A. Afsar Khan 1, A. Sohail 2, S. Rashid 1, M. Mehdi Rashidi 3,4 and N. Alam Khan 5

Transcription

1 Journal of Applied Fluid Mechanics Vol. 9 No. 3 pp Available online at ISSN EISSN DOI:.69/acadpub.jafm Effects of Slip Condition Variable Viscosity and Inclined Magnetic Field on the Peristaltic Motion of a Non-Newtonian Fluid in an Inclined Asymmetric Channel A. Afsar Khan A. Sohail S. Rashid M. Mehdi Rashidi 34 and N. Alam Khan 5 Department of Mathematics & Statistics FBAS IIUI Islamabad 44 Pakistan Department of Mathematics Comsats Institute of Information Technology Lahore 54 Pakistan 3 Shanghai Key Lab of Vehicle Aerodynamics and Vehicle Thermal Management Systems Address: 4 Cao An Rd. Jiading Shanghai 4 China 4 ENN-Tongji Clean Energy Institute of advanced studies Shanghai China 5 Department of Mathematical Sciences University of Karachi Karachi 757 Pakistan Corresponding Author asohail@ciitlahore.edu.pk Received December 4 4; accepted March 5) ABSTRACT The peristaltic motion of a third order fluid due to asymmetric waves propagating on the sidewalls of a inclined asymmetric channel is discussed. The key features of the problem includes longwavelength and low-reynolds number assumptions. A mathematical analysis has been carried out to investigate the effect of slip condition variable viscosity and magnetohydrodynamics MHD). Followed by the nondimensionalization of the nonlinear governing equations along with the nonlinear boundary conditions a perturbation analysis is made. For the validity of the approximate solution a numerical solution is obtained using the iterative collocation technique. Keywords: Variable viscosity; Slip conditions; Peristaltic flow; Third order fluid; Inclined asymmetric channel.. INTRODUCTION During the past three decades most of the contributions on peristaltic motion dealt with the blood and other physiological fluids as the Newtonian fluids. In the recent years a growing interest towards the peristaltic flow of non- Newtonian fluid has been devoted by the biological and engineering practice. This approach is adequate when peristaltic mechanism involved in lymphatic vessels small blood vessels and transport of spermatozoa in the cervical canal is considered. It is of no surprise that majority of the physiological fluids behave like non-newtonian fluid. Keimanesh et al. ) studied the third grade non-newtonian fluid flow between two parallel plates. The peristaltic mechanism has been studied by several researcher experimentally numerically and analytically Latham 966; Shapira et al. 969; Hayat et al. ; Nadeem & Akbar ; Mekheimer & Elmaboud ; Noureen 3; Haroun 7; Sud et al. 977; Srivastava & Agrawal 9). Recent investigations [5] and many more have contributed to the study of peristaltic action. Shapiro and Jaffrin Shapira et al. 969) discussed peristaltic pumping with long wavelength at low Reynolds number. The long wave length approximation is an assumption that the width of the channel is small compared to wavelength of peristaltic waves. The biomagnetic fluid dynamics is the branch of fluid dynamics which studies the peristalsis of fluids. The most characteristic biomagnetic fluid is blood. Due to the complex interaction of the intercellular protein the hemoglobin and the red cell membrane the blood behaves as a magnetohydrodynamic MHD) fluid. The study of the MHD property of blood has potential for therapeutic use in the diseases of blood vessels and heart and help in controlling blood pressure. Rashidi et al. 4) recently investigated the entropy generation in MHD and slip flow over a rotating porous disk with variable prop-

