CC BY
12
% 1И..1..й11?
2022. Т. 40. С. 15—33
Онлайн-доступ к журналу: http://mathizv.isu.ru
Серия «Математика»
Research article УДК 518.517
MSC 34E10, 37N25, 76D05, 76M45, 76Z99, 92B99 DOI https://doi.org/10.26516/1997-7670.2022.40.15
Rotational Reaction over Infected Covid-19 on Human Respiratory Tract in the Presence of Soret Effect with Hall Current
T. Padmavathi1, S. Senthamilselvi1, S. S. Santra2, V. Govindan3, M. Altanji4, S. Noeiaghdam5'6®
1 Vels University, Pallavaram, India
2 JIS College of Engineering, Kalyani, India
3 Phuket Rajabhat University, Phuket, Thailand
4 College of Science, King Khalid University, Abha, Saudi Arabia
5 Irkutsk National Research Technical University, Irkutsk, Russian Federation
6 South Ural State University, Chelyabinsk, Russian Federation El noiagdams@susu.ru
Abstract. Corona virus infects the ciliated cells in the human nasal epithelium. Lung disease and diabetes are enlarged risks of severe breakdown against COVID-19. Cilia are hair-like construction enhanced from the celluloid into the pleural fluid surrounding the cell. Infection is detected in the lungs, a pleural disorder generates a Pleural Effusion. The present paper developed mathematical model intent to analyze the rotational consequence cause of COVID-19 on the human respiratory tract surrounded by the porous medium in the presence of the Soret effect with hall current under the force of the magnetic field. The respiratory tract mechanism of biological flows with the physiological process is observed. The non-dimensional governing equations are solved using the perturbation technique and the numerical computation results are exposed in the form of graphs. The effects of several parameters such as the Hartmann number, Schmidt number, Prandtl number, Soret parameter on the velocity, temperature and concentration fields are determined with their significance.
Keywords: cilia, COVID-19, hall current, magnetic field, pleural fluid, rotation parameter, Soret effect
Acknowledgements: M. Altanji appreciates for the Deanship of Scientific Research at King Khalid University for funding through the research group program under grant number R.G.P2/150/43.
For citation: PadmavathiT., SenthamilselviS., SantraS. S., GovindanV., AltanjiM., Noeiaghdam S.Rotational Reaction over Infected Covid-19 on Human Respiratory Tract
in the Presence of Soret Effect with Hall Current. The Bulletin of Irkutsk State University. Series Mathematics, 2022, vol. 40, pp. 15-33. https://doi.org/10.26516/1997-7670.2022.40.15
Научная статья
Влияние вращения с учетом эффекта Соре и тока Холла в дыхательных путях человека при заражении COVID-19
Т. Падмавати1, С. Сентхамилсельви1, Ш. С. Сантра2, В.Говиндан3, М.Алтанджи4, С. Нойягдам5'6н
1 Университет Велса, Паллаварам, Индия
2 Инженерный колледж JIS, Кальяни, Индия
3 Университет Пхукета Раджабхат, Пхукет, Таиланд
4 Университет Короля Халида, Абха, Саудовская Аравия
5 Иркутский национальный исследовательский университет, Иркутск, Российская Федерация
6 Южно-Уральский государственный университет, Челябинск, Россия И noiagdams@susu.ru
Аннотация. Вирус короны поражает реснитчатые клетки эпителия носа человека. Заболевания легких и диабет повышают риск серьезного срыва в борьбе с COVID-19. Реснички представляют собой волосоподобную конструкцию, увеличенную из целлулоида в плевральную жидкость, окружающую клетку. Инфекция обнаруживается в легких, нарушение плевры приводит к плевральному выпоту. В исследовании разработана математическая модель, предназначенная для анализа вращательного воздействия COVID-19 на дыхательные пути человека, окруженные пористой средой, при наличии эффекта Соре с током Холла под действием силы магнитного поля. В дыхательных путях наблюдается механизм биологических потоков с физиологическим процессом. Безразмерные управляющие уравнения решаются с использованием метода возмущений, а результаты численных вычислений представлены в виде графиков. Влияние нескольких параметров, таких как число Хартмана, число Шмидта, число Прандтля, параметр Соре, на поля скорости, температуры и концентрации определяется их значимостью.
Ключевые слова: реснички, COVID-19, ток Холла, магнитное поле, плевральная жидкость, параметр вращения, эффект Соре
Благодарности: М. Алтонджи благодарит Декана научных исследований Университета Короля Халида за финансирование в рамках программы исследовательской группы в рамках гранта № R.G.P2/150/43.
