Sådhanå (2020) 45:7 Ó Indian Academy of Sciences

https://doi.org/10.1007/s12046-019-1240-8Sadhana(0123456789().,-volV)FT3](0123456789().,-volV)

Dynamical behaviour of a porous liquid layer of an Oldroyd-B model flowing over an oscillatory heated substrate

SAMEH A ALKHARASHI* and AZIZAH ALRASHIDI

Applied Sciences Department, College of Technological Studies, PAAET, Adailiyah, Kuwait e-mail: [email protected]; [email protected]; [email protected]

MS received 27 July 2019; revised 7 October 2019; accepted 25 October 2019

Abstract. The present work aims to study the time-dependent thin porous film flow of an Oldroyd-B model on a heated infinite long flat plate. The fluid and the substrate are both at rest initially. Suddenly, the plate is jolted into motion in its own plane with an oscillatory velocity. Further, an insoluble surfactant is located at the free surface but not in the bulk of the fluid. Inversion of Laplace transform is applied to obtain numerical solutions to the problem. Due to the difficult analytical inversions back to the real-time domain, the need to use numerical inverse Laplace transforms arises, and a numerical approach for this purpose is mentioned and applied. The analytical solution of the special case of the isothermal liquid film when the Reynolds number is vanishing small is obtained and discussed. The flow rate and skin friction are investigated and plotted. Depending on the selected parameters, it is revealed that relaxation time constant lowers the velocity, while the effect of retardation time is opposed to that of relaxation time. It is noted that the Pe´clet number and capillary number enhance the heat transfer rate, whereas the converse is true for elasticity number. It is also observed that the motion of the free surface grows gradually with the increase of Darcy number. The Reynolds number is found to enhance the flow rate and lower skin friction. In the special case when the Reynolds number is vanishing small, it has been shown that the capillary number has an effect, unlike the elasticity number.

Keywords. Oldroyd-B fluid; oscillatory velocity; Laplace transformation; porous media; surfactants; heat transfer.

1. Introduction In our analysis, we use the concept of Laplace transform, which is a very useful technique for solving the differential An abundance of literature deals with the Newtonian fluids equations. It is very important to analytically perform the flow in laminar channels and moving on substrates with a inverse Laplace transform of the obtained solutions, but it is free surface. The Newtonian fluid is the simplest to be a complicated step. However, the numerical inversion solved, not only numerically but also analytically. How- method of Durbin [1] based on a Fourier series expansion is ever, Newtonian fluid applications are very limited in most a successful formula to obtain the unknown functions in the chemical and mechanical industries. Therefore, the study of physical domain. non-Newtonian fluids is more useful in many applications To review some previous studies on this issue, several in mathematics and physics; in this study we involve one of models have been suggested in the literature by many the most popular models for such fluids, which is known as researchers. The velocity profile and pressure gradient of the Oldroyd-B fluid model. the unsteady state unidirectional flow of an Oldroyd-B fluid Studying liquid layers, especially of non-Newtonian in a circular duct are considered in paper [2]. The authors ones, has been extensively investigated in recent years, due solved two basic flow situations, which are suddenly star- to its various applications in modern technology and ted, with a constant acceleration. These two results are industries. Such fluids have applications in large industrial applied to a practical case, that is a trapezoidal piston materials, for examples, a coating of paper or plastic, soap motion and oscillatory flow are also considered. A gener- and cellulose solutions, solutions and melts, certain oils and alized Oldroyd-B fluid with fractional derivative between asphalts, condensers, heat exchangers, food and chemical two parallel plates is studied in work [3]. Analytical solu- industries and many other fields. tions corresponding to two types of unsteady unidirectional flows are obtained. The velocity distributions are deter- mined by means of discrete Laplace transform and finite Fourier sine transform. They used the fractional calculus *For correspondence approach to solve the problem. 7 Page 2 of 16 Sådhanå (2020) 45:7