2 A. Afsar Khan et al. / JAFM Vol. 9 No. 3 pp erties. The influence of moving magnetic field on blood flow was studied by Stud et al. Sud et al. 977) and they observed that the influence of suitable moving magnetic field increases the speed of blood. Srivastava and Agrawal Srivastava & Agrawal 9) considered the blood as an electrically conducting fluid and constitutes a suspension of red cell in plasma. Mekheimer Hayat et al. 7) studied peristaltic flow of a magnetic field in non-uniform channels. Hayat et al. Akram & Nadeem 3) considered the effect of magnetic field on the peristaltic flow of a third order fluid in a symmetric channel. Akram and Nadeem Busuioc et al. ) presented the influence of induced magnetic field and heat transfer on the peristaltic transport of a Jeffrey s fluid in an asymmetric channel. Rashidi et al. ) studied the entropy generation in MHD and slip flow over a rotating porous disk. Khan et al. 4) recently studied effect of heat transfer on peristaltic motion of Oldroyd fluid in the presence of inclined magnetic field. In real system there is always certain amount of slip and no slip conditions are no longer valid. Many investigations Nadeem & Akram ) have been carried out to see the effect of slip on the peristaltic motion of fluids. Moreover the peristaltic flow propagates in symmetric as well as asymmetric channels where the peristaltic waves propagate with different amplitude and phase on the channel walls). All the above mentioned studies indicate that much attention in the past has been given to discuss the peristaltic flow of fluids with constant viscosity. Few attempts have been made to discuss the influence of variable viscosity on the peristaltic flow see Khan et al. ) and the references therein) however the present paper discusses the effects of variable viscosity and slip conditions on the peristaltic flow of a third order fluid in an inclined asymmetric channel. Analysis has been carried out in the presence of inclined magnetic field. The paper is organised as follows in section we have presented a detailed mathematical formulation of the problem. In section 3 the series solution using the perturbation analysis is presented. In section 4 the numerical approach is outlined. In section 5 the results are discussed with the aid of graphical representation. In section 6 the conclusions are made the benefit of the current research is discussed and the future work is proposed.. MATHEMATICAL FORMULATION Let us investigate the flow of an incompressible third-order fluid with variable viscosity in an inclined asymmetric channel of width d + d. The fluid under consideration is conducting in the presence of an applied magnetic field B. Both the magnetic field and channel are inclined at an angle θ and γ respectively. A schematic diagram is presented in Fig. to visualize the problem. The sinusoidal waves travelling down its walls are defined as [ ] ) π H Xt = d + a cos X ct) λ [ ] ) π H Xt = d b cos X ct) + ϕ λ ) In the above equations a b are the wave amplitudes λ is the wavelength and ϕ ϕ π) is the phase difference. ϕ = corresponds to symmetric channel with waves out of phase and ϕ = π the waves are in phase. The equations of motion are given by U X + U = ) Y ) Let D = t +U X +V Y ρd U = P X + S XX X + S XY Y + ρgsinγ σb cosθu cosθ V sinθ) 3) ρd V = P Y + S XY X + S YY Y ρgcosγ +σb sinθu cosθ V sinθ) 4) in which U V are the velocity in the fixed frame ρ is constant density P is the pressure σ is the electrical conductivity and g is the acceleration due to gravity. Expression for extra stress tensor S in a third order fluid is defined by the following relation S = ) µ + β 3 tra A + α A + α A + β A 3 +β 3 A A + A A ) 5) where µ is the coefficient of shear viscosity α i i = ) and β i i = 3) are the material constants. Busuioc et al. ) proved a mathematical theory of existence and uniqueness of solutions for the constitutive law 5 with some restrictions on the material coefficients. In this discussion we have considered some limitations on these coefficients in a manner similar 3

3 A. Afsar Khan et al. / JAFM Vol. 9 No. 3 pp Eqs. 3) and 4) gives ReδD ψ ψ y ) ψ = M cos θ y + cosθ M cosθsinθδ ψ x p x +δ S xx x + S xy y + Re Fr sinγ) Fig.. Schematic diagram of the inclined flow. to Busuioc et al. ; Ali et al. 9). The first three Rivlin Ericksen tensors can be written as A = L + L t A n+ = da n dt + A n L + L t A n n =. where L = V and t represent transpose. The coordinates and velocities in the laboratory XY ) and wave xy) frames are related by the following equations x = X ct y = Y u = U c v = V pxy) = P XY t ). 6) Defining the non-dimensional quantities x = x λ y = y d u = u c v = v cδ δ = d λ h = H d h = H d ϕ = b a M = p = d p cλµ S = d µy) Sx) µy) = µ c µ σ µ t = ct λ ) B d Re = ρcd µ v = ψ x u = ψ y γ = β c µ d γ 3 = β 3c µ d Fr = c. 7) gd Reδ D ψ ψ x ) ψ = M cosθsinθ y + [ M sin θδ ψ x p δ y δ S xy x S yy y Re cosγ 9) Fr ψ y x ψ x where D ψ = y ]. ) Eqs. - 9) after invoking long wavelength and small Reynolds number approximation yields S xy y p x = S xy in which M cos θ y M cos θ ψ y ) ψ y = ) +)+ Re Fr sinγ) S xy = µy) ψ ) 3 y + Γ ψ y. ) In order to seek the effect of variable viscosity on peristaltic flow we let µy) = αy) α << 3) α is the viscosity parameter Γ is Deborah number Γ = γ + γ 3 ). 4) The dimensionless mean flow q is defined as in which where F = h x) h x) h = + acosx q = F + + d ψ y dy = ψh ) ψh ). h = d bcosx + ϕ). 5) 33

4 A. Afsar Khan et al. / JAFM Vol. 9 No. 3 pp The boundary conditions in term of stream function ψ are defined as ψ = F ψ y + βs xy = at y = h ψ = F ψ y βs xy = at y = h 6) where β is a dimensionless slip parameter. The dimensionless pressure rise p is define by d p p = dx 7) dx 3. PERTURBATION ANALYSIS In order to solve the present problem we expanded the flow quantities in a power series of the small Deborah number Γ and small viscosity parameter α ψ = ψ + Γψ +... F = F + ΓF +... p = p + Γp +... ) ψ = ψ + αψ +... ψ = ψ + αψ +... F = F + αf +.. F = F + αf +... p = p + αp +... p = p + αp ) If we substitute Eq. 9) into Eqs. ) to 6) and separate the term of different order in Γ and α λ we obtain the following systems of differential equations for the stream function and pressure gradients together with boundary conditions. CASE-I ) d p dx = 3 ψ y 3 m ψ y + + Re Fr sinγ) CASE-II d p dx = 3 ψ y 3 m ψ y ) y ψ y y 3) 4 ψ y 4 = m ψ y + y y ψ y ) 4) with the corresponding boundary conditions at y = h : ψ = F ψ y + ψ β y βy ψ y = at y = h : ψ = F ψ y ψ β y + βy ψ y =. 