Ссылка для цитирования: PadmavathiT., SenthamilselviS., SantraS. S., Govin-dan V., Altanji M., Noeiaghdam S. Rotational Reaction over Infected Covid-19 on Human Respiratory Tract in the Presence of Soret Effect with Hall Current // Известия Иркутского государственного университета. Серия Математика. 2022. Т. 40. C. 15-33. https://doi.org/10.26516/1997-7670.2022.40.15
1. Introduction
Considering the presence of poisonous synthetic compounds, dirt particles, microbes and infections breathed in air, lungs are unusually impervious to ecological injury. The larynx,the bronchioles, the windpipe, the stem bronchi, and all the aviation routes ramifying seriously inside the lungs, like the intrapulmonary bronchi by the lower aviation route framework comprises and the alveolar channels. Upper aviation helps to breath properly. It consists of nasal passages and nose, the pharynx, paranasal sinuses and the portion of Larynx. Cilia shows a significant part for mucous clearance that is fundamental for typical lung mechanism.
Balamurugan and Karthikeyan [1] investigated the target is hydro attractive oscillatory flow through permeable medium limited by two vertical permeable plates with Dufour and heat source impact within the sight reaction of chemical. Hans-Jurgen Mager et al. [2] considered the intrapleural scattering of powder suspension after instillation in patients with suggestive threatening pleuritis and dissected the impact of pivot on the distribution example of talc in the pleural pit, and the impact of a rotational convention on the general accomplishment of powder pleurodesis. Sharma and Gaur [3] examined the significance of the Soret and Dufour impact within the sight of warm radiation in blood stream in a Newtonian model. Chemical response is additionally read for the blood stream within the sight of gentle stenosis in permeable tightened corridor. In an another paper on Prema and Muthu-cumaraswamy [4] have discussed thermal radiation with the occupation of Hall Effect. Muthucumaraswamy et al. [5; 6] discovered The impacts of rotation on the hydromagnetic free-convection stream of an incompressible liquid sped up limitless vertical plate with different temperature and uniform mass dissemination, within the sight of a substance response also he found that is helpful in attractive control of liquid iron stream in the steel business, fluid metal cooling in atomic reactors, attractive concealment of liquid semi-leading materials and meteorology. Raptis and Singh [7] established the same rotation effects on vertical plate in an existence of the magnetic field. He believed that temperature variation in the temperature plate occurs due to natural convection in the boundary layer. In [8] the authors analysed the consistent MHD limit layer stream cause of a dramatically extending sheet with radiation within the sight of exchange of mass, impacts of Soret and Dufour, and warmth source or sink. Ferdowset et al. [9] inspected the impacts of Dufour and Soret boundaries on the warmth and mass transfer attributes of an upward permeable plate in a permeable medium is broke down. sixth order Runge-Kutta integration program used together with the Nachtsheim- Swigert shooting iteration method. Piccoli et al. [10] figured out an audit of penetrating guidelines have been enforced to accomplish the natural targets furthermore, secure human wellbeing. Free and forced convective flow in pleural fluid with effect of injection has
been discussed by Padmavathi et al. [11]. Also effect of ventricular elasticity due to congenital hydrocephalus and effect of hartmann number on natural convection of pleural effusion can be found in [12; 13].
The current research work investigates on MCC by ciliary movement within the sight of inertial powers by the assistance rotational reaction of mass, force and energy preservation of the human respiratory tract in the presence of soret and hall effect over infected COVID-19 is introduced in the following assumptions and the calculations are made with the support of regular perturbation technique. More information about COVID models based on various functions and parameters can be found in [14; 15].
2. Mathematical formulation
We deal with an unstable non-uniform cilia evolution is essential ciliary dyskinesia. The present model formulated with the assumption of an oscillating flow over a penetrable channel surrounded by both permeable tissue influence of Hall current with uniform heat absorption parameter concealed by homogeneous Magnetic effect has been considered. Free convection stream of an incompressible viscous electrically leading and temperature subordinate warmth engrossing and optimal dainty liquid past a moving limited vertical layer installed in a permeable medium considering soret effect in the presence of hall current. At first time subordinate slip limit circumstance on the analysis of permeable medium at t < 0, the area is believed to be, on the equivalent temperature T and focus С. While t > 0, the temperature of the spot is promptly sped up to T0 and focus upheaved to C0 with perceive to time and kept up with consistent.Due to gravitational force thickness of pollutant mucus liquid is not generally steady and the temperature between the garbage are uniform at some stage in the smooth motion because of lightness pressure. The pollutant microorganisms (COVID-19) are circular (crown like position), non-undertaking, and equivalent size and methodicallly dispensed in the liquid release place. The Reynolds number is appropriated to be adequately small that non-Darcy impacts are insignificant, which is absolutely sensible in the respiratory aviation routes. An acquired issue that prompts hindered mucociliary clearance, rehashed disease of the human chest We have fixed co-ordinate arrangement x-axis along the length of the medium towards skyward direction and z-axis typical to the plate in the mucous liquid. Transverse Magnetic field B0 is applied lateral to z-axis. Both the plate and liquid are in unbending body revolution with uniform precise speed Q about z-axis.In order to kept One stage is fixed and another stage is swaying with uniform speed. In this model cilia act as a resistance parameter. Resistant cells are particularly described by an increment in the number and additionally length of cilia with modified underlying components.
