{"id":4577,"date":"2022-10-01T14:34:21","date_gmt":"2022-10-01T14:34:21","guid":{"rendered":"https:\/\/www.hnjournal.net\/?page_id=4577"},"modified":"2022-10-01T14:34:25","modified_gmt":"2022-10-01T14:34:25","slug":"3-10-2","status":"publish","type":"page","link":"https:\/\/www.hnjournal.net\/en\/3-10-2\/","title":{"rendered":"ANALYTICAL SOLUTION ON \ufffcCOUETTE FLOW"},"content":{"rendered":"

SULIMAN SHEEN 1 <\/sup> , ABDELFATAH ABASHER2<\/sup><\/strong><\/p>\n

<\/a>1<\/sup>Deanship of the Preparatory Year,Prince Sattam bin Abdulaziz University ,Alkharj,Saudi Arabia<\/em><\/p>\n

EMAIL:sulimanmaleeh@gmail.com<\/p>\n

2<\/sup>Mathmatics Department , Faculty of Science ,Jazan Univercity,Jazan, Saudi Arabia <\/em><\/p>\n

EMAIL:amoaf84@gmail.com<\/p>\n

HNSJ, 2022, 3(10); https:\/\/doi.org\/10.53796\/hnsj3102<\/p>\n

Download
\n<\/a><\/p>\n

Published at 01\/10\/2022 Accepted at 05\/09\/2022 <\/strong><\/p>\n

Abstract <\/strong><\/p>\n

In this paper, we obtain basic flow solutions for stationary <\/a>viscous flow between two rotating coaxial cylinders by solving the Navier -Stokes’s equations in the cylindrical coordinates<\/a> system <\/a> for viscous incompressible fluid,simplyfied the equations and obtained analyticaly is a Zero \u2013 Order Bessel’s Function in one variable.<\/p>\n

Key Words: <\/strong>viscous flow, rotating, Navier -Stokes’s equations , coaxial cylinders , pressure, stationary solution, perturbation equations, couette flow.<\/p>\n

\u0639\u0646\u0648\u0627\u0646 \u0627\u0644\u0628\u062d\u062b<\/p>\n

\u062d\u0644 \u062a\u062d\u0644\u064a\u0644\u064a \u0644\u0627\u0646\u0633\u064a\u0627\u0628 \u0643\u0648\u062a\u064a<\/p>\n

\u0633\u0644\u064a\u0645\u0627\u0646 \u0634\u064a\u06462<\/sup> \u0639\u0628\u062f \u0627\u0644\u0641\u062a\u0627\u062d \u0623\u0628\u0634\u06311<\/sup><\/strong><\/p>\n

1 <\/sup>\u062c\u0627\u0645\u0639\u0629 \u0627\u0644\u0623\u0645\u064a\u0631 \u0633\u0637\u0627\u0645 \u0628\u0646 \u0639\u0628\u062f \u0627\u0644\u0639\u0632\u064a\u0632\u060c \u0627\u0644\u062e\u0631\u062c \u060c \u0627\u0644\u0645\u0645\u0644\u0643\u0629 \u0627\u0644\u0639\u0631\u0628\u064a\u0629 \u0627\u0644\u0633\u0639\u0648\u062f\u064a\u0629<\/p>\n

\u0627\u0644\u0628\u0631\u064a\u062f \u0627\u0644\u0625\u0644\u0643\u062a\u0631\u0648\u0646\u064a: sulimanmaleeh@gmail.com<\/p>\n

2<\/sup> \u0642\u0633\u0645 \u0627\u0644\u0631\u064a\u0627\u0636\u064a\u0627\u062a \u060c \u0643\u0644\u064a\u0629 \u0627\u0644\u0639\u0644\u0648\u0645 \u060c \u062c\u0627\u0645\u0639\u0629 \u062c\u0627\u0632\u0627\u0646 \u060c \u062c\u0627\u0632\u0627\u0646 \u060c \u0627\u0644\u0645\u0645\u0644\u0643\u0629 \u0627\u0644\u0639\u0631\u0628\u064a\u0629 \u0627\u0644\u0633\u0639\u0648\u062f\u064a\u0629<\/p>\n