5) CASE-III d p dx = 3 ψ y 3 M cos θ ψ y + ) 3 y ψ y y 6) M cos θ ψ y + 4 ψ y 4 = 6 y subject to boundary conditions at y = h : ψ = F ψ y + β ψ y = β at y = h : ψ = F ψ y β ψ y = β ) ψ 3 ψ y y 3 ) 3 ψ y ψ y ) 3. ) ) 7) 4 ψ y 4 = m ψ y ) where m = M cosθ along with the boundary condition ψ = F ψ y + ψ β y = at y = h ψ = F ψ y ψ β y = at y = h. ) 34

5 A. Afsar Khan et al. / JAFM Vol. 9 No. 3 pp Series solution of the Problem The approximate solution is given as. F ψ = y h ) h h ) F ) + h h ))S L h h ) + F + h h ))mcosh[my h )] cosh[my h )]) L + F + h h ))m cosh[mh h )]) L + F + h h ))βm sinh[my h )] + sinh[my h )]) L F + h h ))βm sinh[mh h )] L [ F +α + y h F S 3 ) h h ) + S ) cosh[mh ] cosh[mh ]) Lh h ) + y h )S F + S 3 + S S ) S L L + S L cosh[my] cosh[mh ]) + h h )S 3 + S ) L h h )F S h h ) S mcosh[my h )] cosh[my h )]) L LL S h h ) S m cosh[mh h )]) LL S h h) S βm sinh[my h )] + sinh[my h )] LL + S h h ) S βm sinh[mh h )] LL [ F +Γ + y h F K 3 ) h h ) + K ) cosh[mh ] cosh[mh ]) Lh h ) + y h )S F + K 3 + K K ) S L L + K L cosh[my] cosh[mh ]) + h h )K 3 + K ) h h )F L L K h h) S mcosh[my h )] cosh[my h )]). LL d p dx = Re sinγ Fr m m F h h ) F + h ) h ))S Lh h ) α Γ S L K L Scosh[mh] cosh[mh]) Lh h ) h h S3 + S S Lh h ) S L L S 3 + S ) cosh[mh] cosh[mh] + msinh[mh]) S Lh h ) βm cosh[mh]h h ) Kcosh[mh] cosh[mh]) Lh h ) K3 h h ) + S K Lh h ) + K L L K 3 + K ) cosh[mh] cosh[mh] + msinh[mh]) K Lh h ) βm cosh[mh]h h ). 9) 35

6 A. Afsar Khan et al. / JAFM Vol. 9 No. 3 pp NUMERICAL SOLUTION The complex behaviour of the non-newtonian fluids can be dealt more swiftly with the help of numerical solvers such as discussed by Sohail et al. 4; Rashidi et al. ) and many more. In this paper Eqs. ) and ) subject to the boundary conditions Eqs. 6) were solved numerically using a finite difference code. This technique implements a collocation formula the collocation polynomial provides a continuous solution which is fourth order accurate homogeneously in the preferred domain. The nonlinear equations are solved iteratively by linearization therefore our numerical approach relies upon the linear equation solvers of Matlab. 5. GRAPHS AND DISCUSSION The main aim of this section is to study the behavior of involved key parameters on pressure rise per wave length p axial velocity u and trapping. Therefore we have prepared Figs. to. Figs. a and b represent the comparison of pressure rise for both perturbation and numerical solutions. It is observed from Fig. a that the perturbation and numerical solutions are almost identical in the pumping region while in Fig b there is a difference between the perturbation and numerical solution. The comparison between perturbation and numerical solutions on axial velocity are discussed in Figs. c and d bottom row left to right). Fig dc shows a difference between perturbation and numerical solution. It is observed from Fig. d that there is a good agreement between numerical and perturbation solution near the walls of the channel. The