Dismissing the Joulean heat scattering and applying Boussinesq's approximation the administering conditions of the stream field are composed Dr.K. Balamurugan et al., [1] and extended the study with mucous resistance parameter term as follows.
Figure 1. Physical configuration of the mucus layer
dw
—— = 0; wo = constant (2.1)
dy
ff + wo if + 2Qw = v (0) + gß(T - Tc ) + gß*(C - Cc) - -TÎrS) (u + mw) + ^-^u
-(1+m2) v ' -
dw dw ( d2w\ aß2 KCNC
-+wo — -^ = «-^(ITm2T(w-mu) + —w M (dT dT\ kt fd2T\ jc
\dT + wodT) = KT {W) + Uëp(T(2.4)
ic ic d2c dckt fi2T\
de +wo ddC = DHy2 + DK W) (2.5)
Boundary condition purposes are fundamental for characterizing an issue and simultaneously essential significance of fluid flow transport. Here,the lung limit is viewed as a moving no-slip, impervious divider and the speed parts are chosen to such an extent that there is no mass transition through any of the control surfaces on the moving dividers. It is hard to endorse a practical limit condition at the parts which was hindered because of decrease of the size of the respiratory framework. We used Initial and Boundary Condition for the present mucous fluid flow Momentum, Temperature and Concentration transportation are given below,
u = 0, w = 0, T = To + e(To — Tc) cos Xt, C = Co + e(Co — Cc) cos Xt at y = 0
u = Uo(1 + cosXt),w = 0, T = Tc,C = Ccat y = 1.
To transform the dimension-less form of fundamental governing equations and the limit conditions in dimensional structure, the accompanying non-dimensional amounts are presented.
y tw0 wL u - T — Tc - C — Cc y = —, - = ——, w =-, u = — ; 0 = - —
V L ' wo' Uo' To — Tc Co -Cc'
where u, w and x, y are velocity co-ordinates of the mucus flow along with the cartesian coordinates, p is density of the mucus fluid, v is kinematic viscosity of the mucus fluid, a is magnetic permeability of the fluid, fi-Coefficient of volumetric thermal expansion, Kc is co-efficient of Stoke's resistance, Nc is the number of density of the foreign particles. ft * Coefficient of volumetric concentration expansion, indicates gravitational acceleration, Dc is mass diffusion of the mucus fluid co-efficient. cp is Specific heat at Constant pressure, Tc is temperature wall of mucus layer, Cc is concentration wall of mucus layer, T0 is Steam temperature of mucus, C0 is Steam concentration of mucus, ,JC is co-efficient of Heat absorption. J is Heat absorption parameter, S is Suction parameter, Kt represents Thermal conductivity, Reis Reynolds Number, t is Non-dimensional time coordinate, w0 is constant Injection/Suction, m is Hall parameter. U0 represents Uniform Speed of the tissue, A is Periodicity of oscillatory motion, is concertation of Electromagnetic Field. M2 is Hartmann Number, Q is coefficient of Rotation parameter, Sc is Schmidt number, Rpm is Resistance parameter, Pe is Peclet Number, Sris Soret Number, Lrepresents length between two-tissue region in the respiratory tract. Temperature of fluid flow in the respiratory tract, 0 is transport diffusivity of the mucosal fluid, is perturbation parameter. Transformed dimensional form of general governing equation as follows, upcoming dimensionless equation we are removing" symbol for our comfort zone.
du du 1 d2u M2
dt + dU+2QW = Redf +ReGr O+ReGc0— Re(1 + m2)(u+mw) —R?m
(2.6)
dw dw ^ 1 d2w M2
~dt + % — 2Qu = ReW — Re(1 + m2)(mu — w) — (2J)
dd dd 1 d2Q TQ
dt + dy = P~eW2 + (2.8)
дф дф_ 1 д2ф Sr д20 dt + Oy = ReSc ду2 + ~КеЩ2
An accounting transformed initial and boundary conditions are,
u = 0, w = 0, 9 = 0 = 1 at y = 0 u = 1, w = 0, 0 = 0 = 0 at y = 1.
Dimensionless parameter followed by
vgpjTa —TC)L
Uo w02 vg/3*(Co - Cc)L
Uow2
Gr =-——— [Crash of Numberj;
Gc =-c—2—"— [Solutal Crash of Numbe rj
Re = WoL [Reynolds Numbe rj; Q = [Rotation paramet erj
M2 aB2L2
M2 = -^v; M2 = —0— [Hartmann Numbe rj
Re(1 + m2) ^
Pe = PCpw°L [peclet Numbe rj; J = ^L [Heat Sour cej Kt pCpWo
Sr = DfTCo -T)) S^t Numbe.rj; = ^
vTm(Co -Cc) P
[ Re s i s t an c e paramet erj.
