Ex a c t a n alysi s of h e a t c o nv e c tio n in visc o el a s ti c F E N E-P flui d s t h r o u g h iso t h e r m al slit s a n d t u b e s N o r o u zi, M , D a g hi g hi, S a n d B e g, OA h t t p:// dx. doi.o r g/ 1 0. 1 0 0 7/ s 1 1 0 1 2-0 1 7-0 7 8 2-2 Ti t l e Ex a c t a n aly sis of h e a t c o nv e c tio n in vis c o el a s tic F E N E-P flui d s t h r o u g h iso t h e r m al slit s a n d t u b e s A u t h o r s N o r o u zi, M , D a g hi g hi, S a n d B e g, OA Typ e Articl e U RL This ve r sio n is a v ail a bl e a t : h t t p:// u sir.s alfo r d. a c. u k/id/ e p ri n t/ 4 3 9 5 8/ P u b l i s h e d D a t e 2 0 1 7 U SIR is a di git al c oll e c tio n of t h e r e s e a r c h o u t p u t of t h e U niv e r si ty of S alfo r d. W h e r e c o py ri g h t p e r mi t s, full t e x t m a t e ri al h el d in t h e r e p o si to ry is m a d e fr e ely a v ail a bl e o nli n e a n d c a n b e r e a d , d o w nlo a d e d a n d c o pi e d fo r n o nc o m m e r ci al p riv a t e s t u dy o r r e s e a r c h p u r p o s e s . Pl e a s e c h e c k t h e m a n u s c ri p t fo r a n y fu r t h e r c o py ri g h t r e s t ri c tio n s. Fo r m o r e info r m a tio n, in cl u di n g o u r p olicy a n d s u b mi s sio n p r o c e d u r e , pl e a s e c o n t a c t t h e R e p o si to ry Te a m a t: u si r@ s alfo r d. a c. u k . AbstractIn this article, two exact analytical solutions for heat convection in viscoelastic fluid flow through isothermal tubes and slits are presented for the first time. Herein, a Peterlin type of finitely extensible nonlinear elastic (FENE-P) model is used as the nonlinear constitutive equation for the viscoelastic fluid. Due to the eigenvalue form of the heat transfer equation, the modal analysis technique has been used to determine the physical temperature distributions. The closed form solution for the temperature profile is obtained in terms of a Heun Tri-confluent function for slit flow and then the Frobenius method is used to evaluate the temperature distribution for the tube flow. Based on these solutions, the effects of extensibility parameter and Deborah number on thermal convection in FENE-P fluid flow have been studied in detail. The fractional correlations for reduced Nusselt number in terms of material modulus are also derived. Here, it is shown that by increasing the Deborah number from 0 to 100, the Nusselt number is enhanced by 8.5% and 13.5% for slit and tube flow, respectively.
In the present work, analytical solutions are presented for thermal convection of the linear Phan-Thien-Tanner fluid (LPTT) in slits and tubes of constant wall temperature by taking account of the viscous dissipation term. Unlike the similar previous studies in which the advection term was neglected in the heat transfer equation, it is considered in this investigation. A continuous relation between the Nusselt number and the Brinkman number is obtained. Expressions for the temperature distribution are derived in closed form and in terms of a Frobenius series for the slit and tube flows, respectively. Based on these solutions, the effects of fluid elasticity and Brinkman number on thermal convection of LPTT fluid flows are studied in detail. It is shown that at negative Brinkman numbers (fluid cooling), increasing the Deborah number leads to a decrease in the Nusselt number, but an increase in the centerline temperature. Nonetheless, this trend is opposite for positive Brinkman numbers (fluid heating), i.e., an increase in the Nusselt number and a decrease in the centerline temperature. Also, there is a Brinkman number beyond which the Nusselt number is smaller than zero, meaning that there is weak heat convection in the flow. Also, the results confirm that the extensibility parameter affects the temperature profile in the same way as the Deborah number. Keywords Viscoelastic fluid • Phan-Thien-Tanner model • Forced convection • Constant wall temperature tube and slit • Viscous dissipation List of symbols A Area of cross section, m 2 Br Brinkman number, Br = U 2 ∕k(T w −T m) c p Specific heat, J kg −1 K −1 d h Hydraulic diameter, d h = 2R for tube flow and d h = 4H for slit flow, m Deformation rate tensor, Eq. (8) De Deborah number, defined as De = U∕R for tube case and De = U∕H for slit case F Dimensionless function h Heat transfer coefficient, W m −2 K −1 H Half of the distance between two parallel plates, m J Equals 0 for slit and 1 for tube k Conductivity coefficient, W m −1 K −1 K Equals 1.5 and 2 for slit and tube, respectively Nu Nusselt number, Nu = hd h ∕k p Pressure, Pa P Perimeter of cross section, m R Tube radius, m T Fluid temperature, K u Axial velocity, ms −1 U Mean velocity, ms −1 Velocity vector y Radial (tube) or transverse (slit) direction, m z Axial direction, m Greek symbols Equals 1 for pure entropy elasticity and 0 for pure energy elasticity 1 1 =1.5U N ∕U for slit and 1 =2 U N ∕U for tube Constant, Eq. (12) Extensibility coefficient Constant viscosity coefficient, Pa s Relaxation time, s Density, Kg m −3 ∇ Upper-convected derivative of stress tensor, Eq. (9) Viscous dissipation, Eq. (15)
This study experimentally investigates the entry of hydrophobic/hydrophilic spheres into Newtonian and Boger fluids. By considering solution of 82% glycerin and 18% water and solution of 80% glycerin, 20% water and 100 ppm polyacrylamide, Newtonian and Boger fluids are made, respectively. It has been tried that liquids' surface tension, density, and viscosity are almost the same. Thus, all dimensionless numbers are approximately the same at a similar impact velocity except for the elasticity number. A PcoDimaxS highspeed camera captures the spheres' trajectory from the impact to the end of the path. Regarding the range of released height ([Formula: see text]), the impact velocities are approximately in the range of [Formula: see text]. The role of fluid elasticity in combination with the sphere surface wettability on the air cavity formation/evolution/collapse is mainly studied. Also, the kinetics of the sphere motion (velocity, acceleration, and hydrodynamic force coefficient) is studied. The results show that air drawn due to the sphere's impact with the Newtonian liquid is more, and the pinch-off takes place later. Also, shedding bubbles are cusped-shaped in the Boger fluid, while in the Newtonian fluid, they are elliptical. In addition, the most significant impact of surface wettability is observed in the Newtonian fluid. Finally, the results reveal that the sphere in the Newtonian fluid can move faster and travel a longer distance in a specific time interval. The differences observed are closely related to the viscoelastic fluid's elasticity property and extensional viscosity.
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