In the name of Allah the most beneficent the most merciful

Transcription

1 In the name of Allah the most beneficent the most merciful

2 Transient flows of Maxwell fluid with slip conditions

3 Authors Dr. T. Hayat & Sahrish Zaib

4 Introduction

5 Non-Newtonian Newtonian fluid Newtonian fluids are satisfactory to describe the flow phenomenon of gases and liquids containing small molecule. Fluids for which the relation between stress and shear rate is not linear are known as non-newtonian fluid. Some examples of non-newtonian fluids are biological fluids, slurries, suspensions and liquid crystals. Some key factors which are dominant in non-newtonian fluids are Shear rate dependent viscosity Ability to creep Ability to relax stresses Presence of Normal stress differences in simple shear flows Presence of yield stress Elongation viscosity Extra tension

6 Classification of viscoelastic fluid Rate type Differential type Integral type Maxwell Model

7 Maxwell Model The general linear viscoelastic model of a Maxwell fluid was first proposed by James Clerk Maxwell in This model is important because of delayed stress relaxation.

8 Stress relaxation t Consider a material subjected to step strain at time t=0 The stress usually decreases from its initial value rapidly at first than more gradually finally approaching a limiting value. If this limiting value of stress in order to maintain a strain is not zero then the material is a solid. If this value is zero and approach to zero is sufficiently rapid than the material is a fluid. The decrease in value of stress in order to maintain a constant value of strain is called stress relaxation. t Limiting stress Relaxation time t Limiting stress Relaxation time t

9 Limitations of linear viscoelasticity It is recommended for the description of stresses in fluid motion that involve small displacement gradients. It cannot describe normal stress phenomenon. For these nonlinearities upper convected Maxwell model is recommended which was given by James. G. Oldroyd. For abrupt changes in flow geometry and large deformations convected Maxwell model should be used. Examples are glycerin and crude oil.

10 Slip Effects The no-slip condition can create a major error in the viscosity data if the fluid exhibits wall slips. At micro and nano scale apparent violation of the noslip boundary condition is seen. Boundary conditions for hydrodynamics was widely discussed during 19 th century. Navier introduced a linear boundary condition which was later proposed by Maxwell. Component of fluid velocity tangent to the surface U is proportional to the rate of strain at the surface.

11 U n u u T. 1 nn Where n denotes the normal to the surface, directed into the liquid. is the slip length. For a pure shear flow slip length can be interpreted as the fictitious distance below the surface. Where the no-slip boundary condition would be satisfied.

12 No- slip is valid when particles close to a surface do not move along with a flow when adhesion is stronger than cohesion. But this is only true macroscopically. In hydrodynamics no-slip boundary fails for hydrophobic surfaces. No-slip also fails for large contact angles. A droplet resting on a solid surface and surrounded by a gas forms a characteristic contact angle θ. If the solid surface is rough, and the liquid is in intimate contact with the solid asperities, the droplet is in the Wenzel state. If the liquid rests on the tops of the asperities, it is in the Cassie Baxter state.

13 No-slip also fails for rough surfaces. Slip occurs over a gap near the surface. Also no-slip does not holds at a very low pressure. No- slip fails for moving contact lines. No- slip is not valid for polyethylene and rubber compounds. No-slip also fails for suspensions. Effect of slip depends on length scale of the flow. Applications Flows in porous media Micro and nano fluidics Friction studies Biological fluids Extrusion

14 Governing Equations The equations governing the flow of an incompressible fluid are divv 0, (1) V t V. V divt, (2) where t is the time, ρ is the density and the velocity V for flow in the x-direction is V u y,t,0,0, (3) in which u is the velocity in the x-direction and an extra stress tensor S in Maxwell fluid satisfies the following relation

15 1 1 D Dt S A 1. (4) In above expression μ is the dynamic viscosity, λ1 is the relaxation time and the first Rivilin-Ericksen tensor A1 can be written as A 1 V V, (5) where is the gradient operator and asterik denotes the matrix transpose. The Cauchy stress tensor T is T pi S, where p is the pressure and I is the identity tensor. (6) In view of Eqs. (3)-(5) we have 1 1 t S xy u y. (7)

16 Stokes Flow We consider the unsteady flow of an incompressible Maxwell fluid over a rigid plate at y=0. We select x-axis parallel to the plate. Here y-axis is taken perpendicular to x-axis. The fluid fills the half space y>0. Initially (at t=0) both fluid and plate are at rest. For t>0, the plate starts oscillations in its own plane. The fluid far away from the plate is at rest. The continuity equation (1) is identically satisfied and Eqs. (2), (3) and (7) give 1 1 t u t 2 u y 2, (8)

17 where ν(=(μ/ρ)) is the kinematic viscosity. Here we consider the existence of slip defined by the relative velocity between the speed of plate and the velocity of fluid at the plate u(0,t) which is assumed to be proportional to the shear rate at the plate. Accordingly the boundary and initial conditions are 1 1 t u 0,t u 0, t y u y,t 0, u y,0 0, 1 1 t U w, (9) (10) U w U 0 cos t, (11) U w U 0 sin t, (12) where γ is the slip parameter.the parameter γ is of dimension of lenght and ω is the frequency of vibration.