Mukhopadhyay and Haldar [4] have illustrated the two- substrate. Initially, the plate and the fluid are both at rest, dimensional flow of a viscoelastic fluid represented by and then it flows suddenly because of the application of an Walters’ B’’ model. They illustrated the effect of the vis- oscillatory velocity in its own plane. The coordinate system coelastic parameter and the Marangoni number on the dif- is selected such that the x-axis coincides with the plate ferent zones, the amplitude of the disturbances on a along with the flow and the y-axis is perpendicular to it (see sub/supercritical region and the nonlinear phase speed in the figure 1). Further, an insoluble surfactant is located at the supercritical region. The subject of paper [5] is the acceler- free surface with a concentration Cðx; tÞ, which changes ated flows for a viscoelastic fluid governed by the fractional according to the transport relation [9, 10]: Burgers’ model. The flow induced by a variable accelerated À Á o o o À1o plate is solved exactly using the Fourier sine transform and tðHCÞþ xðHCuÞ¼Ds x H xC : ð1Þ the fractional Laplace transform. The purpose of work [6]is In this equation and throughout the paper later, the partial the mechanism of Marangoni instability in evaporating thin derivatives of any function are denoted by the subscripts t, liquid films due to soluble surfactant. By the techniques of qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 linear stability and numerical simulation, a thin-film analysis x and y. Also, in this equation, H¼ 1 þðoxhÞ and Ds is is applied and evolution equations for the film thickness and the surface diffusion coefficient where it is assumed that the the surfactant concentration are derived and analysed. It is film is sufficiently thin, so the effect of buoyancy can be found that the stabilizing effects of diffusion and interfacial neglected. mobility are not likely to become significant unless the films The relation between the surfactant concentrations are much thinner. Cðx; tÞ and the interfacial tensions cðC; TÞ can be approx- The stability problem of the steady uniform flow of a imated to be a linear law with the temperature, in which the power-law fluid layer down an inclined porous layer is impacts of Marangoni are imposed, so that investigated in paper [7], in which a two-sided model is proposed. The generalized Darcy’s law is employed in cðC; TÞ¼c0ðC0; TsÞÀcT ðT À TsÞÀcCðC À C0Þ; ð2Þ deriving the governing equations due to the flow in the porous layer. The linear and non-linear stability are investigated, in where c0 signifies the reference value of the surface tension, c oc=oT [ 0 for typical liquids and which the influence of the dimensionless governing param- T ¼À jT¼Ts c oc=oC . The assumption of the linear variation eters is discussed. In paper [8] the authors investigated the C ¼À jC¼C0 steady flow of an Oldroyd-B fluid, where they used suit- of the surface tension is very much compatible with the able transformations to reduce a system of nonlinear partial experimental data for many fluids [9]. differential equations to a system of ordinary differential equations. It is revealed that relaxation time constant and the Prandtl number enhance the heat transfer rate. It is noted that for a larger retardation time constant, the velocity is enhanced 2.2 Fundamental equations and the temperature is lowered. The constitutive equations governing the unsteady heated The present work aims to solve an Oldroyd-B fluid film motion of an incompressible Oldroyd-B fluid are descri- flow on a heated infinite substrate. The flow is through porous bed by momentum balance equation of the viscoelastic media, in which insoluble surfactants are located at the sur- nature liquid layer through a porous medium, the equation face. The Laplace transform and the methods of numerical of continuity that expresses the conservation of mass and inversion of the Laplace transform are applied to obtain the the energy equation for the temperature profile solutions in the original space and to find the effects of dif- [4, 5, 11–15]: ferent fluid parameters. Exact and analytical solutions are À Á à À Á obtained in the special case of the isothermal liquid film when q 1 þ kot ot þðu ÁrÞ u ¼À 1 þ kot rp the Reynolds number is vanishingly small. The paper runs as À Á l/ À Á ð3Þ þ l 1 þ ho r2u À 1 þ ho u þ qg; follows. In the second section, the problem is clearly stated eff t q t and the relevant equations of motion are given. The method of solution is carried out in the fourth section, in which rÁu ¼0; ð4Þ numerical and analytical solutions and their analysis are  à 2 presented. The final section summarizes the results. qcp ot þðu ÁrÞ T ¼jr T; ð5Þ

where q refers to the density of the fluid film, u ¼ðu; vÞ denotes the velocity vector, rðo ; o Þ indicates the 2. Physical model and formulation x y gradient operator, k is the characteristic relaxation time, p is the pressure, l is the effective viscosity of the 2.1 Problem definition eff porous medium, h denotes the deformation retardation We consider a horizontal film flow of an incompressible time, l is the viscosity coefficient of the fluid, / is the Oldroyd-B fluid situated on a heated infinite long flat porosity, q is a parameter called permeability of the Sådhanå (2020) 45:7 Page 3 of 16 7

Figure 1. Sketch of the two-dimensional geometry of the fluid-heated porous film, where q is the density, / is the porosity, q is the permeability of the porous medium, l is the viscosity, k is relaxation time, h denotes the deformation retardation time and the parameter T is the absolute temperature.

porous medium and g is the gravity force. The parameters p À n Á P Á n À cðC; TÞr Á n ¼ pair ; ð9Þ T, j and cp appearing in Eq. (5) denote, respectively, the absolute temperature, thermal conductivity and the speci- where pair represents the pressure afforded by the ambient fic heat at a constant pressure of the liquid. Also, we air, which is taken to be constant. Any deformed surface consider that the fluid viscosity and the effective viscosity is characterized by a unit outward normal vector, which at are equal to each other based on the Brinkman approxi- any pointÀ onÁ the free surface becomes À1 mation [16–18]. n ¼H À oxh; 1 , and P is the stress tensor given in In these equations, to avoid cumbersome algebra, the o À1 o the element form as Pij ¼ leff ð1 þ k tÞ ð1 þ h tÞ material properties such as bulk viscosity, relaxation time, À Á o u þ o u , i; j ! x; y. retardation time, density and thermal conductivities are xj i xi j The balance of tangential stress on the deformable free assumed to be constant (do not vary with temperature). This surface is influenced by the air stress and the Marangoni assumption may be made depending on the type of problem effects via being examined. On the planar surface and the free surface, it is impera- n Á P Á t ¼rcðC; TÞÁt; ð10Þ tive that the initial and boundary conditions for the flow and heat transfer are added to complete the problem definition where t is the unit vector along the tangential direction at (2)–(5). For time t [ 0, the plate at y ¼ 0 starts suddenly to that point such that n Á t ¼ 0. As a thermal boundary con- slide in its own plane with an oscillatory velocity. In this dition expresses the continuity of heat flux from the free situation, we can write surface to the atmosphere, Newton’s law of cooling is adopted: uð0; tÞ¼U0 sin xt; t [ 0; and uðy; 0Þ¼0; 0\y\‘: jo T þ h ðT À T Þ¼0; ð11Þ ð6Þ n g s As the substrate is uniformly heated, the temperature where hg signifies the uniform heat transfer coefficient boundary condition reads describing the rate of heat transport between the liquid and the surrounding air, which is assumed to remain