\u0627\u0644\u0628\u0631\u064a\u062f \u0627\u0644\u0627\u0644\u0643\u062a\u0631\u0648\u0646\u064a: : amoaf84@gmail.com<\/a><\/p>\n

HNSJ, 2022, 3(10); https:\/\/doi.org\/10.53796\/hnsj3102<\/p>\n

\n

\u062a\u0627\u0631\u064a\u062e \u0627\u0644\u0646\u0634\u0631: 01\/10\/2022\u0645 \u062a\u0627\u0631\u064a\u062e \u0627\u0644\u0642\u0628\u0648\u0644: 05\/09\/2022\u0645<\/p>\n

\u0627\u0644\u0645\u0633\u062a\u062e\u0644\u0635 <\/strong><\/p>\n

\u0641\u064a \u0647\u0630\u0647 \u0627\u0644\u0648\u0631\u0642\u0629 \u0646\u062d\u0635\u0644 \u0639\u0644\u064a \u062d\u0644\u0648\u0644 \u0627\u0644\u0627\u0646\u0633\u064a\u0627\u0628 \u0627\u0644\u0627\u0633\u0627\u0633\u064a\u0629 \u0644\u0627\u0646\u0633\u064a\u0627\u0628 \u062b\u0627\u0628\u062a \u0628\u064a\u0646 \u0627\u0633\u0637\u0648\u0627\u0646\u062a\u064a\u0646 \u0645\u062a\u062d\u062f\u062a\u064a\u0646 \u0627\u0644\u0645\u062d\u0648\u0631 \u0628\u064a\u0646\u0647\u0645\u0627 \u0645\u0627\u0626\u0639 \u0644\u0627 \u0627\u0646\u0636\u063a\u0627\u0637\u064a- \u0644\u0632\u062c .\u0630\u0644\u0643 \u0628\u062d\u0644 \u0645\u0639\u0627\u062f\u0644\u0627\u062a \u0646\u0627\u064a\u0641\u0631- \u0627\u0633\u062a\u0648\u0643\u0633 \u0641\u064a \u0627\u0644\u0627\u062d\u062f\u0627\u062b\u064a\u0627\u062a \u0627\u0644\u0627\u0633\u0637\u0648\u0627\u0646\u064a\u0647. \u062a\u0645 \u062a\u0628\u0633\u064a\u0637 \u0627\u0644\u0645\u0639\u0627\u062f\u0644\u0627\u062a \u0648\u062d\u0635\u0644\u0646\u0627 \u062a\u062d\u0644\u064a\u0644\u064a\u0627 \u0639\u0644\u0649 \u062f\u0627\u0644\u0629 \u0628\u064a\u0633\u064a\u0644 \u0645\u0646 \u0627\u0644\u0631\u062a\u0628\u0629 \u0627\u0644\u0635\u0641\u0631\u064a\u0629 \u0641\u064a \u0645\u062a\u063a\u064a\u0631 \u0648\u0627\u062d\u062f.<\/p>\n

<\/a>1.INTRODUCTION<\/strong><\/p>\n

The<\/a> Navier -Stokes’s equations for the velocity <\/a> and the pressure can be written in the form<\/p>\n

in [1]<\/p>\n

Where<\/p>\n

And<\/p>\n

in [2]<\/p>\n

The continuity equation in the cylindrical coordinates is given by,<\/p>\n

In [3]<\/p>\n

These aforementioned equations allow a <\/a>stationary solution of the form<\/p>\n

Thus, Navier -Stokes’s equations<\/p>\n

Reduced to<\/p>\n

and<\/p>\n

in [ 3]<\/p>\n

2.THE <\/strong><\/a>PERTURBATION EQUATIONS AND THE NORMAL MODE<\/strong><\/p>\n

In order to investigate the solutions of the flow system described by equa\u00adtions (1.9) We consider an infinitesimal of the basic flow is given by (1.8) by assuming that the perturbed flow is given by<\/p>\n