Proposed Complex Velocity form is F = u + iw, M2 = ^^^ (1 — im), we express the Equation (2.2) and (2.3) perhaps joined into a solitary condition of the structure as,
dF dF 1 d2F
— + ^ = — ^ + ReGr0 + ReGc0 — M2F — 2/QF — R^F (2.10) dt dy Re dy2
Upon initial and boundary conditions in dimensionless form as follows, F = 0, at y = 0; F = 1 + | (eiXt + e_iAi) at y = 1.
3. Method of solution
A few techniques can be utilized to assess cilia design and capacity in many people. Tests of aviation route ciliated cells for study can be acquired after death or from living subjects straightforwardly from the lung by means of bronchoscopy biopsy. Mucociliary clearance depend upon profoundly planned integrate beating of cilia beyond various ciliated cells. Fundamental arrangement of non-dimensional conditions are coupled, nonlinear partial differential conditions and these can't be evaluated with out of or-der.which can be tackled analytically. This should be possible by expecting
the preliminary answers for the velocity, temperature and diffusion of the fluid flow with the analytical method using basic perturbation technique.
F(y, t) = Fo(y) + 2éxtFi(y) + 2е-xtF2(y)
%, t) = 0Q(y) + 2 eiXt el(y) + 2 e-iAi e2(y)
¿(y, t) = Mv) + 2eiAVi(y) + 2^Mv)
we are impelling to communicate the Momentum, Temperature and Diffusion conditions be subsequent to
Base, first and second part of energy equation:
2 w +J-Pe%=° ^
-j-2 - Pe^r + (J + iX)Pe°i = 0 (3-2)
ay2 ay
^ - pe(^l + (J - i\)pe01 = 0. (3.3)
ay2 ay
In order to corresponding boundary conditions are transformed as,
9q = 9i = 6>2 = 1 at y = 0; в0 = 6>i = в2 = 0 at у = 1.
Base, first and second part of concentration equation:
« - ReSc^p. = -SrSc^ (3.4) ay2 ay ay2
^ - ResM1 - (i\ScRe)<pi = -SrSc^ (3.5) ay2 ay ay2
^ - ReSc^p- + (гXScRe)<i = -SrSc^. (3.6) ay2 ay ay2
In order to initial and boundary conditions are,
<o = pi = <2 = 1 at у = 0; <o = pi = <2 = 0 at у = 1.
Base, first and second part of momentum equation: d2 F F
d-F0 — — (M2 — 2IÙ — Rpm)Fo = —Re2Gr do — Re2Gc0o
d 2 d
(3.7)
d2 F d F
—1 — Red-1 — (M2 — 2IÙ + IXRe — Rvm)Fi = —Re2Gr Oi — Re2Gc0i dy2 dy
(3.8)
d2 F d F
—2 — Re—2 — (M2 — 2/Q + IXRe — Rpm)F2 = —Re2Gr 62 — Re2Gc02. dy2 dy
(3.9)
In order to boundary conditions,
Fo = Fi = F2 = 0 at y = 0; Fo = Fi = F2 = 1 at y = 1.
Subject to the limit conditions, are settled by the typical irritation procedure and the arrangements are inferred as follows:
do = Aiemiy + A2 e™2y (3.10)
Qi = A3 e™3V + A4 e™4y (3.11)
O2 = A5em5y + Ae e™6y (3.12)
00 = A7em7y + As emsy + A25emiy + A26 em2 y (3.13)
01 = A9em9y + Aio emioy + A27e™3y + A28 e™4y (3.14) = A11 emiiy + A12 e™i2y + A29 e™5y + Aao e™6y (3.15)
Fo = A13 emi3y + A14 emi4y + A31 e™iy + A32 e™2y + A33 emry
+ A34emsy + A35 emiy + A36 e™2^ (3.16) Fi = A15 e™i5y + Aie emi6V + A37e™3^ + A38 e™4V + A39 emsï
+ A4oemi0y + A41 e™3 y + A42 e™4y (3.17) F2 = A17 e™i7y + Ai8 emisy + A43 e™5^ + A44 e™6y + A45 emiiy
+ A46 e™i2y + A47 e™5 y + A48 em6y (3.18)
More details can be found in Appendix 1.
4. Results and discussion
For actual comprehension of the present problem calculations are proposed mathematically for various boundaries like Hall parameter, Reynolds Number, Schmidt Number, Soret Number, Hartmann Number, Peclet
Number, Heat absorption Number, Rotation parameter, Grashof Number, Modified Grashof Number,speeding up boundary and time upon the idea of the stream and transport.Velocity F, Temperature T and Concentration C have been concentrated systematically and processed consequences of the insightful arrangements, introduced by equations (3.10) to (3.18) and the results are shown graphically from Figs.2 to 18. In the current research work following default boundary esteems are embraced for calculations: Pe = 10; t = 0.2; J = 1; Re = 0.01; Sc = 0.56; Sr = 0.5; Ô = 1;M2 = 0.5; Gr = 0.5; Gc = 0.5; e = 0.02; A = 0.3; Rpm = 1.
