SULIMAN SHEEN 1 , ABDELFATAH ABASHER2
1Deanship of the Preparatory Year,Prince Sattam bin Abdulaziz University ,Alkharj,Saudi Arabia
EMAIL:sulimanmaleeh@gmail.com
2Mathmatics Department , Faculty of Science ,Jazan Univercity,Jazan, Saudi Arabia
EMAIL:amoaf84@gmail.com
HNSJ, 2022, 3(10); https://doi.org/10.53796/hnsj3102
Published at 01/10/2022 Accepted at 05/09/2022
Abstract
In this paper, we obtain basic flow solutions for stationary viscous flow between two rotating coaxial cylinders by solving the Navier -Stokes’s equations in the cylindrical coordinates system for viscous incompressible fluid,simplyfied the equations and obtained analyticaly is a Zero – Order Bessel’s Function in one variable.
Key Words: viscous flow, rotating, Navier -Stokes’s equations , coaxial cylinders , pressure, stationary solution, perturbation equations, couette flow.
عنوان البحث
حل تحليلي لانسياب كوتي
سليمان شين2 عبد الفتاح أبشر1
1 جامعة الأمير سطام بن عبد العزيز، الخرج ، المملكة العربية السعودية
البريد الإلكتروني: sulimanmaleeh@gmail.com
2 قسم الرياضيات ، كلية العلوم ، جامعة جازان ، جازان ، المملكة العربية السعودية
البريد الالكتروني: : amoaf84@gmail.com
HNSJ, 2022, 3(10); https://doi.org/10.53796/hnsj3102
تاريخ النشر: 01/10/2022م تاريخ القبول: 05/09/2022م
المستخلص
في هذه الورقة نحصل علي حلول الانسياب الاساسية لانسياب ثابت بين اسطوانتين متحدتين المحور بينهما مائع لا انضغاطي- لزج .ذلك بحل معادلات نايفر- استوكس في الاحداثيات الاسطوانيه. تم تبسيط المعادلات وحصلنا تحليليا على دالة بيسيل من الرتبة الصفرية في متغير واحد.
The Navier -Stokes’s equations for the velocity and the pressure can be written in the form
in [1]
Where
And
in [2]
The continuity equation in the cylindrical coordinates is given by,
In [3]
These aforementioned equations allow a stationary solution of the form
Thus, Navier -Stokes’s equations
Reduced to
and
in [ 3]
2.THE PERTURBATION EQUATIONS AND THE NORMAL MODE
In order to investigate the solutions of the flow system described by equations (1.9) We consider an infinitesimal of the basic flow is given by (1.8) by assuming that the perturbed flow is given by
Assuming that the various perturbations are axisymmetric and independent of , and From (1.1) – (1.3) we gain the following linearized equations as
In [3]
and
where is defined by
And the equation of continuity reduces to
By analyzing the disturbance into normal modes. We assume that the disturbances are of the following form
in[4]
where k is the wave number of the disturbance in the axial direction, and p is a constant which can be complex.
Substituting (2.7) in equations (2.2) – (2.6), we get
In[3]
substitute in equation (2.10), we obtain
= (2.14)
From equation (2.14), we find
Substituting (2.15) in equation (2.8), yields
By multiplying equation (2.9) by ), and multiplying equation (2.16) by P
We obtain,
Now, summation equation (2.17) to equation (2.18), we obtain
angular velocity is
in[1]
where is a real function
By Substituting (2.22) at equation (2.21) We get:
Equation (2.24) is a Zero – Order Bessel’s Function, with the solution
in[5]
3.RESULT AND CONCLUSION
(1) Bessel’s Function are closely associated with problems processing circular or cylindrical symmetry, because of their close association with cylindrical domains.in[5]
(2) The solutions of Bessel’s equation are called cylinder functions. Bessel’s Function of the first kind and second kind are special cases of cylinder functions.
REFERENCES
[1] S. Chandrasekhar. ‘Hydrodynamic and Hydromagnetic Stability‘, Dover, New York, 1961.
[2] Murray R.Spiegel , “Vector Analysis and An Introduction To Tensor Analysis”,Rensselaer Polytechnic Institute ,1959.
[3] Hua-Shu Dou, Boo Cheong Khoo2 , and Khoon Seng Yeo,” Instability of Taylor-Couette Flow between Concentric Rotating Cylinders”, Inter. J. of Thermal Science, 2008.
[4] Kyungyoon Min and Richard M. Lueptow,” Hydrodyna ic stability of viscous flow between rotating Porous cylinders with radial flow”, Department of Mechanical Engineering, Northwestern University, Evanston, Illinois 60208 (Received 28 January 1993; accepted 1 September 1993.
[5]LARRY C. ANDREWS , ‘Special functions of mathematics for engineers’, Bellingham, Washington USA ,1998.
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