Variable-Properties-Of-Mhd-Third-Order-Fluid-With-Peristalsis.pdf

Results in Physics (2016) 963–972 Contents lists available at ScienceDirect Results in Physics journal homepage: www.journals.elsevier.com/results-in-physics Variable properties of MHD third order fluid with peristalsis T Latif a,⇑, N Alvi a, Q Hussain a, S Asghar a,b a b Department of Mathematics, COMSATS Institute of Information Technology, Islamabad 44000, Pakistan Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia a r t i c l e i n f o Article history: Received October 2016 Received in revised form November 2016 Accepted November 2016 Available online 12 November 2016 Keywords: Peristaltic flow Third order fluid Temperature dependent properties Viscous dissipation a b s t r a c t This article addresses the impact of temperature dependent variable properties on peristaltic flow of third order fluid in a symmetric channel The MHD fluid and viscous dissipation effects are taken into account Assumptions of long wavelength and low Reynolds number are employed to model the problem The governing nonlinear coupled equations are solved using perturbation method Approximate solutions are obtained for the stream function, temperature and pressure gradient The results are graphically analyzed with respect to various pertinent parameters Ó 2016 Published by Elsevier B.V This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/) Introduction Peristaltic flow is of great interest for engineers, mathematicians and biologists alike The popularity lies in its wide range of applications in both physiology and industry Peristaltic transport is the movement of biological fluids within the hollow vessels that successively contract and expand It is responsible for the movement of food through the digestive tract, urine from kidney to bladder, blood through arteries, bile juice in bile duct, sperm through vas deferens and eggs in the fallopian tube Peristalsis was first discussed by Bayliss and Starling [1] in medical terms in 1899 They described it as the motility in which there is both contraction and relaxation which propels the bolus of food through esophagus and intestines Mathematical analysis of peristalsis was started long after it was done in physiology Latham [2] provided the engineering analysis of peristaltic action for the first time in 1966 Shapiro [3] carried on Lathams work and investigated on renal peristaltic flows in infants Subsequently significant developments in peristaltic transport investigations were made by Fung [4], Barton and Raynor [5] and Weinberg et al [6] Peristaltic techniques are of practical significance in the industry, bioengineering and medical devices Using this mechanism, several industrial peristaltic pumps like roller, hose, finger and blood pumps, heart-lung machines and dialysis machines have been designed Peristalsis has been the subject of many recent research works as mentioned in the references [7–11] ⇑ Corresponding author Biological and industrial fluids are non-Newtonian in nature Newtonian fluids can be considered while studying peristalsis in the ureter but this approach is not good for peristaltic flow analysis in blood vessels, lymphatic vessels, intestine, reproductive tracts and esophagus Numerous researchers are now engaged in studying peristalsis in non-Newtonian fluids particularly viscoelastic fluids because of its applications in physiology, engineering, medical science and industry[12–18] Blood and almost all semi-solid edible things like bread, jam and yogurt possess both viscous and elastic properties [19,20] Third order fluid is a sub-class of viscoelastic fluids and it comprehensively