Fig. 2-(a) shows that increased Peclet Number increasing Concentration field in the mucous layer. It is possible that more modest Peclet numbers could exist for carbon dioxide or inactive gases, yet the "rate" constants for these blood-gasses are regularly huge. For Peclet numbers more noteworthy than fifty, the end-fine arrangements are basically indistinguishable.
y
(e) Sc
Figure 2. Deviating Parameters Effect on Concentration Field
Fig. 2-(b) displays that Respiratory contaminations can be brought about by chill air through expanded bronchial irritation brought about by relationship of provoke aspect, for example, cold and diseases are the pair ready to destabilize the patient to dissipate quicker, it may be supplanted
very well. Effect of concentration field is increases as heat absorption parameter increases.
Fig. 2-(c) Portrayed that enhancing Reynolds number diminishing the concentration wave field on the ciliary tip. Mucus layer needs inertial force of the center to sustain the velocity. During inward breath, higher momentum of wind current happened in automated oxygenating comparatively ordinary breathing in the bronchioles, which infers stream was towards the deadline of the section.In opposition, at the time of expiration, the velocity of wind stream during ordinary breathing was quicker than during automated oxygenation as shown in Figure 2-(c).
Fig. 2-(d) represented that increasing Soret number dominating the Concentration field. The Soret effect appears in suspended mixtures of microbe particles and mucous fluids. This phenomenon is caused by the temperature gradient, so that the movement of fluid molecules in the hot zone and high energy levels in this region displaces the small particles toward the frozen region of the ciliated sector.
Fig. 2-(e) exhibit that rising Schmidt number lessen the concentration profile on the mucous layer.
It is investigated that symplectic wave rule over the antiplectic wave for the mucociliary freedom in the aviation routes. Figs. 3-(a) to 3-(j) established that increasing various dimensional parameters are (M2, Sc, Gr, Gc, Pe, Re, J, Sr, Q, Rpm) decreases magnitude of velocity field.rotate the model to see the connection between the stomach and the lungs. The lower respiratory framework, or lower respiratory section, comprises of the windpipe, the bronchi and bronchioles, and the alveoli, which make up the lungs. Typical breathing Fig. 3-(e) were seen as far as vortices age that might be because of the distinction in stream rate and Reynolds number.
Fig. 4-(a) represents that occurrence of cool air in the transportation of mucociliary progress heat absorption might be play vital role to reduce the temperature in the respiratory tract. As it also defines that J increases temperature profile decreases. Kim et al., [16] we intend to address these limits by inspecting the divider and tissue impact through a liquid design association and demonstrated the entire lung including stream examples and qualities inside the respiratory framework.
Fig. 4-(b) shows that increasing Peclet number decreasing the temperature distribution.In this unique circumstance, the cell societies manage the cost of an exploratory framework with a tunable Peclet number, which can be differed by changing either the viscoelasticity of the bodily fluid layer, the driving conditions for bodily fluid vehicle in the model by adjusting mucus fluid, the constraining conditions at the bodily fluid epithelial connection point.
(i) Gc (j) Rpm
Figure 3. Deviating Parameter Effect on Velocity Field 5. Conclusion
An examination of the impacts of Soret number, hall current,substance reaction parameter also, magnetic parameter, rotation parameter on tem-
(a) J (b) Pe
Figure 4- Deviating Parameters Effect on Temperature Field
peramental hydromagnetic free convection stream with heat and mass move of a thick, incompressible electrically conducting passed through a moving vertical mucous layer permeable medium is investigated analytically with the help of regular perturbation technique. Significant discoveries are as per the following:
— Concentration profile diminished while enhancing Reynolds number.
— Boost up the concentration profile expanding Peclet number parameter.
— Lessen the concentration profile rising Schmidt number.
— Dominating the Concentration field increasing Soret number.
— Decreases magnitude of velocity profile with increasing dimensional parameters
— Augmenting Peclet number diminishing the temperature dissemination.
— Heat absorption parameter increases temperature profile decreases and concentration profile increases.
Sidra Shaheen Et al., [17] The present problem shows that mucociliary leeway in the aviation route can be constrained by the consistency and versatility of the mucus fluid within the sight of velocity, warm, absorption and inertial development.