variation of pressure rise with q is studied in Figures 3 4 and 5 for various values of Γ α β and M Figure 3) θ ϕ γ and Fr Figure 4) b and a Figure 5). At lower Deborah numbers the fluid flow is associated with Newtonian viscous flow. At higher Deborah numbers Γ > ) the material behaves like non-newtonian fluid increasingly dominated by elasticity. Fig. 3a studies the effect of Γ on p. The solid line curve is for Newtonian fluid Γ = ). It is found that the pumping curve for third order fluid lies below the Newtonian fluid in the pumping region while in free pumping there is no difference in Newtonian and third order fluid. Fig. 3b and 3c show the effect of α and β on p. It is noted that magnitude of pressure rise decreases by increases α and β. In Fig. 3d we describe the influence of M. It reveal that pumping rate decreases by increasing M. Fig. 4a is the graph of p for different values of θ. It indicates that the pumping rate increases with an increase in θ for both free pumping and copumping regions. To observe the effect of phase difference ϕ on p we have presented in Fig. 4b. It shows that in pumping region p > ) the pumping increases with an increase in ϕ. However in copumping and free pumping regions this situation is quite opposite. Fig 4c describes the influence of angle of inclination γ. It shows that the pressure rise increases in all regions with an increase in γ. Fig. 4d depicts the variation of p with Fr. It is noticed that with an increase in Froude number the pumping rate decreases in all regions. The observations regarding the effects of lower and upper wave amplitudes b and a respectively on p are quite opposite to that as in the case of ϕ Figs. 5a and 5b. Fig. 6a studies the effect of Γ on the velocity u versus y. We observe that there is no effect of Γ on the velocity at the centre of the channel as the curves coincide. Fig. 6b represents the graph of the velocity u for three values of α. It is clear from the figure that the velocity increases by increasing α. The variation of u with y for different values of β and M are studied in Fig. 7a and Fig. 7b. It is obvious from the figures that increasing β and M led to decrease in the velocity. 6. CONCLUSIONS AND FUTURE WORK This paper presented a detailed analysis of the consequence of slip conditions and crucial fluid parameters including the variable viscosity and inclined magnetic field that effect the peristaltic transport of a third order fluid in an inclined asymmetric channel. Detailed understanding of these flow parameters can help to understand the complex rheology of biological fluids such as the peristaltic mechanism involved in lymphatic vessels narrow blood vessels and intestine. The perturbation theory has been applied to approximate the solution of Eqs. ) and ) with boundary conditions 6). For the confirmation of the perturbation solution we presented a graphical analysis of the variation in fluid velocity and pressure gradient relative to dimensionless mean flow based on a collection of judiciously selected parametric values. The main results can be summarized as: a) The pressure rise increases in all regions with an increase in γ which is the angle at which the channel is inclined to the horizontal) while it decreases in all regions with increasing Fr which is the ratio of the charac- 36