represents the properties possessed by non-Newtonian fluids Continuum physicists and mathematicians have keen interest in third order fluid model Study of third order fluid flow problems provides a good deal of physical insight about elastic effects and develops the better understanding of mathematical procedures required for coping with nonlinear viscoelastic problems The ordered fluids provide good insight into some problems that cannot be studied by generalized Newtonian fluids or by linear viscoelastic models The third order fluid model has been used to examine flows in a wide variety of systems such as flow in eccentric annuli, flow in a channel, flow in a converging tube, radial flow between parallel disks, boundary layer flows and the motion of suspended, orientable particles Few researchers discussed the peristaltic action on third order fluid and are mentioned in [21–23] Several investigators studied the effects of MHD (magnetohydrodynamics) on peristaltic flows due to its importance in industry and medical sciences Such considerations have played key role in the design of MHD power generators, solidification processes of E-mail address: tanzeela@live.fr (T Latif) http://dx.doi.org/10.1016/j.rinp.2016.11.016 2211-3797/Ó 2016 Published by Elsevier B.V This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) 964 T Latif et al / Results in Physics (2016) 963–972 metals, cancer therapy, magnetic resonance imaging (MRI), petroleum industry and nuclear industry Recent work on MHD flows is cited in [10,12,14,16,24] Partial slip effect in peristaltic flow of third grade fluid is studied by Hayat and Mehmood [25] Heat transfer is the shifting of thermal energy from a body to the relatively cold ambient objects It is not the physiological manifestation of the phenomenon of peristaltic flow with heat transfer, but it is the mathematical description that has harnessed its potential to be implemented in industry Heat transfer occurs in many engineering processes like chemical distillatory processes, channel type solar energy collectors, heat exchangers, paper making and food processing The importance of studying heat exchange in biological systems is quite significant It has obvious involvements in hemodialysis, treatment of liver cancer and lung cancer Various researchers [26–29] have documented the effects of heat transfer for both Newtonian and non-Newtonian fluids Why this paper: In the existing literature, much attention has been given to the constant fluid viscosity which is not an efficient approach It is acceptable to consider viscosity to be constant for the isothermal fluids but if there is viscous dissipation, one cannot ignore the effects of variation of viscosity due to temperature difference [30] Its role is highly significant in peristaltic movement of physiological fluids such as polymer solutions, honey, blood and syrups The viscosity of these fluids cannot remain constant when there is temperature variation Few attempts have been made in which viscosity is taken as function of space coordinates [31,32] but it is more realistic to consider the temperature dependent viscosity as the temperature difference exists in biological systems In addition to that temperature dependent thermal conductivity is also another important fragment that must be accounted for Hence in the present paper, our emphasis is on the significance of temperature dependent viscosity and thermal conductivity This paper deals with the study of peristaltic flow of third order fluid with temperature dependent viscosity and thermal conductivity under the effect of magnetic field In this work viscous dissipation effects are also taken