# Exact Solution of the Navier–Stokes Equation Describing Nonisothermal Large-Scale Flows in a Rotating Layer of Liquid with Free Upper Surface

## Abstract

We present an analytic representation of an exact solution of the Navier–Stokes equations that describe flows of a rotating horizontal layer of a liquid with rigid and thermally isolated bottom and a free upper surface. On the upper surface, a constant tangential stress of an external force is given, and heat emission governed by the Newton law occurs. The temperature of the medium over the surface of the liquid is a linear function of horizontal coordinates. We find the solution of the boundary-value problem for ordinary differential equations for the velocity and temperature. and examine its form depending on the Taylor, Grashof, Reynolds, and Biot numbers. In rotating liquid, the motion is helical; account of the inhomogeneity of the temperature makes the helical motion more complicated.

- Authors:

- Perm State University (Russian Federation)

- Publication Date:

- OSTI Identifier:
- 22771261

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Mathematical Sciences

- Additional Journal Information:
- Journal Volume: 230; Journal Issue: 5; Conference: International symposium on differential equations, Perm (Russian Federation), 17-18 May 2016; Other Information: Copyright (c) 2018 Springer Science+Business Media, LLC, part of Springer Nature; http://www.springer-ny.com; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1072-3374

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 42 ENGINEERING; 97 MATHEMATICAL METHODS AND COMPUTING; BOUNDARY-VALUE PROBLEMS; EXACT SOLUTIONS; FLOW RATE; HEATING; LIQUIDS; NAVIER-STOKES EQUATIONS; REYNOLDS NUMBER; STRESSES; SURFACES; TEMPERATURE DEPENDENCE

### Citation Formats

```
Shvarts, K. G., E-mail: kosch@psu.ru.
```*Exact Solution of the Navier–Stokes Equation Describing Nonisothermal Large-Scale Flows in a Rotating Layer of Liquid with Free Upper Surface*. United States: N. p., 2018.
Web. doi:10.1007/S10958-018-3796-Y.

```
Shvarts, K. G., E-mail: kosch@psu.ru.
```*Exact Solution of the Navier–Stokes Equation Describing Nonisothermal Large-Scale Flows in a Rotating Layer of Liquid with Free Upper Surface*. United States. doi:10.1007/S10958-018-3796-Y.

```
Shvarts, K. G., E-mail: kosch@psu.ru. Tue .
"Exact Solution of the Navier–Stokes Equation Describing Nonisothermal Large-Scale Flows in a Rotating Layer of Liquid with Free Upper Surface". United States. doi:10.1007/S10958-018-3796-Y.
```

```
@article{osti_22771261,
```

title = {Exact Solution of the Navier–Stokes Equation Describing Nonisothermal Large-Scale Flows in a Rotating Layer of Liquid with Free Upper Surface},

author = {Shvarts, K. G., E-mail: kosch@psu.ru},

abstractNote = {We present an analytic representation of an exact solution of the Navier–Stokes equations that describe flows of a rotating horizontal layer of a liquid with rigid and thermally isolated bottom and a free upper surface. On the upper surface, a constant tangential stress of an external force is given, and heat emission governed by the Newton law occurs. The temperature of the medium over the surface of the liquid is a linear function of horizontal coordinates. We find the solution of the boundary-value problem for ordinary differential equations for the velocity and temperature. and examine its form depending on the Taylor, Grashof, Reynolds, and Biot numbers. In rotating liquid, the motion is helical; account of the inhomogeneity of the temperature makes the helical motion more complicated.},

doi = {10.1007/S10958-018-3796-Y},

journal = {Journal of Mathematical Sciences},

issn = {1072-3374},

number = 5,

volume = 230,

place = {United States},

year = {2018},

month = {5}

}