7 A. Afsar Khan et al. / JAFM Vol. 9 No. 3 pp α = α =.3 α=.5 p q 3 β = β=. β=.4 p q M= M=. M=.4 p Fig.. a) Comparison of p for fixed Γ=. ϕ=.3 α=. M=.5 θ= π 4 Fr = β =.4 γ= π 6 b =.7 a =.4 d = Re = 4.; b) Comparison of p for fixed M=.5 Γ=. ϕ=. β =.4 θ = π 6 Fr = α =. γ= π 6 b =.7 a =.4 d = Re = 5.; c) Comparison of u for fixed Γ=. ϕ =.4 α =.4 M =.5 θ = π 6 Fr = β =.5 b =.4 a =.4 d = q =.; d) Comparison of u for fixed Γ=. ϕ =.4 α =. M =.5 θ = π 6 Fr = β =.5 b =.4 a =.4 d = q = 4.; q Fig. 3. a) Variation of Γ on p when ϕ =.3 α =. M =.5 θ = π 4 Fr = β =.4 γ = π 6 b =.7 a =.4 d = Re = 4.; b) Variation of α on p when ϕ =. Γ=. M =.5 θ = π 4 Fr = β =.4 γ = π 6 b =.7 a =.4 d = Re = 5.; c) Variation of β on p when Γ=. ϕ =. M =.5 θ = π 6 Fr = α =. γ= π 6 b =.7 a =.4 d = Re = 5.; d) Variation of M on p when Γ=. ϕ =. β =. θ = π 6 Fr = α =. γ = π 6 b =.7 a =.4 d = Re = 5. 37

8 A. Afsar Khan et al. / JAFM Vol. 9 No. 3 pp Fig. 5. Left to right a) Fig k Variation of b on p when Γ=. ϕ =.3 M =.5 α =. θ = π 6 Fr = β =.3 γ = π 6 a =.4 d = Re = 4; b) Fig l Variation of a on p when Γ=. ϕ =.3 M =.5 α =. θ = π 6 Fr = β =.4 γ = π 3 b =.7 d = Re =. Γ = Γ =. Γ=.4.5 u y.5 α = α =. α=. u.5 Fig. 4. a) Variation of θ on p when Γ=. ϕ =.3 M =.5 α =. Fr = β =.4 θ = π 6 γ = π 6 b =.7 a =.4 d = Re = 4.; b) Variation of ϕ on p when Γ=. α =. M =.5 θ = π 6 Fr = β =.4 γ = π 4 b =.7 a =.4 d = Re = 4.; c) Variation of γ on p when Γ=. α =. M =.5 θ = π 6 Fr = β =.4 α =. b =.7 a =.4 d = Re = 5.; d) Variation of Fr on p when ϕ =.3 Γ=. M =.5 α =. θ = π 4 β =.4 γ = π 6 b =.7 a =.4 d = Re = y Fig. 6. From left to right a) Variation of Γ on u when ϕ =.4 M =.5 θ = π 6 β =.5 q = a =.4b =.4 d = ; b) Variation of α on u when Γ =. ϕ =.4 M =.5 θ = π 6 β =.5 q = a =.4b =.4 d =.;

9 A. Afsar Khan et al. / JAFM Vol. 9 No. 3 pp β = β=. β=.4 ACKNOWLEDGMENTS The authors would like to thank the reviewers and the chief editor for their useful comments. u u y M=.5 M= M= y Fig. 7. a) Variation of β on u when Γ=. ϕ=.4 M =.5 θ = π 6 α =. q = a =.4b =.4 d=.; b) Variation of M on u when ϕ =.4 Γ=. θ = π 6 β =.5 q = a =.4 b =.4 d =. teristic velocity to a gravitational wave velocity). This shows that an increased inclination of channel increases the pressure whereas increased characteristic velocity will reduce the pressure. b) The axial velocity increases with increasing θ ϕ and α while it decreases with increasing β and M i.e. the Hartman number which is the ratio of magnetic effect to viscous effect). We conclude from this contrasting velocity profile under the influence of flow parameters that the fluid flow is influenced by the inclination the slip condition and the electromagnetic parameter. Our future work will take into account the thermal effects which have great importance in the peristalsis such as heat conduction in biological tissues heat convection due to the blood flow through the pores of the tissues and radiation between surface and its environment. Many researchers studied the peristalsis of viscous and non-newtonian fluids with and without heat transfer in the peristaltic flow phenomenon. Recently these effects in both Newtonian and viscoelastic fluid flows have been investigated. The peristaltic flow of a third order fluid in an inclined asymmetric