Nomenclature
MCC ^ Mucuociliary Clearance HCoV ^ Human Covid
u,w ^ x and y velocity co-ordinates of the mucus flow
cp ^ Specific Heat at Constant pressure
Tc ^ Wall temperature of mucus
Cc ^ Wall concentration of mucus
T0 ^ Steam temperature of mucus
C0 ^ Steam concentration of mucus
J ^ Heat absorption paramter
S ^ Suction paramter
Kt ^ Thermal conductivity
Re ^ Reynolds Number
t ^ Non-dimensional time coordinate
M2 ^ Hartmann Number
Q ^ Rotation parameter
Sc ^ Schmidt number
Rpm ^ Resistance parameter
Pe ^ Peclet Number
Sr ^ Soret Number
Sc ^ Schmidt Number
F ^ Velocity of the fluid flow in the respiratory tract 9 ^ Temperature of fluid flow in the respiratory tract f ^ transport diffusivity of the mucosal fluid
Appendix 1
Pe + JTPef—4JPe Pe -J (P e)2 - 4JPe
mi =-*—2-; m2 =--—2-
Pe + д/(Pe)2 - 4Pe(J + iX) Pe - ^(Pe)2 - 4Pe(J + iX) m3 =-2-; m4 =-2-
Pe + д/(Pe)2 - 4P e (J - i\) Pe -J (P e)2 - 4Pe( J - iX) m5 =-2-; me =-2-
m7 = 0; m8 = Sc * Re
(Sc * Re) + д/(Sc * Re)2 + 4гXScRe
mg = mio = mii = mi2 =
2_'
(Sc * Re) - д/(Sc * Re)2 + 4iXScRe 2
(Sc * Re) + д/(Sc * Re)2 - 4iXScRe 2 ; (Sc * Re) - ^(Sc * Re)2 - 4iXScRe 2
Re + д/(Re)2 + 4(M2 + 2Ш) Re - ^(Re)2 + 4(M2 + 2Ш)
mi3 =---; mi4
mi5 = mie =
2 ' 2
Re + д/(Re)2 + 4(M2 - 2Ш + ¿XRe) 2 ; Re - ^(Re)2 + 4(M~2 - 2Ш + iXRe) 2
Re W(ße)2 + 4(M2 - 2iQ - i\Re)
m17 =-2-;
Re - V(Re)2 + 4(M2 - 2iQ - iXRe) mi8 =-2-
emi em3
Ai = 1 - A2; A2 =-; A3 = 1 -A4; A4 =-;
' emi_em2 em3_em4
A5 = 1 - A;
ems
A = em5 - em6 ; A7 = 1 A8-A25-A26;
1 - A25 (em7 - em2) - A26 (em7 - emi) A8 = -
8 e>m7_ems
A9 = 1 -Ai0 -A27 -A28;
1 - A27(em9 - em4) - A28(em9 - em3)
10
A.o
emg _ emio
Aii = 1 -A12 -A29 -A30;
1 - A29(emi1 - em6) - A30(emi2 - em4)
A
i2
mii mi2
Re2 Gr Re2Gc
Re 2Gr Re 2Gc . 1 +
, , -------------- , M2 +2iQ M2+2iQ,
Ai3 = Ai4 - -rz——TTTs - T7—TTTTs ; Ai4 =
M2 + 2iQ M2 + 2iQ' 14 emi4 - emi3
Re2 G Re2 G
M2 - 2iQ + iXRe M2 - 2iQ + iXRe
Re2 Gr Re2Gc
Ai5 = Ai6
Ä 1 + M2-2iQ+i\Re M2-2iQ+i\Re
Ai6 = -
16 emi6 — emis
Re2 G Re2 G
Ai7 = Ai6 -
1 +
M2 - 2iQ - iXRe M2 - 2iQ - iXRe
Re2 Gr Re2Gc
a M2-2iQ-i\Re M2-2iQ-iXRe
Ai8 = -
18 emis — emi7
( m2)2 mi ( mi)2 m2 ( m4)2 m3
Ai9 —-; A20 —-; A21 —-
19 mi m2 20 mi m2 2i m3 m4
. (m3)2 em4 (m6)2ems (m5)2em6 A22 —-; A23 —-; A24 —-
22 em3 — em4 23 em5 — em6 24 ems — em6
Sr * Sc * Ai9 Sr * Sc * A20 A25 = -7—^—77;—-7; A26 =
A27
A28
(m2)2 - ( Sc * Re * m2)' (mi)2 - (Sc * Re * mi)
S * S * A2i (m4)2 - ( Sc * Re * m4) - iXScRe' S * S * A22
(m3)2 - (Sc * Re * m3) - iXScRe
Бг * Бс * А23
А-29 = Т- \2-
А30 А31 А32 А33 А34 А35 А36 А37 = -
^38 = -
А39 А40 А41 А42 А43 А44 А45 А46
(те)2 — (Бс * Ее * т6) — гХБсЕе' Бг * Бс * А24 (т5)2 — (Бс * Ке * т5) — гХБсЕе Ее2СтАх
(т1)2 — (Ее * т{) — (М2 + 2гП)'
Ке2СгА2_
(т,2)2 — (Ее * т,2) — (М2 + 2гП)
Ке2СсА7 (т,7)2 — (Ее * т,7) — (М2 + 2гП)1 Ее2СсА8
(Ш8)2 — (Ее * т8) — (М2 + 2гй) Ее2СсА25
(т,2)2 — (Ее * т,2) — (М2 + 2гй)1 Ее2СсА26
(т1)2 — (Ее * т\) — (М2 + 2гй) Ее2СтАъ