T ¼ Tw at y ¼ 0: ð7Þ constant, and Ts is the ambient temperature above the layer. The kinematic condition signifies that fluid on the free surface stays on the interface and relates the motion of the free interface to the fluid velocities at the free 2.3 Non-dimensional equations surface: To derive the dimensionless equations, it is customary to employ scaling defined from the film geometry. Hence, we oth þ u Árðh À yÞ¼0: ð8Þ can use the unperturbed film thickness ‘ as the character- The jump of normal stress across the free surface is bal- istic length, to define the following dimensionless anced by surface tension and given by quantities: 7 Page 4 of 16 Sådhanå (2020) 45:7 pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffiffiffi l ‘g to the Biot number, which measures the heat transfer ðx; yÞ!‘; u ! ‘g; t ! ‘=g; p; P ! ; ‘ ð12Þ mechanism at the free surface; if Bi equals zero, thep surfaceffiffiffiffiffiffiffi 3 T À Ts ! DT; C ! C0; is then thermally insulated. The symbol Pes ¼ ‘ g=Ds denotes the surfactant surface Pe´clet number. where DT ¼ Tw À Ts. In terms of these dimensionless variables, the equations of motion and energy as well as the associated boundary conditions become 3. Basic state and the perturbed flow À Á à À Á Re 1 þ kot otu þðu ÁrÞu ¼À 1 þ kot À Á À Á ð13Þ When the flow is stationary and of constant film thickness, 2 1 rp þ 1 þ hot r u À 1 þ hot u À j; and when the substrate is uniformly heated, the velocity Da field and the heat distribution are decoupled from each  à 2 other. In this case, the base state solutions of the velocity Pe otT þðu ÁrÞT ¼r T; ð14Þ and the heat equation are obtained by the zero-order where the continuity equation remains invariant under these governing equations and the corresponding boundary conditions. The primary flow is obtained by assuming a dimensionlesspffiffiffiffiffiffiffi quantities. In these equations, Re ¼ quiescent initial state; therefore, the base state velocity in ðq ‘3gÞ=l refers to the Reynolds number, which is used to the fluid layer is zero, and the flow is steady and fully help predict flow patterns in different fluid flow situations. developed. Da q=‘2 The parameter ¼ is the Darcy number, which is In base state flow, the solution of the energy equation is the ratio of the permeability of the medium to the length obtained using Eq. (14) and the corresponding boundary j squared, while the symbol is a unit vector along the condition (Eq. (15)); it has the form vertical axis. The quantity Pe ¼ PrRe denotes the Pe´clet number and Pr ¼ðlcpÞ=j is Prandtl number. The Prandtl 1 þ Bið1 À yÞ T ðyÞ¼ ; ð21Þ number plays a strong role in determining the extent of the b 1 þ Bi thermal boundary layer as against the momentum boundary layer, and is strictly a function of the properties of the fluid and the hydrostatic pressure is given by and not flow properties. p ¼ p^ À y; ð22Þ The non-dimensional boundary conditions at the sub- 0 0 strate y ¼ 0 are then where p^0 denotes the constant pressure of air. Since we deal with a linear stability analysis, such that u ¼ v ¼ 0; T ¼ 1: ð15Þ the interfacial displacement is small, we perturb the flow At the free surface y ¼ hðx; tÞ, we have the following around the afore-mentioned base solutions. Hence, the dimensionless conditions: fluid motion is describe by the linearized equations toge- À Á à À ÁÈ à ther with the relevant appropriate linearized boundary 1 þ ko p À p þ 2HÀ2 1 þ ho ðo hÞ2 À 1 conditions. To this end we linearize the system, assuming t  air Ã É t x all variables as the sum of the base state and a small oxu À oyu þ oxv oxh À Án o perturbation. We also apply the corresponding boundary À3 o À1 o2 ¼ÀH 1 þ k t MaT À Neð1 À CÞÀCa x h; conditions at the mean elevation of the interface, y ¼ 1, by expanding the variables in Taylor series; hence, in the ð16Þ perturbed state, the dimensionless film thickness can be À ÁÈ ÃÂ Ã É 2 evaluated as 1 þ hot oyu þ oxv 1 ÀðoxhÞ À 4oxuoxh À ÁÈ Â ÃÉ ð17Þ ¼ÀH 1 þ kot NeoxC þ Ma oxT þ oyToxh ; hðx; tÞ¼1 þ gðx; tÞ; ð23Þ