into account The problem is modeled in the Section ‘‘Basic equations” and the governing equations are obtained using long wave length and low Reynolds approximations In Section ‘‘Problem description”, approximate asymptotic solutions are achieved by applying perturbation technique Section ‘‘Perturbation solution” deals with the graphical illustrations of various parameters Concluding remarks are presented in the Section ‘‘Graphical results and discussion” Basic equations The basic equations governing the fluid flow and heat transfer are expressed as follows: Continuity equation:   V ¼ 0; r ð1Þ Momentum equation: dV  s  ỵ J  B0 ; q  ẳr dt 2ị Energy equation: dT r   ðk  TÞ; qcp  ẳ s  L ỵ r dt Fig Pressure rise DP k versus flow rate Q for / = 0.4 ð3Þ 965 T Latif et al / Results in Physics (2016) 963–972 in which V represent the fluid velocity, q is the fluid density, t is the  VÞ is the gradient time, d=dt is the material time derivative, Lð¼ r  is the thermal vector, cp is the specific heat, T is the temperature, k conductivity of fluid, J is Joule current, and B0 is the magnetic field  is given by Cauchy stress tensor s s ẳ PI ỵ S; 4ị where P is the fluid pressure, I is the identity tensor, and S is the extra stress tensor for third order fluid defined by [33]:
  ỵ b3 trA21 A1 ỵ a1 A2 ỵ a2 A21 ỵ b1 A3 Sẳ l
 ỵ b2 A1 A2 ỵ A2 A1 ; 5ị Anỵ1 ẳ dAn ỵ An L ỵ LT An ; dt @X ỵ @V @Y ẳ 0; 8ị @U @U @U q  ỵU ỵV @t @X @Y n ẳ 1; 2: ð6Þ  is the fluid viscosity, A1  A3 are the first In above equations, l three Rivlin–Ericksen tensors whereas and bi ; ði ¼ 1; 2; 3Þ are the material constants @V @V @V q  þU þV @t @X @Y qcp Problem description Consider a two-dimensional channel ðH < Y < HÞ of half width a filled with an electrically conducting third order fluid The channel walls are flexible and subjected to the constant temperature T w The flow in the channel is generated when sinusoidal waves of small amplitude b propagate on the channel walls with constant speed c The shape of the walls can be described mathematically as ð7Þ in which k is the wavelength whereas X and Y are axes along and perpendicular to the channel walls respectively If U is the Xcomponent of velocity, V is the Ycomponent of velocity, r is the fluid electrical conductivity and B0 ¼ ð0; B0 ; 0Þ is signifying the applied magnetic field in the direction normal to the flow then the Eqs (1)–(3) in component form can be written as @U with A1 ẳ L ỵ L T ;   2p Y ẳ HX; tị ẳ a ỵ b cos X  ctị ; k ! ẳ ! @T @T @T ỵU þV @t @X @Y ¼ ! @P @X @P @Y @ ỵ ỵ @SXX @X @SXY @X  @T ẳ k @X @X  ỵ ỵ ! ỵ @SXY @Y @SYY @Y @ @Y  rB20 U; ;  @T k @Y 9ị 10ị ! ỵ SXX ! @U @V @V @U ỵ SYY ỵ SXY ỵ ; @X @Y @X @Y ð11Þ Note the effects of induced magnetic field are not taken into account under the assumption of negligibly small magnetic Reynolds number [15] Also SXX ; SXY ; SYY are the components of the extra stress tensor as already mentioned in [34] Fig Pressure gradient dp=dx versus x for / = 0.4, Q = 0.5 966 T Latif et al / Results in Physics (2016) 963–972  appearing in the Eqs  and thermal conductivity k The viscosity l (8)–(11) are assumed to vary linearly with temperature and given by the following equations [30,17]:     l ¼ l0  cðT  T w ị ; k ẳ k0 ỵ bT  T w Þ ; ð12Þ in which l0 and k0 are respectively the fluid dynamic viscosity and thermal conductivity at constant temperature T ¼ T w , and c and b are constants Þ moving with velocity c with respect Defining a wave frame ð x; y  to the fixed frame X; Y; t by the transformation x ¼ X  ct;  ¼ Y; u  ðx; y ị ẳ UX; Y; tị  c; y  v x; yị ẳ VX; Y; tị; px; yị ẳ PX; Y; tÞ; ð13Þ  ; v and p  respectively represent the velocities and preswherein u sure