channel with slip condition variable viscosity heat transfer and inclined magnetic field will be a challenging problem and will be analyzed both analytically and numerically in our future work. REFERENCES Afsar Khan A. R. Ellahi M. Mudassar Gulzar Moshen Sheikoleslami 4). Effects of heat transfer on peristaltic motion of Oldroyd fluid in the presence of inclined magnetic field J. Magn. Magn. Mater Akram S. S. Nadeem 3). Influence of induced magnetic field and heat transfer on the peristaltic motion of a Jeffery fluid in an asymmetric channel: closed form solutions Journal of Magnetism and Magnetic Materials 3. Ali N. Y. Wang T. Hayat and M. Oberlack 9). Slip effects on the peristaltic flow of a third grade fluid in circular cylinder Journal of Applied Mechanics 76. Busuioc A. V. D. Iftimie and M. Paicu ). On steady third grade fluids equations Nonlinearity 7) 6. Haroun M. H. 7). Non-linear peristaltic flow of a fourth grade fluid in an inclined asymmetric channel Commut. Material Sci Hayat T. A. Afsar M. Khan and S. Asghar 7). Peristaltic transport of a thied order fluid under the effect of a magnetic field Comput. Math. Appl Hayat T. A. Afsar and N. Ali ). Peristaltic transport of a Johnson- math. Segalman fluid in an asymmetric channel Math. Comput. Model Keimanesh M. M. M. Rashidi Ali J. Chamkha R. Jafari ). Study of a third grade non-newtonian fluid flow between two parallel plates using the multistep differential transform method Computers and Mathematics with Applications 6 ). Khan A. A. R. Ellahi K. Vafai ). Peristaltic transport of Jeffery fluid with variable viscosity through a porous medium in an asymmetric channel Advances in Mathematical physics 5. Latham T. M. 966). Fluid Motion in a Peristaltic Pump Thesis Massachusetts Institute of Technology Cambridge. Mekheimer Kh. S. Abd.Y. Elmaboud ). The influence of heat transfer and magnetic field on peristaltic transport of 39

10 A. Afsar Khan et al. / JAFM Vol. 9 No. 3 pp a Newtonian fluid in a vertical annulus: application of an endoscope Phys. Lett. A Nadeem S. N. S. Akbar ). Numerical solutions of peristaltic flow of Williamson fluid with radially varying MHD in an endoscope Int. J. Numer. Meth. Fluids. Nadeem S. S. Akram ). Slip effects on peristaltic flow of Jeffery fluid in an asymmetric channel under the effect of induced magnetic material Int. J. Numer. Meth. Fluids Noureen S. 3). Mixed convection peristaltic flow of third order nanofluid with an induced magnetic field PLoS ONE. Rashidi M. M. O. Anwar Bg M. T. Rastegari ). A generalized differential transform method for combined free and forced convection flow about inclined surfaces in porous media Chemical Engineering Communications 99 ) Rashidi M. M. N. Kaviani S. Abelman. 4). Investigation of entropy generation in MHD and slip flow over a rotating porous disk with variable properties International Journal of Heat and Mass Transfer 7. Shapira A. H. M. Y. Jaffrin and S. L. Weinberg 969). Peristaltic pumping with long wavelengths at low Reynolds number J. Fluid Mech Sohail A. H. A. Wajid and M. M. Rashidi 4). Numerical Modeling Of CapillaryGravity Waves Using The Phase Field Method Surface Review and Letters.3. Srivastava L. M. R. P. Agrawal 9). Oscillating flow of a conducting fluid with a suspension of spherical particles J. Appl. Mech Sud V. K. G. S. Sekhon and R. K. Mishra 977). Pumping action on blood flow by a magnetic field Bull. Math. Bio