v ¼ oth þ uoxh; ð18Þ where gðx; tÞ1 is a perturbed value to the stationary À Á film thickness. Similarly, in order to perturb the other o o À1o À1o physical quantities around the static case, we introduce a tðHCÞþ xðHCuÞ¼Pes x H xC ; ð19Þ small harmonic disturbance for the velocities, the pres- oyT À oxhoxT þHBiT ¼ 0: ð20Þ sure, the temperature and the surfactant concentration as pffiffiffiffiffi follows: Herein, Ma ¼ðcTpDTffiffiffiffiffiÞ=ðl ‘gÞ represents the Marangoni 9 u ¼ 0 þðup; vpÞ; > number, Ca ¼ðl ‘gÞ=c0, labelled as the capillary num- => p p p ; ber, expresses the effect of surface tension, Ne ¼ ¼ 0 þ p pffiffiffiffiffi > ð24Þ T ¼ Tb þ Tp; > ðcCC0Þ=ðl ‘gÞ denotes the elasticity number encapsulat- ;> ing the effect of surface surfactants and Bi ¼ðq‘Þ=j refers C ¼ 1 þ Cp; Sådhanå (2020) 45:7 Page 5 of 16 7 where the subscript p denotes the perturbed quantities. At the surface y ¼ 1, the balance of the normal and the Since we deal with an incompressible and two-dimensional tangential tensor becomes flow, the solutions of this problem can be obtained by o3fþR o fþR fþRÀ1UðsÞ¼0; ð33Þ introducing a stream function wðx; y; tÞ. This function is y 11 y 12 0 related to the velocity perturbations by the relations o2   ^ yf þR21f þR22g þR23CðsÞ¼0; ð34Þ up ¼ oyw, vp ¼Àoxw, and automatically satisfies the con- tinuity equation (Eq. (4)). where The present problem can be solved using the Laplace ^ = ; ^ = ; transformation technique. Indeed, the Laplace transform R11 ¼R11 R0 R12 ¼ R12 R0 is a very useful tool for solving differential equations. 1 þ sh þ ðÞD3k2ð1 þ shÞþsð1 þ skÞRe a R^ ¼À ; This is an operation that transforms a function of time 11 21ðÞþ sh þ sð1 þ skÞReDa t to a function of frequency domain s. A general 4 ^ k ð1 þ skÞðÞD1 þ BiðÞ1 þ CaMa a form of the Laplace transform of original function f in R12 ¼À ; 2sðÞ1 þ Bi CaðÞ1 þ sh þ sð1 þ skÞReDa space and time is defined with respect to time by the ð1 þ shÞDa integral R0 ¼ ; Z 21ðÞþ sh þ sð1 þ skÞReDa 1 2 fðx; y; sÞ¼Lðf ðx; y; tÞÞ ¼ f ðx; y; tÞeÀstdt; ð25Þ k ðÞsð1 þ shÞþBiðÞ sð1 þ shÞÀð1 þ skÞMa R21 ¼ ; 0 sð1 þ shÞðÞ1 þ Bi where s  0 is the generally complex-valued Laplace ikð1 þ skÞMa R ¼ ; transform parameter, and the over-bar refers to a trans- 22 1 þ sh formed quantity. Applying Laplace transformation of ikð1 þ skÞNe R ¼ ; Eqs. (13) and (14), and considering Eq. (24), we reach the 23 1 þ sh stream function and heat equations: while the surfactants condition and Newton’s law read    1 sð1 þ skÞ 2 1 4 2 þ r wðx; y; sÞ¼ r wðx; y; sÞ; o  i k ^ ReDa ð1 þ shÞ Re yf À s þ CðsÞ¼0; ð35Þ k Pes ð26Þ   oyg þ Big ¼0; ð36Þ Pe sT ðx; y; sÞþBið1 þ BiÞÀ1o w ¼r2T ðx; y; sÞ: p x p where, in these conditions, we assume that ikx ð27Þ CpðsÞ¼C^ðsÞe þ c:c. In the light of these boundary   conditions, the functions f , g are of the form Let us seek the solutions of w and Tp in the form fðy; sÞ¼c eky þ c eÀky þ c eyc þ c eÀyc; ð37Þ wðx; y; sÞ¼UðsÞy þ fðy; sÞeikx þ c:c:; ð28Þ 1 2 3 4 n h i ikBiPe ikx gðy; sÞ¼ K ðc e2ky þ c Þecy þ K ðc eð2cþkÞy þ c ekyÞ eðkþcþdÞy T x; y; s g y; s e c:c: 29 1 Bi K K c 1 2 k 3 4 pð Þ¼ ð Þ þ ð Þ ð þ Þ c k o c e2ðkþcþdÞy c e2ðkþcÞy eÀ½2ðkþcÞþdŠy; Towards this end, substituting Eqs. (28) and (29)in þ 5 þ 6 Eqs. (26) and (27) and simplifying, we get two new equa- ð38Þ tions for f and g: where n o 1 sReðÞ1 þ sk 2 2 o4fÀ 2k2 þ þ Kk ¼ k À d ; Kc y Da 1 þ sh sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n o ð30Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 sReðÞ1 þ sk 2 2 2 sð1 þ skÞRe 1 2 o2fþ k2 k2 þ þ f ¼ 0; ¼ c À d ; c ¼ k þ þ ; d ¼ k þ sPe: y Da 1 þ sh 1 þ sh Da À Á o2 2 À1  yg À k þ sPe g þ kPeBið1 þ BiÞ f ¼ 0: ð31Þ Finally, we have the following solutions: È É   ky Àky cy Àcy ikx Now we obtain the corresponding boundary conditions to wðx; y; sÞ¼UðsÞy þ c1e þ c2e þ c3e þ c4e e þ c:c:; the system (30) and (31), using the linearized version of ð39Þ conditions (16)–(20). Thus, the associated conditions in n h i ikBiPe terms of f and g can have the following form at the plate T ðx; y; sÞ¼ K ðc e2ky þ c Þecy þ K ðc e2cy þ c Þeky eðkþcþdÞy p 1 Bi K K c 1 2 k 3 4 y ¼ 0: ð þ Þ c k o 2ðkþcþdÞy 2ðkþcÞy ikxÀ½2ðkþcÞþdŠy þ c5e þ c6e e þ c:c:;  o  f ¼ yf ¼ 0and alsog ¼ 0: ð32Þ ð40Þ 7 Page 6 of 16 Sådhanå (2020) 45:7