corresponding to the wave reference frame We introduce the non-dimensional variables and parameters: x x¼ ; k v  a2 p v ¼ ; p¼ ; ckl0 dc  w a H b l ; d¼ ; h¼ ; /¼ ; w¼ ; l¼ ca k a a l0 rffiffiffiffiffiffi a  r qca b c2 S¼ ; c1 ¼ ; B a; Re ¼ S; M ¼ l0 l0 c l0 l0 a  k b c2 b c2 c2 ¼ 2 ; c3 ¼ ; k ¼ ; k0 l0 a l0 a h¼  y y¼ ; a T  Tw ; Tw  ¼ bT w ;  u u¼ ; c Pr ¼ @w ; u¼ @y l0 c p k0 v ; Br ¼ @w ; ¼ @x l0 c2 T w k0 ; a ¼ cT w ; where d; M, Re, ci i ẳ  3ị, Pr, Br, a;  and w are used to denote the wave number, Hartman number, Reynolds number, material parameters, Prandtl number, Brinkman number, viscosity parameter, thermal conductivity parameter and stream function respectively After using Eqs (13) and (14) into Eqs (8)–(11) and then adopting the long wavelength and low Reynolds approximation (see for detail [35]), we arrive at !3   @p @ @2w @2w @w  M2  ahị ỵ 2C ỵ1 ; 0ẳ ỵ @x @y @y @y @y 0¼ @p ; @y ð16Þ !3 @2 @2w @2w 2@ w  M ; ¼ ð1  ahị ỵ 2C @y @y2 @y2 @y !2 !4   2 @ @h @ w @ w 5; ỵ 2C ỵ hị ỵ Br41  ahị 0ẳ @y @y @y2 @y2 14ị 15ị 17ị 18ị in which Cẳ c2 ỵ c3 ị is the Deborah number Note that continuity Eq (8) is vanished automatically and the compatibility Eq (17) is obtained by taking the cross differentiation of Eqs (15) and (16) The physical boundary conditions with respect to wave frame are w ¼ 0; @2w ¼ 0; @y2 Fig velocity u versus y for / = 0.4, Q = 1.8, x = 0.1 @h ẳ 0; @y at y ẳ 0; 19ị 967 T Latif et al / Results in Physics (2016) 963–972 w ¼ F; @w ¼ 1; @y h ¼ 0; at y ẳ hxị ẳ ỵ / cos2pxị: ð20Þ It may be noted that the present analysis in the absence of heat transfer and constant viscosity of the fluid reduces to the problem in Reference [34] The dimensionless time mean flow rates Q and F in respective fixed and wave reference frames can be related by the relation Q ẳ F ỵ 1; 21ị with Z Fẳ hðxÞ @w dy: @y ð22Þ In view of perturbation technique, we write
 f ẳ f 00 ỵ Cf 01 ỵ C2 f 02 ỵ a f 10 ỵ Cf 01 ỵ C2 f 02 ; C < 1; a < 1; ð24Þ where f is any flow quantity Using Eq (24) in the governing Eqs (15), (17), (19), (20) and (23), collecting the coefficients of like powers of a and C and dropping the terms of OðC2 Þ; OðaCÞ and OðaC2 Þ, we get a system of linear equations Solving the resulting system for w; dp=dx and h, we get w ẳ a2 sinhMyị ỵ a1 y ỵ Cẵa6 sinhMyị ỵ a7 sinh3Myị ỵa3 a4 coshMyị ỵ a5 ịy ỵ aẵa8 a14 sinhMyị ỵ a9 sinh3Myịị ỵa8 a10 þ a11 coshðMyÞÞy þ ða13 sinhðMyÞÞy2 þ a12 coshðMyÞy3 Perturbation solution  ð25Þ The system of Eqs (15)–(18) comprises nonlinear coupled differential equations whose closed form solution is difficult to find So, we construct the series solution by utilizing the asymptotic analysis In order to achieve this, the viscosity parameter a and the thermal conductivity parameter  are taken asymptotically small and of the same order of magnitude It may also be recalled that the viscosity parameter c and the thermal conductivity parameter b are of the same dimension 1=T Thus, the energy equation can be written as: !2 !4   2 @ @h @ w @ w ẳ 0: ỵ Br41  ahị ỵ 2C ỵ ahị @y @y @y2 @y2 23ị " 3 dp 32h M  12hM ỵ 24 sinh2hMị ẳ aBrM5 F ỵ hị dx 384sinhhMị  hM coshhMịị # sinh4hMị  48hM cosh2hMị ỵ 384ðsinhðhMÞ  hM coshðhMÞÞ " # M ðF þ hÞ ð12hM  sinhð2hMÞ þ sinhð4hMÞÞ C 16sinhhMị  hM coshhMịị ỵ M3 F ỵ hị coshhMị sinhðhMÞ  hM coshðhMÞ Fig Temperature profile h versus y for / = 0.4, Q = 1.8, x = 0.1 ð26Þ 968 T Latif et al / Results in Physics (2016) 963–972
 h ¼ b1 2M2 h  yịh ỵ yị  cosh2hMị ỵ cosh2Myị Z DP k ẳ  ỵC b2 b3 y2 ỵ b2 b4 ỵ hMb5 cosh2Myị ỵ 5coshMh ỵ 4yịị   dp dx: dx ð28Þ The heat transfer coefficient Z at y ẳ hxị is shown below: ỵ5coshMh  4yịịị ỵ 45sinhMh ỵ 2yịị ỵ sinhM3h ỵ 2yịị @h @h : @x @y ỵ5sinhMh  2yịị ỵ sinhM3h  2yịịị Zẳ 5sinhMh ỵ 4yịị  5sinhMh  4yịịị