(т,3)2 — (Ее * Ш3) — (М2 — 2Ш + гХЕе)1 Ее2СгА4
(т4)2 — (Ее * Ш4) — (М2 — 2Ш + гХЕе) Ее2СсА9
(т9)2 — (Ее * т9) — (М2 — 2гй + гХЕе)1 Ее2СсАю
(тю)2 — - (Ее -- * тю) — (М2 — 2гП + гХЕе) Ее2СсА27
(т3)2 — (Ее * т3) — (М2 — 2гй + гХЕе)1 Ее2СсА28
(т4)2 — (Ее * т4) — (М2 — 2Ш + гХЕе) Ее2СтАъ
(т5)2 — (Ее * т5) — (М2 — 2гй — гХЕе)1 Ее2СгАе
(те)2 — (Ее * те) — (М2 — 2Ш — гХЕе) Ее2СсАц
(тп)2 — - (Ее -- * тп) — (М2 — 2гП — гХЕе) Ее2СсАХ2
(т12)2 — (Ее * ту2) — (М2 — 2т — гХЕе)
A47
Re2GcA
-29
A48
(m5)2 - ( Re * m5) - (M2 - - iXRe) Re2GcA30
(me)2 - ( Re * me) - (M2 - 2iÜ - iXRe)
References
1.
2.
3.
4.
5.
6.
7.
8. 9.
10.
Balamurugan Dr.K., Karthikeyan R. Radiation Effects of MHD Oscillatory Rotation Flow Through a Porous Medium Bounded by Two Vertical Porous Plates in the Presence of Hall Current and Dufour Effect with Chemical Reaction. International Journal of Mathematics and Its Applications, 2016, vol. 4, iss. 1-B, pp. 47-56.
Mager H.-J., Maesen B., Verzijbergen F., Schramel F. Distribution of Talc Suspension During Treatment of Malignant Pleural Effusion With Talc Pleurodesis. Lung Cancer, 2002, vol. 36, pp. 77-81.
Sharma M., Gaur R.K., Sharma B.K. Radiation Effect on MHD Blood Flow Through A Tapered Porous Stenosed Artery with Thermal and Mass Diffusion. Int. J. Of Applied Mechanics and Engineering, 2019, vol. 24, iss. 2, pp. 411-423. https://doi.org/10.2478/Ijame-2019-0025.
Alias Prema K.M., Muthucumaraswamy R. Rotation Effects on Flow Past an Exponentially Accelerated Vertical Plate with Variable Temperature and Uniform Mass Diffusion in the Presence of Thermal Radiation. J. Applied Sciences Research, 2015, vol. 11, iss. 14, pp. 111-120.
Muthucumaraswamy R., Lal T., Ranganayakulu D. Rotation Effects on MHD Flow Past an Accelerated Vertical Plate with Variable Temperature and Uniform Mass Diffusion. Journal of Engineering, vol. 11, fascicule 1. (ISSN 1584-2665) Muthucumaraswamy R., Dhanasekar N., Easwara Prasad, G. MHD and Rotation Effects on Flow Past an Accelerated Vertical Plate With Variable Temperature and Mass Diffusion in the Presence of Chemical Reaction. Int. J. of Applied Mechanics and Engineering, 2013, vol. 18, iss. 4, pp. 1087-1097. https://doi.org/10.2478/ijame-2013-0068.
Raptis A., Singh A.K. Rotation Effects On MHD Free-Convection Flow Past An Accelerated Vertical Plate. Mechanics Research Communications, 1985, vol. 12, iss. 1, pp. 31-40.
Reddy B., Sudhakar M., Rushi B., Varma S. Soret and Dufour effects on MHD
boundary layer flow over stretching sheet with heat source/sink. Global Journal of
Pure and Applied Mathematics, 2015, vol. 11, iss. 5, pp. 2897-2916
Ferdows M., Tzirtzilakis E., Kaino K., Chen, C.-H. Soret and Dufour Effects on
Natural Convection Heat and Mass Transfer Flow in a Porous Medium Considering
Internal Heat Generation. Int. J. Appl. Math. Stat., 2008, vol. 13, iss. D08, pp.
36-48.