ik ÀÁfunction fðx; y; sÞ. As stated earlier, due to the difficult gðx; sÞ¼À c ek þ c eÀk þ c ec þ c eÀc eikx þ c:c: s 1 2 3 4 analytical inversions back to the real-time domain, the need ð41Þ to use numerical inverse Laplace transforms arises. We will implement a suitable numerical approach to obtain the Expressions for the mathematical coefficients c1; c2; ...; c6 unknown functions in the physical domain based on a are presented in the Appendix. At this step, we can define Fourier series expansion that is adopted in [1, 23], which two physical quantities, and the first is the flow rate or gives us an expression for the numerical inversion of sometimes called volume velocity or volumetric flow rate. Laplace transform; hence, the function f(x, y, t) is approx- As a physical concept, the flow rate depends on the area of imated by the formula the channel and the velocity of the fluid, which is given by h n   the volume of fluid that passes per unit time. The rate of ett 1 XN np  f ðx; y; tÞ¼ fðx; y; tÞþRe f x; y; t þ i flow Qðx; sÞ is usually expressed as s 2 s n¼0 Z Z  oi hðx;sÞ hðx;sÞ n ÀÁÀÁ npt  o   hk Àhk exp i ; 0\t\2s: Q ¼ udy ¼ ywdy ¼ hUðsÞþ c1 e À 1 þ c2 e À 1 s 0 0 ÀÁÀÁo hc Àhc ikx ð45Þ þ c3 e À 1 þ c4 e À 1 e : ð42Þ Here, the function f refers to w, T, g, Q or S and the parameter N is a sufficiently large integer and should be The other physical quantity of engineering interest is the chosen such that  skin friction Sðx; sÞ, which occurs due to the friction n    o between the fluid and the solid moving substrate. To obtain tt np npt e Re f x; y; t þ i exp i \; ð46Þ the skin friction, the shear stress is evaluated at the lower s s boundary: where  is a prescribed small positive number that corre- ÀÁ 2 ikx 2 2 sponds to the degree of accuracy to be performed. The S¼ o u ¼ o w ¼ e k ½Šþc þ c c ½Šc þ c : y y¼0 y y¼0 1 2 3 4 optimal choice of t was obtained according to the criteria ð43Þ described in [24, 25]. From [3, 11, 23], s ¼ 2tmax ¼ 200 where tmax is the time up to which the results are to be Expressions (42) and (43) are in agreement with the results achieved, tt ¼ 9 and N ¼ 700; several numerical tests were obtained previously (see Refs. [19–21]). conducted to select these values. In all considered cases, the At this juncture, the problem reduces to the problem of formula (45) provides stable and convergent results. With determining the inverse Laplace transformation of the the help of a suitable personal computer, the computations solutions to the physical domain, which is discussed in the were performed. The convergence of numerical solutions is next section. discussed in a simple trial and error manner, by enlarging the number the truncation constant N, while searching for stability in the numerical parameters of the computed 4. Methods of solution solutions. Having validated the numerical theory, figures 2, 3, 4, 4.1 Inversion of Laplace transform 5, 6, 7 and 8 illustrate the influences of some dimen- The main goal of this section is to perform the inverse sionless quantities on the behaviour of heat transfer and elevation as well as the velocity profile in the numerical Laplace transformation of the functions w, T , g, Q and S p solutions, obtained by fixing the values of all the physical given by (39)–(43). However, the expressions of the solu- parameters fixed except one having varying values for tions for such functions are usually complicated and cannot comparison. Figure 2 illustrates the influence of both time be inverted analytically, except for some limiting cases that and horizontal position on the surface wave evolution; in permit an analytical inversion as mentioned later. We then this graph the two planes g x and g t are illus- must resort to numerical calculations. The complex inver- ð À Þ ð À Þ trated corresponding, respectively, to the parts (a) and (b). sion formula for Laplace transform fðx; y; sÞ of a function In this figure, four curves are presented for the sake of f(x, y, t) is given by the following complex integral [22]: comparison. It can be shown from this figure that there is Z tþi. a symmetric motion about x ¼ 0. In another point of view, 1 st f ðx; y; tÞ¼ lim fðx; y; sÞe ds; ð44Þ the parts of figure 3 provide graphical demonstration of 2pi .!1 tÀi. the influence of the increasing value of Darcy number on where the integration is done along the vertical line t ¼ the liquid free surface in a three-dimensional space ReðsÞ in the complex plane, wherein Re is used to denote interface. Inspection of these parts reveals that the motion the real part of the present terms. The parameter t is greater of the free surface grows gradually with the increase of than the real part of all singularities of the transformed Darcy number. Also, through the horizontal position in the Sådhanå (2020) 45:7 Page 7 of 16 7

Figure 2. Effect of (a) time and (b) horizontal position on the surface wave evolution in the two planes (g À x) and (g À t), for a system having Bi ¼ 0:7, Ma ¼ 1, Pe ¼ 2, Ca ¼ 0:05; Pes ¼ 0:5; k ¼ 0:3; h ¼ 0:1; Ne ¼ 0:5, Re ¼ 5; x=2 and U0 ¼ 0:5.

Figure 3. Three-dimensional spatial evolution of free-surface deformation for the system at Re ¼ 10, Ma ¼ 1, Bi ¼ 0:7, Pe ¼ 2, Ca ¼ 0:8, Pes ¼ 0:5, U0 ¼ 0:3, k ¼ 0:3, h ¼ 0:1, Ne ¼ 0:5 and x ¼ 0:2, whereas the values Da ¼ 0:005, 0.01 and 0.5 are selected for the parts (a), (b) and (c), respectively.

fluid motion, an oscillating behaviour is observed for sometimes perturbs the wave motion where a part of its increasing Darcy number. On the other hand the increase kinetic energy may be absorbed, and hence destabilizing in the permeability of the medium due to Darcy number behaviour is sound. 7 Page 8 of 16 Sådhanå (2020) 45:7

Figure 4. Variation of temperature distribution inside the film with time, for the same system as defined in figure 2: (a) at Ca ¼ 0:9, Pe ¼ 3 and Ne ¼ 1; 3; 5, (b) at Pe ¼ 3, Ne ¼ 10 and Ca ¼ 0:7; 0:8; 1 and (c) at Ca ¼ 0:9, Ne ¼ 10 and Pe ¼ 2, 2.5, 3.

Figure 4(a), (b) and (c) shows, respectively, the flow behaviour during motion. Figure 5 displays snapshots influences of elasticity, capillary and Pe´clet numbers in of instantaneous streamlines of the stream function, in the plane (T À t). In contrast with the results investi- which the streamlines graphs are plotted by fixing the gatedinfigure4(a)–(c), it is worthwhile to notice that value of all the physical quantities except the Pe´clet increasing both Pe´clet and capillary numbers leads to an number, which has values equal to 0.1, 2 and 5 at t equal increase in the maximum values of the fluid tempera- to 50. Inspection of figure 5 reveals that the flow consists ture, while the opposite influence is observed for of two regular cells in figure 5(a), which transforms into increasing the elasticity number. This result is in two malformed cells in figure 5(b) that reach almost four agreement with that obtained by Hayat et al [8]. On the irregular contours in the last part of this graph. A con- other hand, the numerical results in figure 4(a), where clusion that may be made from the comparison among Ne increases stepwise from 1 to 5, and figure 4(b), figure 5(a)–(c) is that the Pe´clet number has a destabiliz- where Ca = 0.7, 0.8 and 1, show that the curves meet ing influence on the movement of the fluid film. The effect the t-axis at three convergent values of t, while the lines of the variations of Pe´clet number on the isothermal converge at the same point (t =83.7)infig- contours is shown in figure 6(a)–(c). It can be seen that as ure 4(c) when Pe has the values 2, 2.5 and 3. the Pe´clet number increases, the contours gradually turn For a better understanding of the effect of Pe´clet out to be four sets of cells as shown in figure 6(b) and (c). number Pe on the behaviour of the transient dimensionless This increase in the number of cells can be explained by streamline and the corresponding temperature profile, the convection heat transfer, which increases on increas- figures 5 and 6 are depicted. The streamlines are an ing the Pe´clet number. This result is consistent with that effective tool for visualizing the qualitative impression of obtained in Ref. [26]. Sådhanå (2020) 45:7 Page 9 of 16 7