2 ỵb2 y 48hM sinhMh ỵ 2yịị ỵ 48hM sinhMh  2yịị Graphical results and discussion ỵ48M coshMh ỵ 2yịị  48M coshMh  2yịịị ỵaẵb6 b10 ỵ hMb11 coshMh ỵ 2yịị ỵ coshMh  2yịịịb12 cosh2Myị 12coshM3h ỵ 2yịị ỵ 9coshMh ỵ 4yịị  12coshM3h  2yịị ỵ9coshMh  4yịịị ỵ 36sinhMh ỵ 2yịị ỵ 36sinhM3h ỵ 2yịị 9sinhMh ỵ 4yịị ỵ 36sinhMh  2yịị ỵ 36sinhM3h  2yịị 9sinhMh  4yịịị ỵ b6 b7 y4  b6 y2 b8 ỵ 16M2 18sinhMh ỵ 2yịị ỵsinhMh  2yịịịị  b6 y2 18hMcoshMh ỵ 2yịị ỵcoshMh  2yịịịịị  b6 b9 ysinh2Myị
b6 y3 64hM sinhMh ỵ 2yÞÞ 64hM sinhðMðh  2yÞÞ  64M coshðMðh þ 2yÞÞ i þ64M3 coshðMðh  2yÞÞ ð27Þ The constants and bi appearing in the solution expressions (25)–(27) are given in the Appendix A The dimensionless pressure rise per wavelength is evaluated by the definition ð29Þ The effect of various parameters on pressure rise per wavelength, pressure gradient, velocity, temperature, heat transfer coefficient and streamlines are discussed here Pertinent parameters adhered to are a (viscosity parameter and thermal conductivity parameter), Br (Brinkman number), C (Deborah number) and M (Hartman number) The most important aspect of peristalsis is pumping against the pressure rise To discuss this phenomenon the graphs of pressure rise per wavelength DP k against volume flow rate Q for different values of involved parameters are plotted in Fig The DP k versus Q plane is divided in to different regions depending upon the algebraic signs of DP k and Q The region, where Q < and DPk > 0, is known as the retrograde pumping region In this region, the flow of fluid is because of the pressure gradient and its direction is opposite to the wave propagation The peristaltic pumping region occurs when Q > and DPk > The resistance of the pressure gradient in this region is overcome by the peristalsis of the walls and fluid moved in the forward direction When DPk ¼ then we have free pumping zone and the corresponding volume flow rate Q is known as free pumping flux In copumping region, Q > and DPk < 0, the pressure difference assists the flow due to peristalsis of the walls The effect of a on DPk is shown in Fig 1a It is impor- Fig Heat transfer coefficient Z versus x for / = 0.4, Q = 1.8 T Latif et al / Results in Physics (2016) 963–972 tant to note that increasing a reduces viscosity and raises thermal conductivity and for a ¼ 0, we attain the fluid with constant viscosity and thermal conductivity Fig 1a shows that in retrograde region, the pressure rise per wavelength decreases with the increase in a However its behavior is opposite in the copumping region The effects of Br on DPk are shown in Fig 1b and these are similar to that of a Figs 1c and 1d present the effects of C and M respectively and it is noted that DPk increases with an increase in these parameters in retrograde region whereas it behaves in opposite manner in copumping region It can be seen that under all sorts of variations, DP k shows no deviation in the peristaltic pumping region Fig illustrates the influence of a, Br, C and M on the pressure gradient dp=dx within one wavelength i.e., x ½0:5; 0:5 Also, the pressure gradient is small at x ¼ 0, which is the wider part of the channel and this is physically justified as fluid can easily pass without the assistance of large pressure gradient While much greater pressure gradient is required at the narrow part of the channel in order to maintain the same flux of the fluid to pass through it From Figs 2a and 2b, it is evident that in the narrow part of the channel, where x ½0:5; 0:2 [ ½0:2; 0:5, the pressure gradient is decreasing for increasing values of a and Br However, no variation can be seen in the wider