Assuming that the various perturbations are axisymmetric and inde\u00adpendent of , and From (1.1) – (1.3) we gain the following linearized equations as<\/p>\n

In [3]<\/p>\n

and<\/p>\n

where is defined by<\/p>\n

And the equation of continuity reduces to<\/p>\n

By analyzing the disturbance into normal modes. We assume that the disturbances are of the following form<\/p>\n

in[4]<\/p>\n

where k is the wave number of the disturbance in the axial direction, and p<\/em> is a constant which can be complex.<\/p>\n

Substituting (2.7) in equations (2.2) – (2.6), we get<\/p>\n

\n

In[3]<\/p>\n

\n

substitute in equation (2.10), we obtain<\/p>\n

= (2.14)<\/p>\n

\n

From equation (2.14), we find<\/p>\n

\n

<\/a> Substituting (2.15) in equation (2.8), yields<\/p>\n

By multiplying equation (2.9) by ), and multiplying equation (2.16) by P<\/p>\n

We obtain,<\/p>\n

\n

Now, summation equation (2.17) to equation (2.18), we obtain<\/p>\n

\n

But, , therefore <\/a> – (2.20)<\/p>\n

angular velocity is<\/p>\n

in[1]<\/p>\n

where is a real function<\/p>\n

By Substituting (2.22) at equation (2.21) We get:<\/p>\n

\n

\n

Equation (2.24) is a Zero \u2013 Order Bessel’s Function, with the solution<\/p>\n

in[5]<\/p>\n

3.RESULT AND CONCLUSION<\/strong><\/p>\n

(1) Bessel’s Function are closely associated with problems processing circular or cylindrical symmetry, because of their close association with cylindrical domains.in[5]<\/p>\n

(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.<\/p>\n

\n

<\/a> either<\/p>\n

REFERENCES<\/strong><\/p>\n

[1] S. Chandrasekhar. ‘Hydrodynamic and Hydromagnetic Stability<\/strong>‘, Dover, New York, 1961.<\/p>\n

[2] Murray R.Spiegel , \u201cVector Analysis and An Introduction To Tensor Analysis<\/strong>\u201d,Rensselaer Polytechnic Institute ,1959.<\/p>\n

[3] Hua-Shu Dou, Boo Cheong Khoo2 , and Khoon Seng Yeo,\u201d Instability of Taylor-Couette Flow between Concentric Rotating Cylinders\u201d<\/strong>, Inter. J. of Thermal Science, 2008.<\/p>\n

[4] Kyungyoon Min and Richard M. Lueptow,\u201d Hydrodyna ic stability of viscous flow between rotating Porous cylinders with radial flow<\/strong>\u201d, Department of Mechanical Engineering, Northwestern University, Evanston, Illinois 60208 (Received 28 January 1993; accepted 1 September 1993.<\/p>\n

[5]LARRY C. ANDREWS , ‘Special functions of mathematics for engineers’<\/strong>, Bellingham, Washington USA ,1998.<\/p>\n

<\/h2>\n

\n","protected":false},"excerpt":{"rendered":"

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, […]<\/p>\n","protected":false},"author":2,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"_joinchat":[],"footnotes":""},"class_list":["post-4577","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.hnjournal.net\/en\/wp-json\/wp\/v2\/pages\/4577","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.hnjournal.net\/en\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.hnjournal.net\/en\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.hnjournal.net\/en\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.hnjournal.net\/en\/wp-json\/wp\/v2\/comments?post=4577"}],"version-history":[{"count":1,"href":"https:\/\/www.hnjournal.net\/en\/wp-json\/wp\/v2\/pages\/4577\/revisions"}],"predecessor-version":[{"id":4589,"href":"https:\/\/www.hnjournal.net\/en\/wp-json\/wp\/v2\/pages\/4577\/revisions\/4589"}],"wp:attachment":[{"href":"https:\/\/www.hnjournal.net\/en\/wp-json\/wp\/v2\/media?parent=4577"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}