Figure 5. Streamline contours for a system having the same parameters considered in figure 3: (a) Pe ¼ 0:1, (b) Pe ¼ 2 and (c) Pe ¼ 50.

Figure 7(a) and (b) investigates the velocity develop- the retardation time h on the velocity profile, where it is ments for different relaxation and retardation times, varied stepwise from 1 to 5, are depicted in figure 7(b). respectively. In figure 7(a), the solid, dashed and dotted It is of interest that, in comparison with the relaxation curves illustrate three distinct values of relaxation times time, the retardation time plays an opposite role in flow (k = 1, 3 and 5, respectively), where we fix the retar- development. Precisely, a larger value of h will yield a dation time h to 1. It is assumed that k  h  0 (see Refs. faster growth of velocity fields. In other words, a phys- [3, 18]). It is worthwhile to notice that a larger value of k ical interpretation of this result is that, there is an will result in slower development of the velocity beha- increase in the kinetic energy of the particles of the viour. Furthermore, the velocity is a monotonically fluids. This is due to the increase of the streaming increasing function of time t. Since it is well known in velocity, which leads to an increase in the perturbed fluid dynamics that velocity plays a destabilizing influ- motion. A similar result was reported by Haitao and ence, it is concluded that the relaxation time has a sta- Mingyu [3]. Also, this discussion is in agreement with bilizing role in the film motion. The effects of changing thatmentionedbyQiandJin[27] in their studies of 7 Page 10 of 16 Sådhanå (2020) 45:7

Figure 6. Isothermal contours for the same system as considered in figure 5. unsteady helical flows of a generalized Oldroyd-B fluid 4.2 Analytical solution of a limiting case with fractional derivative. To check the validity of Eqs. (42) and (43), fig- Now, it is of interest to investigate the limiting case of ure 8(a) plots the flow rate Q whereas figure 8(b) illustrates small Reynolds number to obtain the analytical solution. In this limit, we consider a Newtonian viscous isothermal the skin friction S against time t, for some values of Rey- nolds number Re. These parts are plots of flow rate and the flow; also, the Reynolds number is vanishingly small: Re  1; then we can write skin friction against t for Re = 10, 25 and 50, where other rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi parameters are held fixed. From figure 8(a), it is shown that 1 sRe c k2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi O Re2 ; Reynolds number has the effect of increasing the flow rate ¼ þ þ þ ð Þ ð47Þ Da 2 k2 þ 1 of the fluid. In figure 8(b), Reynolds number is shown to Da decrease the skin friction; this behaviour is physically pffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi 2 1 p Æy k þDa acceptable since the Reynolds number is inversely related Æcy Æy k2þ 1 e syRe 2 e ¼e Da Æ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Æ OðRe Þ: ð48Þ to the viscosity, and hence coincides with the fact that the 2 k2 þ 1 skin friction drag is caused by the viscosity of fluids. Thus, Da we conclude that the Reynolds number has opposite effects In this limit, the solution of the stream function is then on the flow rate and skin friction. given by Sådhanå (2020) 45:7 Page 11 of 16 7

Figure 7. Profiles of the velocity u as a function of time t for a system having the same parameters considered in figure 2: (a) at h ¼ 1 where k ¼ 1; 3; 5 and (b) at k ¼ 5 where h ¼ 1, 3, 5.

Figure 8. Planes Q À t and SÀt for the same parameters considered in figure 2, where Re ¼ 10; 25; 50 corresponding to solid, dashed and dotted curves, respectively.

À Á    ikx wðx; y; sÞ¼UðsÞy þ L0 þ L1Re e þ c:c:; ð49Þ The coefficients Ls, a and b that appear in these equations are functions of the basic parameters of the problem. The   where L0 and L1 are functions of the parameter s and the mathematical formulas of these coefficients are lengthy and other physical quantities. Taking the inverse of Laplace not included here. However, the authors are ready to give transformation, the stream function and the surface the values of these constants upon request. It is revealed deflection will be expressed as functions of time: that the solutions (50) and (51) in the special case of van- È ishingly small Reynolds number are in good agreements wðx; y; tÞ¼U0 sinðxtÞy þ L1 sinðxtÞþL2 cosðxtÞ pffiffi pffiffi É ð50Þ with those obtained in the articles [28, 29]. ðaþ bÞt ðaÀ bÞt ikx þL3e þL4e e þ c:c:; In connection with this special case, figure 9(a) and (b) ffiffi illustrates the influence of increasing capillary and the ik È p gðx; tÞ¼À L sinðxtÞþL cosðxtÞþL eðaþ bÞt elasticity numbers on the surface wave profile with the s 1 2 3 evolution in the horizontal position. The values of the other pffiffi É ðaÀ bÞt ikx representative parameters are given in the caption of fig- þL4e e þ c:c: ure 9. The two parts of this graph are illustrated in the ð51Þ (g À x) plane, where the values 0.05, 0.1 and 0.15 are 7 Page 12 of 16 Sådhanå (2020) 45:7

