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Lz2520802095

3. 3. C. Sudhakar, N. Bhaskar Reddy, B. Vasu, V. Ramachandra Prasad / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue5, September- October 2012, pp.2080-2095 2. CONSTITUTIVE EQUATIONS FOR This rheological model is very versatile and robust THE WALTERS-B VISCOELASTIC and provides a relatively simple mathematical FLUID formulation which is easily incorporated into Walters [23] has developed a physically boundary layer theory for engineering applications accurate and mathematically amenable model for [25, 26]. the rheological equation of state of a viscoelastic fluid of short memory. This model has been shown 3. MATHEMATICAL MODEL: to capture the characteristics of actual viscoelastic An unsteady two-dimensional laminar free polymer solutions, hydrocarbons, paints and other convective flow of a viscoelastic fluid past a semi- chemical engineering fluids. The Walters-B model infinite vertical plate is considered. The x-axis is generates highly non-linear flow equations which taken along the plate in the upward direction and the are an order higher than the classical Navier-Stokes y-axis is taken normal to it. The physical model is (Newtonian) equations. It also incorporates elastic shown in Fig.1a. properties of the fluid which are important in extensional behavior of polymers. The constitute u equations for a Walters-B liquid in tensorial form v may be presented as follows: X pik   pgik  pik * (1) Boundary 1 Tw , Cw layer pik *  2    t  t * e1ik  t *  dt * (2) T , C   N    t  t*      e   t  t *  /  d  (3) o Y 0where pik is the stress tensor, p is arbitrary isotropic pressure,g ik is the metric tensor of a fixed coordinate system xi, eik 1 isthe rate of strain tensor and N   is the distribution functionof relaxation times, . The following generalized form of (2) has Fig.1a. Flow configuration and coordinate systembeen shown by Walters [23] to be valid for all classes of motionand stress. Initially, it is assumed that the plate and the x 1pik  x, t   2    t  t  *m *r e1mr  x*t *  dt * (4) * * fluid are at the same temperature T and   x x  concentration level C everywhere in the fluid. At time, t  >0, Also, the temperature of the plate and in which xi  xi * *  x, t, t  denotes the position at * the concentration level near the plate are raised to time t* of the element which is instantaneously at   Tw and C w respectively and are maintained the position, xi, at time, t. Liquids obeying the constantly thereafter. It is assumed that the relations (1) and (4) are of the Walters-B’ type. For such fluids with short memory i.e. low relaxation concentration C  of the diffusing species in the times, equation (4) may be simplified to: binary mixture is very less in comparison to the other chemical species, which are present, and hence the Soret and Dufour effects are negligible. It is also e1ik p*ik  x, t   20e   2k0 (5) assumed that there is no chemical reaction between 1 ik t the diffusing species and the fluid. Then, under the  above assumptions, the governing boundary layer in which 0   N  d defines the limiting equations with Boussinesq’s approximation are 0 viscosity at low shear rates, u v Walters-B’  0 (6)  x y k0    N  d is the Walters-B’ viscoelasticity 0  parameter and is the convected time derivative. t 2082 | P a g e
4. 4. C. Sudhakar, N. Bhaskar Reddy, B. Vasu, V. Ramachandra Prasad / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue5, September- October 2012, pp.2080-2095u u u k Tw  T  u v  g  T   T   g  *  C   C     t  x y T0  2u  3u (7) Typical values of  are 0.01, 0.05 and 0.1 2  k0 2 corresponding to approximate values of y y t  k Tw  T  equal to 3.15 and 30 k for a reference T  T  T  k  2T  u v  (8) temperature of T0 =300 k. t  x y  c p y 2 On introducing the following non-dimensional C  C  C   2C   quantities u v  D 2   cvt  (9) t  x y y y x yGr1/4 uLGr 1/2 X ,Y  ,U  , L L  The initial and boundary conditions are T   T  C   C  tL2 1/2 (11) t   0 : u  0, v  0, T   T , C   C 0   T ,C  , t  Gr ,  Tw  T   Cw  C   t   0 : u  0, v  0, T   Tw , C   Cw at y  0    * (Cw  C )   k Gr1/2 u  0, T   T , C   C at x  0   N ,  0 2 ,  (Tw  T )   L u  0, T   T  , C   C  as y (10)    g  (Tw  T )   Gr  , Where u, v are velocity components in x u0 3 and y directions respectively, t  - the time, g – the   kL Tw  T    Pr  , Sc  ,  acceleration due to gravity,  - the volumetric  D Tw coefficient of thermal expansion,  - the * Equations (6), (7), (8), (9) and (10) are reduced to volumetric coefficient of expansion with the following non-dimensional form concentration, T  -the temperature of the fluid in U U  0 (12) the boundary layer, C  -the species concentration X Y  in the boundary layer, Tw - the wall temperature, U U U  2U  3U (13) U V   T  NC   2  T - the free stream temperature far away from the t X Y Y 2 Y t   plate, C w - the concentration at the plate, C - the free stream concentration in fluid far away from the plate,  - the kinematic viscosity,  - the thermal T T T 1  2T diffusivity,  - the density of the fluid and D - the U V  (14) species diffusion coefficient. t X Y Pr Y 2 In the equation (4), the thermophoretic velocity C C C 1  2C U V   Vt was given by Talbot et al. [39] as t X Y Sc Y 2 T kv T   C T  2T  VT   kv   C 2  (15) Tw y 1  Tw Gr 4  Y Y Y  Where Tw is some reference temperature, the value The corresponding initial and boundary conditions of kv represents the thermophoretic diffusivity, and are k is the thermophoretic coefficient which ranges in t  0 : U  0, V  0, T  0, C  0 value from 0.2 to 1.2 as indicated by Batchelor and t  0 : U  0, V  0, T  1, C  1 at Y  0, Shen [40] and is defined from the theory of Talbot et al [39] by U  0, V  0, C  0 as X  0 (16) 2Cs (g /  p  Ct K n ) 1  K n (C1  C2e  Cs / Kn )    U  0, V  0, C  0 as Y   k (1  3Cm K n ) 1  2g /  p  2Ct K n  Where Gr is the thermal Grashof number, Pr is the fluid Prandtl number, Sc is the Schmidt A thermophoretic parameter  can be defined (see number, N is the buoyancy ratio parameter,  is the Mills et al 2 and Tsai [41]) as follows; viscoelastic parameter and  is the thermophoretic parameter. To obtain an estimate of flow dynamics at the barrier boundary, we also define several important 2083 | P a g e