11 A. Afsar Khan et al. / JAFM Vol. 9 No. 3 pp APPENDIX m = M cosθ L = msinh[mh ] sinh[mh ]) + βm cosh[mh ] + cosh[mh ]) L = m cosh[mh h )]) βm sinh[mh h )] +h h ) m sinh[mh h )] ) + h h )βm 3 cosh[mh h )] +h h )β m 4 sinh[mh h )] P = F + h h ))mcosh[mh ] cosh[mh ]) L + βm sinh[mh ] sinh[mh ]) F + h h )) L P = F + h h ))msinh[mh ] sinh[mh ]) L + βm cosh[mh ] cosh[mh ]) F + h h )) L P 3 = P + P P P 4 = P + P P P 5 = P P P 6 = P P S = 3P h sinh[mh ] S = 3P h sinh[mh ] 3P h cosh[mh ] 3P h cosh[mh ] + P mh sinh[mh ] + P mh sinh[mh ] P mh cosh[mh ] P mh cosh[mh ] 3h cosh[mh ] + 3h cosh[mh ]) h S 3 = P + P m sinh[mh ] h sinh[mh ) ] ) 3h sinh[mh ] 3h sinh[mh ] h +P P m sinh[mh ] h cosh[mh ) ] S 4 = 3P cosh[mh ] h P sinh[mh ] P mh cosh[mh ] + + P m h + 3P sinh[mh ] cosh[mh] P m h sinh[mh ] S 5 = 3P cosh[mh ] h P sinh[mh ] P mh cosh[mh ] + P m h cosh[mh ] + 3P sinh[mh ] + P m h sinh[mh ] S 6 = mp sinh[mh ] + Pm h cosh[mh ] + P mcosh[mh ] P m h sinh[mh ] + P m 3 h sinh[mh ] P m 3 h cosh[mh] 39

12 A. Afsar Khan et al. / JAFM Vol. 9 No. 3 pp S 7 = mp sinh[mh ] + P m h cosh[mh ] + P mcosh[mh ] P m h sinh[mh ] + P m 3 h sinh[mh ] P m 3 h cosh[mh ] S = βr + βs 6 + S 4 ) S 9 = βr βs 7 + S 5 ) S = S S 9 S = m cosh[mh h )]) βm sinh[mh h )] ) S = cosh[mh ] cosh[mh ] h h ) + msinh[mh ] + βm cosh[mh ] R = h m P cosh[mh ] h P m sinh[mh ] R = h m P cosh[mh ] h P m sinh[mh ] R 4 = m 6 cosh[mh ]) 3 P 3 sinh[mh ]) 3 P 3 3P sinh[mh ] 6 cosh[mh ]) P + 3P cosh[mh ]P sinh[mh ]) R 5 = m 6 cosh[mh ]) 3 P 3 sinh[mhh ]) 3 P 3 3P sinh[mh ] 6 cosh[mh ]) P + 3P cosh[mh ]P sinh[mh ]) K = m4 P 3 [sinh[3mh ] mh cosh[mh ]] +m 4 P 5 [ sinh[3mh ] + 9mh cosh[mh ] m4 P 4 [cosh[3mh ] + mh sinh[mh ]] ] [ + m 4 P 6 cosh [3mh ] + 9mh sinh[mh ] ] K = m4 P 3 [sinh[3mh ] mh cosh[mh ]] +m 4 P 5 [ sinh[3mh ] + 9mh cosh[mh ] m4 P 4 [cosh[3mh ] + mh sinh[mh ]] ] [ + m 4 P 6 cosh [3mh ] + 9mh sinh[mh ] ] K 3 = m4 P 3 [sinh[3mh ] sinh[3mh ]] m4 P 4 [cosh[3mh ] cosh[3mh ]] m4 P 3 mh cosh[mh ] h cosh[mh ]) + m4 P 4 mh sinh[mh ] h sinh[mh ]) + m4 P 5 9mh cosh[mh ] h cosh[mh ]) +m 4 9mh sinh[mh ] h sinh[mh ]) P 6 + m4 P 5 [3mh ] sinh[3mh ] + cosh[3mh ] cosh[3mh ] [ K 4 = m4 P 3 3mcosh[3mh ] mcosh[mh ] m h sinh[mh ] ] [ m4 P 4 3msinh[3mh ] msinh[mh ] m h cosh[mh ] ] +P 5 m 4 [ 3mcosh[3mh ] m 4 P 6 [ 3msinh[3mh ] + 9mcosh[mh ] + 9msinh[mh ] + 9h m ] sinh[mh ] + 9h m ] cosh[mh ] 39

13 A. Afsar Khan et al. / JAFM Vol. 9 No. 3 pp [ K 5 = m4 P 3 3mcosh[3mh ] mcosh[mh ] m h sinh[mh ] ] [ m4 P 4 3msinh[3mh ] msinh[mh ] m h cosh[mh ] ] +P 5 m 4 [ 3mcosh[3mh ] m 4 P 6 [ 3msinh[3mh ] + 9mcosh[mh ] + 9msinh[mh ] + 9h m ] sinh[mh ] + 9h m ] cosh[mh ] K 6 = P 3m 4 [ 9m sinh[3mh ] 4m sinh[mh ] + m 3 h cosh[3mh ] ] mp [ 4 9m cosh[3mh ] + 4m cosh[mh ] + m 3 h sinh[3mh ] ] + [ 9m P 5 m 4 ] sinh[3mh ] + 9m sinh[mh ] + 9m3 cosh[mh ] m 4 P 6 [ 9m cosh[3mh ] + 9m cosh[mh ] + 9m3 sinh[mh ] ] K 7 = P 3m 4 [ 9m sinh[3mh ] 4m sinh[mh ] + m 3 h cosh[3mh ] ] [ m4 P 4 9m cosh[3mh ] + 4m cosh[mh ] + m 3 h sinh[3mh ] ] P 5 m 4 [ 9m sinh[3mh ] m 4 P 6 [ 9m cosh[3mh ] + 9m sinh[mh ] + 9m3 cosh[mh ] + 9m cosh[mh ] + 9m3 sinh[mh ] K = K 4 + βk 6 + βr 4 ) K 9 = K 5 βk 7 βr 5 ) K = K K 7 ] ] 393