Figure 9. Plane ðg À xÞ in the limiting case of small Re, where Ne ¼ 0:2, U0 ¼ 0:2, k ¼ 0:5, y ¼ 1, Pes ¼ 1, x ¼ 2: (a) at Ne ¼ 0:2 where Ca ¼ 0:05; 0:1; 0:15 and (b) at Ca ¼ 0:15 where Ne ¼ 0:1; 0:2; 0:3. selected for the capillary number in part (a), whereas in part From the plotted charts, we note the following results. In (b), three distinct values of the elasticity number (= 0.1, 0.2 a three-dimensional space interface, the motion of the free and 0.3) are considered, for the sake of comparison. It is surface grows gradually with the increase of Darcy number, clear from figure 9(a) that as the capillary number decrea- in which an oscillating behaviour is observed. Both Pe´clet ses, surface waves elevation increases, while the converse and capillary numbers lead to an increase in the maximum is true due to increasing elasticity number in part (b) of this values of the fluid temperature, while the opposite influence graph. Comparing the results displayed in the two parts of is observed for increasing elasticity number. In the concept figure 9, it can be remarked that the effect of increasing of streamlines behaviours, it is found that the Pe´clet number capillary number on the fluid layer flow is opposed to that has a destabilizing influence in the movement of the fluid of the elasticity number. In a sense acceptable to physicists, film. A larger value of relaxation time will result in slower the capillary number is directly proportional to the vis- development of the velocity behaviour, while the opposite cosity, whereas the elasticity number is inversely propor- is true for larger value of retardation time. That is, the tional, and viscosity is known to increase the stabilization relaxation time has a stabilizing role in the film motion and process. This behaviour is in good agreement with that hence the retardation time plays a destabilizing one. The obtained by Allias et al [30] and Alkharashi [31]. Also, Reynolds number was found to have an effect of increasing similar results are mentioned by Tahir et al [32]. the flow rate, while the opposite was achieved in the case of skin friction. In the limiting case of an isothermal liquid film as the Reynolds number is vanishingly small (Re  1), 5. Summary an analytical solution is obtained. It is worth mentioning that the effect of capillary number is opposed to that of the The motivation of the present work is to investigate the elasticity number on the fluid motion. flow of a two-dimension incompressible Oldroyd-B fluid film on a heated infinite long flat plate through porous media. This is in the presence of interfacial insoluble sur- Acknowledgements factants. The fluid and the plate are initially at rest. Sud- denly, the substrate is subjected to oscillatory velocity in its This work was supported and funded by ‘‘The Research own plane. Numerical solutions of the dimensionless Program of Public Authority for Applied Education and problem have been obtained using the standard Laplace Training in Kuwait’’ (Project No. TS-18-06). transform. To transform the obtained solutions from Laplace space back to the original space, for example, stream function, temperature and surface elevation, the Appendix numerical inversion method of Durbin is applied. That is, semi-analytical solutions in the original space are obtained The variables c1; c2; ...; c6 in Eqs. (37) and (38) may be to illustrate the impacts of different fluid parameters. Exact written as follows: and analytical solutions are acquired in some special cases. Sådhanå (2020) 45:7 Page 13 of 16 7  UðsÞ c1 ¼ ð2kBcPeA35ðÞA25A46 ÀA26A45 Kk c0

þð2kBcPeA35ðÞA26A45 ÀA25A46

þ iðA25A35ðÞ2cA42 þðk À cÞA43 Àðk þ cÞA44

þA26A35ððk þ cÞA44 À 2cA42

þðc À kÞA43ÞþðikNeð2cA32 þðk À cÞA33 Àðk þ cÞA Þþð ðc À kÞA À 2cA : 34 23 22

þðk þ cÞA24ÞA35ÞAðÞÞ45 ÀA46 KkÞKc ;

c0 ¼2kBPeðÞAcA12 À cA11 þ kðÞA13 ÀA14 35ðÞA26A45 ÀA25A46 Kk

þ Kcð2kBPeðÞAcA11 À cA12 À kðÞA13 ÀA14 35ðÞA26A45 ÀA25A46

À iKkðcA14A26A35A41 À kA14A25A35A41 À cA14A25A35A41 þ kA14A26

ÂA35A41 À 2cA11A25A35A42 À kA14A25A35A42 þ cA14A25A35A42

þ 2cA11A26A35A42 þ kA14A26A35A42 À cA14A26A35A42 À kA11A25

ÂA35A43 þ cA11A25A35A43 þ 2kA14A25A35A43 þ kA11A26A35A43

À cA11A26A35A43 À 2kA14A26A35A43 þ kA11A25A35A44 þ cA11A25 2 ÂA35A44 À kA11A26A35A44 À cA11A26A35A44 À ik NeA14A31A45 2 À ikcNeA14A31A45 À 2ikcNeA11A32A45 À ik NeA14A32A45 þ ikcNe 2 2 ÂA14A32A45 À ik NeA11A33A45 þ ikcNeA11A33A45 þ 2ik NeA14A33 2 ÂA45 þ ik NeA11A34A45 þ ikcNeA11A34A45 þ kA14A21A35A45 þ cA14

