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Abstract

This article focuses on the problem of convective heat transport in an incompressible fluid flow inside a cylinder, governed by the Navier-stokes equations. The study provides exact solutions for fluid motion. These solutions help in understanding how fluid behaves in such environments, contributing to the study of fluid dynamics.

Introduction

Fluid mechanics studies the behavior of fluids using models like the Navier-Stokes equations, but due to their complexity, only a few exact solutions are known. Researchers often use experimental and theoretical models to explore key aspects such as velocity distribution and flow patterns, and recent advancements have refined temperature estimates and proven the existence of global solutions to these equations.

This article focuses on finding analytical solutions to the Navier-stokes equations in cylindrical coordinates, crucial for understanding mass, momentum, and heat transport in engineering applications. The solutions to the Navier-Stokes are presented, along with a summary of the findings.

Governing Equations

The fluid layer, confined between two parallel plates separated by a distance , experiences a uniform heat flux . The plates are no-slip boundaries with fixed temperatures T0 and T1. The velocity field u, pressure p, temparatue T , follows the boussinesq approximation, capturing buoyancy effects while assuming density variations only affect bouyancy forces.

${dv \over dt}+v.\bigtriangleup v=-\bigtriangledown p+v\bigtriangledown ^{2}v+{\widehat {k}}g\alpha T$

${dT \over dt}+v.\bigtriangledown T=k\bigtriangledown ^{2}T+\gamma$

$\bigtriangledown .v=0$

with the boundary conditions

v=0 and $\mathrm {T} (Z=0)=T0,T(Z=d)=T1$

we introduce the cylindrical coordinates r, $\varphi$, z which are associated with the cartesian coordinates x, y, z by the relations

$x=r\cos \varphi ,$ $y=r\sin \varphi$, $Z=z$

• the incompressible viscous fluid flows are axisymmetric and helical .

Then the boundary conditions in the non-dimensional form become :

$v=0,$ $T(z=0)={\tilde {T}},$ $T(z=1)=0$

where ${\tilde {T}}=(k/\gamma d^{2})(T0-T1)$

For completeness, we write explicitly the three dimensional equations in cylindrical coordinates (r, z)  :

${\frac {1}{p_{r}}}\left({\frac {\partial v_{r}}{\partial t}}+v_{r}{\frac {\partial v_{r}}{\partial r}}-{\frac {v_{\varphi }^{2}}{r}}+v_{z}{\frac {\partial v_{r}}{\partial z}}\right)=-{\frac {\partial p}{\partial r}}+{\frac {\partial }{\partial r}}\left({\frac {1}{r}}{\frac {\partial \left(rv_{r}\right)}{\partial r}}\right)+{\frac {\partial ^{2}v_{r}}{\partial z^{2}}}$

${\frac {1}{P_{r}}}\left({\frac {\partial v_{\varphi }}{\partial t}}+v_{r}{\frac {\partial v_{\varphi }}{\partial r}}+{\frac {v_{r}v_{\varphi }}{r}}+v_{z}{\frac {\partial v_{\varphi }}{\partial z}}\right)={\frac {\partial }{\partial r}}\left({\frac {1}{r}}{\frac {\partial \left(rv_{\varphi }\right)}{\partial r}}\right)+{\frac {\partial ^{2}v_{\varphi }}{\partial z^{2}}}$

${\frac {1}{P_{r}}}\left({\frac {\partial v_{z}}{\partial t}}+v_{r}{\frac {\partial v_{z}}{\partial r}}+v_{z}{\frac {\partial v_{z}}{\partial z}}\right)=RaT-{\frac {\partial p}{\partial z}}+{\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial v_{z}}{\partial r}}\right)+{\frac {\partial ^{2}v_{z}}{\partial z^{2}}}$

${\frac {\partial T}{\partial t}}-{\frac {1}{r}}{\frac {\partial \psi }{\partial z}}{\frac {\partial T}{\partial r}}+{\frac {1}{r}}{\frac {\partial \psi }{\partial r}}{\frac {\partial T}{\partial z}}={\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial T}{\partial r}}\right)+{\frac {\partial ^{2}T}{\partial z^{2}}}+1$

${\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial \psi }{\partial r}}\right)+{\frac {\partial ^{2}\psi }{\partial z^{2}}}-{\frac {\psi }{r^{2}}}={\frac {1}{r}}\left({\frac {\partial ^{2}\psi }{\partial z^{2}}}+{\frac {\partial ^{2}\psi }{\partial r^{2}}}-{\frac {1}{r}}{\frac {\partial \psi }{\partial r}}\right)$

where $\psi$ is the stream function.