part of the channel From Figs 2c and 2d, one can see that pressure gradient is small for Newtonian and hydrodynamic fluid when compared respectively with third order and hydromagnetic fluid There is an additional comment on Fig 2d 969 Unlike the other three parameters, the variations in M cause the curves to have a visible difference even in the wider part of the channel The Fig shows the impact of a, Br, C and M on velocity The behavior of the velocity near the channel walls is opposite to that of the center but variation is insignificant So, it is for this reason, we have only emphasized the peak velocities that occur at the center of the channel It is evident from Fig 3a that peak velocity has a positive correlation with a This is quite relevant to the physical situation because increase in a causes decrease in viscosity and consequently the resistance to deformation due to stresses becomes low which triggers a rise in velocity It is also manifested in the figure that fluids with variable viscosity and thermal conductivity have more velocity than those with constant viscosity and thermal conductivity The effect of Br on velocity is illustrated in Fig 3b, which demonstrates that fluids with higher Br values have relatively greater peak velocities Like a and Br, C too has a positive correlation with the peak velocity of the fluid as is shown in Fig 3c Moreover the velocity of third order fluid is high as compared to that of Newtonian fluid Of the four parameters, only Hartman M has a negative correlation with the peak velocity of the fluid as is depicted in Fig 3d The decrease in velocity is caused by the drag effects of the Lorentz force which itself increases with M The temperature profiles h are plotted against y for different values of a, Br, C and M, and are displayed in Figs We observe from Fig 4a that a has a negative correlation with h in the flow Fig Streamlines for variation of a for / = 0.2, Q = 0.97, Br = 0.8, C = 0.01, M = 0.8 Fig Streamlines for variation of Br for / = 0.2, Q = 0.97, a = 0.4, C = 0.01, M = 0.8 970 T Latif et al / Results in Physics (2016) 963–972 Fig Streamlines for variation of C for / = 0.2, Q = 0.97, a = 0.5, Br = 0.7, M = 0.7 Fig Streamlines for variation of M for / = 0.2, Q = 0.97, a = 0.5, Br = 0.7, C = 0.01 field In fact a does not have the same correlation with viscosity as it has with thermal conductivity An increase in a causes a decrease in viscosity and in turn decreases the inter-molecular forces and viscous dissipation Ultimately, there is decrease in temperature h On the other hand a has a positive correlation with the thermal conductivity, which also explains the decrease in temperature The reason for such a behavior is that an increase in thermal conductivity increases the ability of the fluid to conduct heat and since the temperature of the fluid is higher than the walls, consequently it decreases Hence, a decrease in viscosity and increase in thermal conductivity decreases the temperature The dependence of h on Br is presented in Fig 4b It is a positive correlation The reason for such a manifestation is that the viscous dissipation causes the conversion of kinetic energy to heat energy and obviously the temperature rises Fig 4c shows that C has a similar effect on h as Br has The dependence of h on M is shown in Fig 4d Rising values of M have decremental effects on the amplitude of h because according to Curie’s law, magnetization is inversely proportional to temperature Variation of heat transfer coefficient Z at the wall y ẳ hxị for various values of sundry parameters is shown in Fig As Z is a cyclic function, therefore, showing the plot for one wavelength