ÂA21A35A45 þ 2cA11A22A35A45 þ kA14A22A35A45 À cA14A22A35A45

þ kA11A23A35A45 À cA11A23A35A45 À 2kA14A23A35A45 À kA11A24A35

ÂA45 À cA11A24A35A45 þA13ðA25A35ðÞðk À cÞA41 þðk þ cÞA42 À 2kA44

ÀA26A35ðÞþðk À cÞA41 þðk þ cÞA42 À 2kA44 ðikNeððk À cÞA31 þðk þ cÞ

ÂA32 À 2kA34ÞÀðÞA ðk À cÞA21 þðk þ cÞA22 À 2kA24 35ÞAðÞÞ45 ÀA46

þA12ðA25A35ðÞþA2cA41 Àðk þ cÞA43 þðk À cÞA44 26A35ððk þ cÞA43

À2cA41 þðc À kÞA44ÞþðikNeðÞþðð2cA31 Àðk þ cÞA33 þðk À cÞA34 k

þcÞA23 À 2cA21 þðÀk þ cÞA24ÞA35ÞAðÞÞþ45 ÀA46 ðikNeðA14ððk þ cÞ

ÂA31 þðk À cÞA32 À 2kA33ÞþA11ðÞÞ2cA32 þðk À cÞA33 Àðk þ cÞA34

þAð 11ðÞÀAðc À kÞA23 À 2cA22 þðk þ cÞA24 14ððk þ cÞA21 þðk À cÞA22

À2kA23ÞÞA35ÞA46ÞÞ;

k  À À À ke 3 3 2 A11 ¼ k ð1 þ skÞDa þ Ca sð1 þ shÞÀ k Bð1 þ skÞMa À s 2k ð1 þ shÞ A0  À1 þsð1 þ skÞReÞÞDaÞ ; B¼ÀBið1 þ BiÞ ; A0 ¼ 2sCaðÞ1 þ sh þ sð1 þ skÞReDa ; Àk À À À À ke 3 3 2 A12 ¼ k ð1 þ skÞDa À Ca sð1 þ shÞþ k Bð1 þ skÞMa þ s 2k ð1 þ shÞ A0 7 Page 14 of 16 Sådhanå (2020) 45:7

 þ sð1 þ skÞReÞÞDaÞÞÞ ; c  À À e 4 4 A13 ¼ k ð1 þ skÞDa þ Ca scð1 þ shÞÀ k Bð1 þ skÞMa A0 À ÀÁ þ sc À 3k2 À c2 ð1 þ shÞ  Àsð1 þ skÞReÞÞDaÞÞ ; Àc  À À e 4 4 A14 ¼ k ð1 þ skÞDa À Ca scð1 þ shÞþ k Bð1 þ skÞMa A0 ÀÀÁ þ sc 3k2 À c2 ð1 þ shÞ  þsð1 þ skÞReÞÞDaÞ ;

k 2 e k À2k A21 ¼ fg2sð1 þ shÞKk þBð1 þ skÞMaðÞ sPe þ Kk ; A22 ¼ e A21; sð1 þ skÞKk c ÈÉÀÁ ÀÁ e 2 2 2 A23 ¼ sk þ c ð1 þ shÞKc þ k Bð1 þ skÞMa sPe þ Kc ; sð1 þ skÞKc À2c A24 ¼e A23; d Àd k 2 Àk 2 c A25 ¼ie kMa; A26 ¼ ie kMa; A31 ¼ ie k ; A32 ¼Àie k ; A33 ¼ ie kc; Àc A34 ¼Àie kc; k2 iekkBðÞk þ Bi Pe ieÀkkBðÞk À Bi Pe A35 ¼s þ ; A41 ¼À ; A42 ¼ ; Pes Kk Kk ieckBðÞc þ Bi Pe A43 ¼À ; Kc Àc ie kBðÞc À Bi Pe d Àd A44 ¼ ; A45 ¼ e ðÞd þ Bi ; A46 ¼ e ðÞBi À d ; Kc

   À UðsÞ UðsÞ 2 c2 ¼ 2kBcPeA35ðÞA26A45 ÀA25A46 Kk c4 ¼ 2k BPeA35ðÞA25A46 ÀA26A45 Kk c0 c0 À 2 þ Kcð2kBcPeA35ðÞA25A46 ÀA26A45 þ Kc 2k BPeA35ðÞþA26A45 ÀA25A46 Kk

À i KkðA25A35ðÞ2cA41 Àðk þ cÞA43 þðk À cÞA44 Â ðiA25A35ðÞðk þ cÞA41 þðk À cÞA42 À 2kA43

þA26A35ððk þ cÞA43 À 2cA41 À iA26A35ððk þ cÞA41 þðk À cÞA42

þðc À kÞA44ÞþðikNeðÞ2cA31 Àðk þ cÞA33 þðk À cÞA34 À 2kA43ÞÀðkNeðÞðk þ cÞA31 þðk À cÞA32 À 2kA33 þðð k þ cÞA À 2cA þ iððk þ cÞA þðk À cÞA 23 21  21 22

þðc À kÞA24ÞA35ÞAð45 ÀA46 ÞÞÞ ; À2kA23ÞA35ÞAð45 ÀA46 ÞÞÞÞ ;     UðsÞ 2 UðsÞ c3 ¼ 2k BPeA35ðÞA26A45 ÀA25A46 Kk c5 ¼ ð2kBPeðA26A35ðÞcA41 À cA42 À kA43 þ kA44 c0 À c0 2 þ Kc 2k BPeA35ðÞþA25A46 ÀA26A45 Kk þ ðikNeðcA31 À cA32 À kA33 ÀÁ Â ðiA25A35ðÞðc À kÞA41 Àðk þ cÞA42 þ 2kA44 þkA34ÞþðÞAcA22 À cA21 þ kA23 À kA24 35ÞA46Þ Kk À Kc

þ iA26A35ððk À cÞA41 þðk þ cÞA42 þ KkKcðiA35ðcA24A42

À2kA44ÞþðkNeðÞðk À cÞA31 þðk þ cÞA32 À 2kA34 À kA24A41 À cA24A41 À 2cA21A42 À kA24A42 þ iððk À cÞA þðk þ cÞA À kA A þ cA A 21 22 21 43 21 43 þ 2kA24A43 À2kA24ÞA35ÞAð45 ÀA46 ÞÞÞ ; Sådhanå (2020) 45:7 Page 15 of 16 7

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