Piccoli A., Agresti V., Balzarini A., Bedogni M., Bonanno R., Collino E. Top-petti A.M. Modeling the effect of COVID-19 lockdown on mobility and NO2 concentration in the lombardy region. Atmosphere 2020, vol. 11, iss. 12, pp. 1-18. https://doi.org/10.3390/atmos11121319
Padmavathi T., Senthamil Selvi S., Santra S.S., Ali R., Govindan V., Noeiaghdam S., Nieto J.J. Free and Forced Convective Flow in Pleural Fluid with Effect of
Injection between Different Permeable Regions. Coatings, 2021, vol. 11, iss. 1313. https://doi.org/10.3390/coatings11111313
12. Balasundaram H., Sathiamoorthy S., Santra S.S., Ali R., Govindan V., Dreglea A., Noeiaghdam S. Effect of Ventricular Elasticity Due to Congenital Hydrocephalus. Symmetry, 2021, vol. 13, iss. 2087. https://doi.org/10.3390/sym13112087
13. Padmavathi T., Senthamilselvi Dr.S. Effect of Hartmann number on Natural Convection of Pleural Effusion. Turkish Journal of Computer and Mathematics Education, 2021, vol. 12, iss. 10, pp. 5672-5681.
14. Hedayati M., Ezzati R., Noeiaghdam S. New Procedures of a Fractional Order Model of Novel coronavirus (COVID-19) Outbreak via Wavelets Method. Axioms, 2021, vol. 10, iss. 122. https://doi.org/10.3390/axioms10020122
15. Noeiaghdam S., Micula S., Nieto J.J. Novel Technique to Control the Accuracy of a Nonlinear Fractional Order Model of COVID-19: Application of the CES-TAC Method and the CADNA Library. Mathematics, 2021, vol. 9, iss. 1321. https://doi.org/10.3390/math9121321
16. Kim J., Pidaparti R.M., Computational Analysis of Lung and Isolated Airway Bifurcations under Mechanical Ventilation and Normal Breathing. Fluids, 2021, vol. 6, iss. 388. https://doi.org/10.3390/fluids6110388
17. Shaheen S., Maqbool K., Beg O.A., Gul F. Thermal analysis of airway mucus clearance by ciliary activity in the presence of inertial forces. SN Appl. Sci., 2021, vol. 3, iss. 461. https://doi.org/10.1007/s42452-021-04439-1
Об авторах About the authors
Падмавати Т., канд. физ.-мат. наук, Университет Велса, Индия, 600117, Паллаварам, pathumyanmar@gmail.com или padmavathi.phd@velsuniv.ac.in, https://orcid.org/0000-0001-6073-0468
Сентхамилсельви С., д-р
физ.-мат. наук, доц., Университет Велса, Индия, 600117, Паллаварам, msselvi230@gmail.com
СантраШ. С., д-р физ.-мат. наук, доц., Инженерный колледж ЛБ, Кальяни, Индия, 741235, Западная Бенгалия, shyam01.math@gmail.com или
shyamsundar.santra@jiscollege.ac.in,
https://orcid.org/
0000-0001-9740-3081
Padmavathi T., Cand. Sci. (Phys.-Math.), Vels University, Pallavaram, 600117, India, pathumyanmar@gmail.com or padmavathi.phd@velsuniv.ac.in, https://orcid.org/0000-0001-6073-0468
Senthamilselvi S., Dr. Sci.
(Phys.-Math.), Assoc. Prof., Vels University, Pallavaram, 600117, India, msselvi230@gmail.com
SantraS.S., Dr. Sci. (Phys.-Math.), Assoc. Prof., JIS College of Engineering, Kalyani, West Bengal, 741235, India,
shyam01.math@gmail.com or
shyamsundar.santra@jiscollege.ac.in,
https://orcid.org/0000-0001-9740-3081
ГовинданВ., д-р физ.-мат. наук, доц., Университет Пхукета Раджабхат, Таиланд, 83000, Пхукет, govindoviya@gmail.com, https://orcid.org/0000-0003-4750-1964
АлтонджиМ., д-р физ.-мат. наук, доц., Научный колледж, Университет Короля Халида, Саудовская Аравия, 61413, Абха
Нойягдам С., д-р (PhD) прикладной математики, доц., Иркутский национальный исследовательский технический университет, Российская Федерация, 664074, г. Иркутск; Южно-Уральский государственный университет, Российская Федерация, 454080, г. Челябинск, snoei@istu.edu, noiagdams@susu.ru, https://orcid.org/0000-0002-2307-0891
GovindanV., Dr. Sci.
(Phys.-Math.), Assoc. Prof., Phuket Rajabhat University, Phuket, Thailand, 83000, govindoviya@gmail.com, https://orcid.org/0000-0003-4750-1964
AltanjiM., Dr. Sci. (Phys.-Math.), Assoc. Prof., College of Science, King Khalid University, Abha 61413, Saudi Arabia
Noeiaghdam S., PhD (Applied Mathematics), Assoc. Prof., Irkutsk National Research Technical University, Irkutsk, 664074, Russian Federation; South Ural State University, Chelyabinsk, 454080, Russian Federation, snoei@istu.edu, noiagdams@susu.ru, https://orcid.org/0000-0002-2307-0891
Поступила в 'редакцию / Received 11.12.2021 Поступила после рецензирования / Revised 22.01.2022 Принята к публикации / Accepted 15.02.2022
