AUTHOR=Asjad Muhammad Imran , Usman Muhammad , Assiri Taghreed A. , Ali Arfan , Tag-ElDin ElSayed M. 

TITLE=Numerical investigation of fractional Maxwell nano-fluids between two coaxial cylinders via the finite difference approach

JOURNAL=Frontiers in Materials

VOLUME=9

YEAR=2023

URL=https://www.frontiersin.org/journals/materials/articles/10.3389/fmats.2022.1050767

DOI=10.3389/fmats.2022.1050767

ISSN=2296-8016

ABSTRACT=<p>This study deals with numerical solution of momentum and heat transfer of fractional ordered Maxwell fluids within a coaxial cylinder. It is well known that the complex dynamics of flow regime can be well-described by the fractional approach. In this paper, a fractional differentiation operator <inline-formula id="inf1"><mml:math id="m1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mi>D</mml:mi><mml:mi>t</mml:mi><mml:mi>α</mml:mi></mml:msubsup></mml:mrow></mml:math></inline-formula> of Caputo was applied for fractional modeling of magneto-hydro-dynamic (MHD) fluid. A set of appropriate transformations was applied to make the governing equations dimensionless. The finite differences were calculated by the discretization of momentum profile <inline-formula id="inf2"><mml:math id="m2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>u</mml:mi><mml:mrow><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>r</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:mrow></mml:math></inline-formula> and heat profile <inline-formula id="inf3"><mml:math id="m3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>T</mml:mi><mml:mrow><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>r</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:mrow></mml:math></inline-formula>. The results obtained for <inline-formula id="inf4"><mml:math id="m4" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>u</mml:mi><mml:mrow><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>r</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:mrow></mml:math></inline-formula> and <inline-formula id="inf5"><mml:math id="m5" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>T</mml:mi><mml:mrow><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>r</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:mrow></mml:math></inline-formula> were plotted against different physical parameters, such as Prandtl number <inline-formula id="inf6"><mml:math id="m6" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi mathvariant="italic">Pr</mml:mi></mml:mrow></mml:math></inline-formula>, the square of Hartmann number <inline-formula id="inf7"><mml:math id="m7" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>, thermal Grashof number <inline-formula id="inf8"><mml:math id="m8" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>G</mml:mi><mml:mi>r</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:math></inline-formula> thermal radiation parameter <inline-formula id="inf9"><mml:math id="m9" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>N</mml:mi><mml:mi>r</mml:mi></mml:mrow></mml:math></inline-formula>, and heat source/sink parameter <inline-formula id="inf10"><mml:math id="m10" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula>. The results were verified by comparing data from the proposed method with MAPLE built-in command results. Subjecting the system to a strong magnetic field led to increasing <inline-formula id="inf11"><mml:math id="m11" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>T</mml:mi><mml:mrow><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>r</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:mrow></mml:math></inline-formula> and decreasing <inline-formula id="inf12"><mml:math id="m12" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>u</mml:mi><mml:mrow><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>r</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:mrow></mml:math></inline-formula>. It was found that increasing <inline-formula id="inf13"><mml:math id="m13" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>G</mml:mi><mml:mi>r</mml:mi><mml:mtext> </mml:mtext><mml:mi mathvariant="normal">a</mml:mi><mml:mi mathvariant="normal">n</mml:mi><mml:mi mathvariant="normal">d</mml:mi><mml:mtext> </mml:mtext><mml:mi mathvariant="italic">Pr</mml:mi></mml:mrow></mml:math></inline-formula> increased the velocity and temperature profiles. Addition of <inline-formula id="inf14"><mml:math id="m14" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>C</mml:mi><mml:mi>u</mml:mi></mml:mrow></mml:math></inline-formula> nanoparticles to a base fluid of <inline-formula id="inf15"><mml:math id="m15" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mi>O</mml:mi></mml:mrow></mml:math></inline-formula> enhanced its heat transfer capability. Also, increasing the angular frequency of inner cylinder velocity resulted in a high velocity profile of fractional Maxell nano-fluids within a coaxial region (cylinder).</p>