18 Laplace Transform Laplace integral is also known as fredholm integral of 1 st kind defined as u s ue st dt, 0 s i This is one sided unilateral Laplace transform technique. Here s is a complex variable which is useful for analyzing casual linear systems provided that integral converges. gives exponentially changing amplitude and is the frequency of sinusoids.

19 Sufficient condition that guarantee the existence of Laplace is function should be piecewise continuous and of exponential order. Named in honor of Pierre Simon De Laplace. Changes a signal in time domain into a signal in frequency domain Complex exponentials are a compact way of representing both sinusoids and exponentials

20 Defining Laplace transform us(y,s) by u s ue st dt, the problem consisting of Eqs. (8)-(11) becomes d 2 u s dy s u s 0,s du s 0,s dy s s u s y,s 0. u s 0, U s s s 2 2, For sinusoidal oscillation, the slip condition gives 1 1 s u s 0,s du s 0,s dy U s s 2 2. (13) (14) (15) (16) (17)

21 The solutions of Eq. (14) satisfying the above boundary conditions are u s y,s U 0 se 1 1 s s s s 1 1 s y, (18) u s y,s U 0 e 1 1 s s s s 1 1 s y. (19)

22 Exact solutions Solutions for cosine oscillations at the wall for positive values of γ Now for exact solutions we have used the residue theory to evaluate the inverse laplace transform of Eq. (18) defined by f t 2 i 1 i e st F s ds i 0 0 increasing exponentials decreasing exponentials s plane

23 We note that in the inverse Laplace transform of Eqs. (18) and (19), there are simple poles at s=-iω and s=iω and branch point at s=0. The value of s given by s=(ν/(γ²-λ1ν)) also represents a simple pole. The solution at branch point is given by u, U 0 u sp U re r r 2 1 sin 1 1 r r 2r 1 1 r cos In above expression η, τ, β and λ1 are 1 2 2r 1 1 r 1 1r r dr. (20) y, t, 2, 1 1. (21)

24 Dimensionless expression of steady periodic velocity is u sp, U 0 1 A cos C 2 1 A 2 B 2 B sin C 2 e D 2, (22) in which A , B , (23) C , D (24)

25 The steady periodic relative velocity (between the velocity of the fluid at the wall and the wall itself) is u rel U 0 u sp 0, U 0 cos B sin A cos 2 A 2 B 2 cos. 1 A 2 B 2 (25) The transient velocity is ut, U re r r 2 1 sin 1 1 r r 2r 1 1 r cos 1 2 2r 1 1 r 1 1r r dr.(26)

26 Solutions for sine oscillations at the wall for positive values of γ Adopting the similar methodology as in the previous subsection we obtain the following results u, U 0 u sp U e r r 2 1 sin 1 1 r r 2r 1 1 r 1 2 2r 1 1 r cos 1 1 r r dr. (27) u sp, U 0 1 A sin C 2 1 A 2 B 2 B cos C 2 e D 2, (28) u rel U 0 u sp 0, sin A sin B cos 2 A 2 B 2 sin U 0 1 A 2 B 2, (29) u t, U e r r 2 1 sin 1 1 r r 2r 1 1 r 1 2 2r 1 1 r cos 1 1 r r dr. (30)

27 Graphical results (Stokes flow) Fig. 1a. Effects of slip parameter on the steady periodic velocity profile (Stokes flow).

28 Fig. 1b. Effects of relaxation time λ1 on the steady periodic velocity profile (Stokes flow).

29 Fig. 2a. Effects of slip parameter on steady wall slip velocity (Stokes flow).

30 Fig. 2b. Effects of relaxation time λ1 on steady wall slip velocity (Stokes flow).

31 Fig. 3a. Effects of slip parameter on the steady periodic velocity profile (Stokes flow).

32 Fig. 3b. Effects of relaxation time λ1 on the steady periodic velocity profile (Stokes flow).

33 Fig. 4a. Effects of slip parameter on the steady wall slip velocity (Stokes flow).

34 Fig. 4b. Effects of relaxation time λ1 on the steady wall slip velocity (Stokes flow).

35 Fig. 5a. Effects of large range of slip parameter on the slip velocity at a specific time (Stokes flow).

36 Fig. 5b. Effects of oscillating slip parameter on the transient velocity profile (Stokes flow)

37 Couette Flow Here we consider an incompressible Maxwell fluid between two plates distant h apart. X and Y-axes are chosen parallel and normal to the plates. The flow is induced by the oscillations of lower plate. The problem statement consists of Eq. (8) and the following boundary and initial conditions. 1 1 t u 0,t u 0,t y 1 1 t U w, (31) 1 1 t u h, t u h, t y 0, (32) Writing u y,0 0. t, U u h, Y y h, 1 1 (33) (34)

38 and dropping asteriks one obtains 1 1 U 1 R 2 U Y 2, (35) 1 1 U 0, h U 0, Y R 0 sin, (36) 1 1 in which U 1, h U Y,0 0, U 1, Y 0, R h2, R 0 U 0 h, (37) (38) (39)