is sufficient For the sake of demonstration, the wavelength over the interval ½0; 1 is chosen From these plots it is seen that Z is positive (negative) on the left (right) of the mean point x ẳ 0:5ị The absolute value, jZ j decreases for increasing values of a (see Fig 5a) and increases with Br, C and M (see Figs 5d) An intriguing phenomenon in transport of fluid is trapping and is presented by sketching streamlines in the Figs 6–9 A bolus is enclosed by splitting of a streamline under certain conditions and it is carried along with the wave in the wave frame This process is called trapping The trapped bolus is found to expand by increasing a, Br, and C (see Figs 6–8) However the size of bolus decreases by the rising effects of M as shown in Fig Conclusion In this paper, peristaltic flow of MHD third order fluid with temperature dependent viscosity and thermal conductivity along with viscous dissipation effects are analyzed and series solutions are obtained through perturbation method The findings of the present study are summarized as follows: It is found that the effects of a (the viscosity and thermal conductivity parameter) are opposite to that of C (third order parameter) on pressure rise per wavelength and pressure gradient a and C cause a good variation in narrow part of channel in contrast to wider part Peak velocity is high for variable viscosity and thermal conductivity as compare to constant viscosity and thermal conductivity Similarly third order fluid has more velocity than Newtonian fluid It is found that the temperature has positive correlation with thermal conductivity and negative correlation with viscosity It is found that the magnitude of heat transfer coefficient Z for a fluid with constant viscosity and constant thermal conductivity is 971 T Latif et al / Results in Physics (2016) 963972 Br2 M4 F ỵ hị higher than that for a fluid with variable viscosity and variable thermal conductivity The Newtonian fluid has less heat transfer coefficient as compared with the third order fluid Besides that, increasing values of a and C show increasing trend in the size of the trapped bolus
 b7 ¼  320hM coshðhMÞ  320M sinhðhMÞ ; Appendix A b8 ẳ 16M2 34 sinhhMị ỵ sinh3hMịị  hM4b15 ỵ cosh3hMịịị; FM coshhMị ỵ sinhhMị ; hM coshhMị  sinhhMị Fỵh ; a2 ẳ  hM coshhMị  sinhhMị M F ỵ hị

;   6hM  4h M coshhMị ỵ b20 ỵ hM cosh3hMị ; b10 ẳ hM16b16 hM ỵ b22 ị ỵ 315 cosh3hMị ỵ 51 cosh5hMịị ỵ b17 ;
 b11 ¼ 40h M2  33 ;
 b12 ẳ 32hM sinhhMị 8h M2 ỵ cosh2hMị ỵ 15 ; 16sinhhMị  hM coshhMịị 12288hM coshhMị  sinhhMịị b9 ẳ 96M a1 ẳ a3 ¼ b6 ¼ ; a4 ¼ 12MðsinhðhMÞ  hM coshhMịị;
 b13 ẳ sinhhMị 12h M ỵ cosh2hMị ỵ 13 ; a5 ẳ M12hM  sinh2hMị ỵ sinh4hMịị;
 b14 ẳ 16h M2  25 sinh3hMị ỵ sinh5hMị ỵ b18; a6 ẳ a7 ẳ M F ỵ hị 16sinhhMị  hM coshhMịị M F ỵ hị a15 ; a16 ;
 b15 ¼ 8h M sinhhMị ỵ  2h M2 coshhMị  3hM sinh3hMị; 16sinhhMị  hM coshhMịị a8 ẳ 3 BrM F ỵ hị
 b16 ẳ 3 4h M2 ỵ sinh3hMị ỵ b21 ỵ 9hM cosh3hMị; ; 384sinhhMị  hM coshhMịị a9 ẳ 3hM coshhMị  sinhhMịị; b17 ẳ 45 sinh3hMị  27 sinh5hMị;
 b18 ¼ 16 12h M þ 18h M þ sinhðhMÞ;
 a10 ẳ M 32h M ỵ 12hM ỵ 48hM cosh2hMị ỵ a18 ;
 2 b19 ẳ 60  72h M  6h M2 ỵ cosh2hMị;
 a11 ẳ 12M 4h M2  coshð2hMÞ  a16 ;
 16BrM M F ỵ hị a ; a12 ¼  16 384ðsinhðhMÞ  hM coshðhMÞÞ
 b20 ẳ sinhhMị 4h M  cosh2hMị  ; a13 ẳ 24M BrF ỵ hị 384ðsinhðhMÞ  hM coshðhMÞÞ a16 ; References a14 ẳ hM4hMa17  cosh3hMịị ỵ sinh3hMị; a15 ẳ sinh3hMị ỵ 3hM4hM sinhhMị  cosh3hMịị; a16 ẳ hM coshhMị  sinhhMị; a17 ẳ sinh3hMị ỵ 2hMcoshhMị  4hM sinhhMịị; a18 ẳ 24 sinh2hMị  sinh4hMị; b1 ẳ  b2 ẳ BrM F ỵ hị 2 8sinhhMị  hM coshhMịị BrM F ỵ hị ; 512ðhM coshðhMÞ  sinhðhMÞÞ
 b21 ẳ 16h M ỵ 18h M2  sinhhMị;
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