inserting , , and into the Navier-Stokes equation and focusing on the time independent velocities such that , we have the following set of equations:The incompressible conditions give the continuity equation:

For the expression of the velocity as in (13) in plane, the independence of the third direction gives the following relation:Therefore, the continuity equation can be reexpressed asFinally, we assume the pressure is also independent of the third direction:

Since the specific choice of the velocities reduces the three-dimensional problems to two-dimensional problems and working in the complex coordinates might have a number of advantages thanks to the numerous available techniques in complex analysis (e.g., conformal mapping), we now introduce the complex coordinates such thatSo, by applying chain rule, we get the following relations:Now, let us rewrite the Navier-Stokes equation in complex coordinates by replacing derivatives in real coordinates with complex coordinates:

The addition and subtraction of the first two equations in (23) givewhere and and as already noticed , , and are assumed to be constants. The third equation becomesIt is interesting to note that the Coriolis effect does not appear in the equation which is related to two-dimensional velocity. Its effect appears on the equations for and the pressure.

Now suppose, for simplicity, that one of ’s is nonzero such that (in other words, ) and . In this case, we haveThe solution would beHowever, we try to make it simpler by taking . Then, the special solution of this equation becomesSo, we have the Navier-Stokes equation in complex coordinates under the Coriolis force with only one nonvanishing component of : The first equation means that is a holomorphic function. In this case, can be written as . By choosing negative sign and putting this holomorphic function with into the above equations, we get So, when only one of ’s, say , is nonvanishing, the Coriolis effect appears only in the pressure equation.

When , the solution in complex coordinates is given by [6]

When we consider two-dimensional flow where the velocities are given by , , and and , we get the result similar to the case obtained when only one component of is nonvanishing. In Figure 1, we plotted and velocity of (31) for a given value of the viscosity 100 when the Coriolis effect does not exist. We see that the velocity vanishes along the -axis.