39 R is an oscillating Reynolds number and R0 denotes the ratio between the lateral Reynolds number to an oscillating Reynold number. The solution in the transformed s-plane is U s Y, s R 0 R 0 s 2 1 h sinh Rs 1 1 s Rs cosh Rs s 1s 2 h cosh Rs 1 1 s Y Rs 1 1 s cosh Rs 1 1 s 1 1 s h 2 Rs sinh Rs 1 1 s h cosh Rs 1 1 s Rs 1 1 s sinh Rs 1 1s 2 h sinh Rs 1 1 s cosh Rs 1 1 s Rs 1 1 s Y. (40) 1 1 s h 2 Rs sinh Rs 1 1 s

40 In the inverse Laplace transform of above equation there are simple poles at s=-i and s=i and infinite number of zeros at s n 2 1 (where jn is a real number and n is the index integer number of the pole changing from one to infinity). Here jn satisfies j n 2 R tan j n 2 h j n 1 1 s n h 2 Rsn. (41) We note that the poles located at s n transient behavior of the velocity. The solution in (Y,τ) plane is 2 1 j n 2 R are responsible for the U Y, R 0 n 1 Re s U s Y, s e s s n Re s U s Y, s e s i Re s U s Y,s e s i, (42)

41 transient velocity profile which for large times decay rapidly i.e. U t Y, R 0 R n 1 F1 j n, Y F 2 j n e j n 2 R 2 1, (43) F 1 j n,y sin j n h 2j n j n 2 R cos j n cos j n Y cos j n h 2j n j n 2 R sin j n sin j n Y, (44)

42 h h 1h R h j n 2 R sin j n F 2 j n j n 2 R j n 2 2 R j n 2 R 2 h cos j n 2j n h 2 j n R R 2 1 (45)

43 The steady periodic velocity through the residues at s=-i and s=i is U Y, R 0 M 1M 4 M 2 M 3 f 2 Y sin f 1 Y cos M 3 2 M 4 2 M 1M 3 M 2 M 4 f 1 Y sin f 2 Y cos M 3 2 M 4 2 M 3M 6 M 5 M 4 f 3 Y cos f 4 Y sin M 3 2 M 4 2 M 5 M 3 M 6 M 4 f 3 Y sin f 4 Y cos M 3 2 M 4 2, (46) f 1 Y sinh cy cos dy, f 2 Y cosh cy sin dy, (47) f 3 Y cosh cy cos dy, f 4 Y sinh cy sin dy, (48)

44 M 1 cosh c cos d asinh c cos d bcosh c sin d, h (49) M 2 sinh c sin d h a cosh c sin d b sinh c cos d, (50) M 3 2 h ccosh c cos d d sinh c sin d M 4 2 h 1 2 h 2 R cosh c sin d sinh c cos d, csinh c sin d d cosh c cos d 1 2 h 2 R sinh c cos d cosh c sin d, (51) (52) M 5 sinh c cos d a cosh c cos d b sinh c sin d, (53) h M 6 cosh c sin d a sinh c sin d b cosh c cos d, (54) h

45 a R , b R , c R , d R (55) The velocity field for unsteady flow is U U t U sp. (56)

46 Graphical results (Couette flow) Fig. 6a. Effects of slip parameter on the steady periodic velocity profile (Couette flow).

47 Fig. 6b. Effects of relaxation time λ1 on the steady periodic velocity profile (Couette flow).

48 Fig. 7a. Effects of oscillating Reynolds number on the steady periodic slip velocity at Y=0 (Couette flow).

49 Fig. 7b. Effects of oscillating Reynolds number for different values of λ1 on the steady periodic slip velocity at Y=0 (Couette flow).

50 Fig.8a. Effects of oscillating Reynolds number on the steady periodic slip velocity at Y=1 (Couette flow).

51 Fig.8b. Effects of oscillating Reynolds number for different values of λ1 on the steady periodic slip velocity at Y=1 (Couette flow).

52 Fig. 9a. Effects of slip coefficient on the steady periodic slip velocity at Y=0 (Couette flow).

53 Fig. 9b. Effects of slip coefficient for different values of λ1 on the steady periodic slip velocity at Y=0 (Couette flow).

54 Fig. 10a. Effects of slip coefficient on the steady periodic slip velocity at Y=1 (Couette flow).

55 Fig. 10b. Effects of slip coefficient for different values of λ1 on the steady periodic slip velocity at Y=1 (Couette flow).

56 Closing remarks The oscillations in the velocity are decreased as the slip parameter increases near the vibrated wall for Stokes and Couette flows. The relaxation time increases the velocity profile for different values of slip and oscillating Reynolds number. The overall effect of vibrational Reynolds number is to increase the velocity profile. At certain times, the slip coefficient increases the fluid slip velocity for different values of vibrational Reynolds number and relaxation time. In Stokes flow the steady periodic velocity in case of cosine oscillations is achieved earlier when compared to the sine oscillations. The time required to reach the steady periodic velocity increases when relaxation time is increased.

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