Analytical Solutions

As the fluid flows are axis symmetric and helical, we can write :

$v(r,z,t)=\operatorname {rot} \left({\frac {\psi (r,z,t)}{r}}e_{\varphi }\right)+\left(rv_{\varphi }\right)e_{\varphi }$

Applying the operator rot to the relation and using the explicit relations between the cylindrical components of the vorticity and velocity, we get :

$\Delta \left({\frac {\psi (r,z,t)}{r}}\right)={\frac {\partial v_{z}}{\partial r}}-{\frac {\partial v_{r}}{\partial z}},$

where $\bigtriangleup$ is the laplacian in cylindrical coordinates r,z. Using the divergence-free condition

$\nabla \cdot v(r,z,t)=0{\text{ i.e. }}{\frac {\partial \left(rv_{r}\right)}{\partial r}}+{\frac {\partial \left(rv_{z}\right)}{\partial z}}=0$

And the stream function $\psi$ for the velocity through the relation

${\begin{aligned}\\&v_{r}(r,z,t)=-{\frac {1}{r}}{\frac {\partial \psi }{\partial z}}(r,z,t),\quad v_{z}(r,z,t)={\frac {1}{r}}{\frac {\partial \psi }{\partial r}}(r,z,t)\end{aligned}}$

${\begin{aligned}&{\text{ then the equation is equivalent to which leads to : }}\\&\left(r^{2}-r\right){\frac {\partial ^{2}\psi }{\partial r^{2}}}+\left(r^{2}-r\right){\frac {\partial ^{2}\psi }{\partial z^{2}}}+(r+1){\frac {\partial \psi }{\partial r}}-\psi =0\end{aligned}}$

${\begin{aligned}&{\text{ Putting }}\psi (r,z,t)={\frac {\alpha f(s)}{r-1}},s={\frac {\alpha z}{r}},\alpha =\alpha (t){\text{, }}\\&\left(s^{2}+\alpha ^{2}\right)f^{\prime \prime }(s)+3sf^{\prime }(s)=0\end{aligned}}$

which produces the solution:

$\psi (r,z,t)={\frac {C_{1}z\varepsilon \left(z-z^{2}\right)}{\alpha (t)(r-1){\sqrt {z^{2}+r^{2}}}}}+{\frac {C_{2}\alpha }{r-1}}$

where $\epsilon (\zeta )$ is the sign function and C1,C2 = const

the equation has the following solution :

$\psi (1,z,t)=0,\forall t\in (0,\tau ),\quad 0<z<1\$

using the boundary conditions $v_{\mid z=0,1}=0$ , $v(r,z,t)=\left(v_{r}(r,z,t),v_{\varphi }(r,z,t),v_{z}(r,z,t)\right.$

we obtain that , C2=0. Without loss of generality we chose c1=1 and the solution takes the form:

${\begin{aligned}&\psi (r,z,t)=\varepsilon \left(z-z^{2}\right){\frac {z}{\alpha (t)(r-1){\sqrt {z^{2}+r^{2}}}}}\\&\psi (1,z,t)=0\end{aligned}}$

Conclusion

In this article, we have investigated the exact solutions to the Navier Stokes equations in a cylinder containing a viscous incompressible fluid.

References

[1] Zadrzynska E.; Zajaczkowski W.M. Global Regular Solutions with Large Swirl to the Navier-Stokes Equations in a cylinder. J. Math. Fluid Mechanics; Vol. 11, 126-169, 2009

[2] Busse, F. The bounding theory of turbulence and its physical significance in the case of turbulent Couette flow. In: Statistical Models and Turbulence, edited by M. Rosenlatt and C. M. Van Atta, Springer Lecture Notes in Physics Vol. 12 (Springer, Berlin, 1972), pp. 103 -126.

[3] White, F.M.(2016). Viscous Fluid Flow. McGraw-Hill.

Article prepared by

1.Sahil Singh (Roll.No - 21134025), IIT BHU (Varanasi)

2.Rajnandan Kumar (Roll.No - 21134024), IIT BHU (Varanasi)

3.Tarun (Roll.No - 21134028), IIT BHU (Varanasi)

4.Vivek Singh (Roll.No - 21134030), IIT BHU (Varanasi)

5.Shubham Raj (Roll.No - 21134027), IIT BHU (Varanasi)