Orr-sommerfeld Stability Analysis of Two-fluid Couette Flow

Orr-sommerfeld Stability Analysis of Two-fluid Couette Flow ORR-SOMMERFELD STABILITY ANALYSIS OF TWO-FLUID COUETTE FLOW WITH SURFACTANT V.P.T.N.C.Srikanth BOJJA1* , Maria FERNANDINO1, Roar SKARTLIEN2 ABSTRACT In the present work, the surfactant induced instability of a sheared two fluid system is examined. The linear stability analysis of two-fluid couette system with an amphiphilic surfactant is carried out by developing Orr-Sommerfeld type stability equations along with surfactant transport equation and the system of ordinary differential equations are solved by Chebyshev Collocation method[1,2]. Linear stability analysis reveals that the surfactant either induces Marangoni instability or significantly reduces the rate at which small perturbations decay. Keywords:Linear stability, Orr-Sommerfeld, Marangoni mode, Amphiphilic surfactant. NOMENCLATURE A complete list of symbols used, with dimensions, is given. Greek Symbols Growth rate Surfactant concentration Mass density, [kg/m3]. Dynamic viscosity, [kg/m.s]. Height of perturbed inteface Surface tension Wave number ,Stream functions Latin Symbols Capillary number Marangoni number Number of Collocation pints Reynolds number Plate/Wall velocity Complex wave spped Width of fluid layer Amplitude of Pressure disturbance Amplitude of surfactant concentration disturbance Amplitude of interface perturbation Viscosity ratio Depth ratio Shear of basic velocity Velocity, [m/s]. Sub/superscripts Index i. Index j. Perturbed quantities Base state quantities INTRODUCTION Two layer channel flows and flows with and without surfactants have been given considerable importance because of its numerous industrial applications. Oil recovery[3], lubricated pipelining[4], liquid coating processes[5] are typical industrial situations where Two layer channel flows are often seen. Surfactants also have wide range of industrial applications for example in enhanced oil recovery[6]. Using Perturbation analysis, the primary instability of the two-layer plane Couette–Poiseuille flow was studied by Yih[7] and his studies revealed that even at small Reynolds numbers, the interface is susceptible to long-wave instability associated with viscosity stratification. Yiantsios Higgins[8] later extended this study for small to large values of wavenumber and confirmed the existence of the shear mode instability. Boomkamp Miesen[9] came up with the method of an energy budget for studying instabilities in parallel two-layer flows, where energy is supplied from the primary flow to the perturbed flow and instability appears at sufficiently long wave numbers through the increase of kinetic energy of an infinitesimal disturbance with time. In the presence of surfactant at the sheared interface, Frenkel Halpern[10,11] discovered that even in the stokes flow limit, the interface is unstable as the surfactant induces Marangoni instability, which was later confirmed by Blyt h Pozrikidis[12]. In the case of Stokes flow, they identified two normal modes, the Yih mode due to viscosity stratification inducing a jump in the interfacial shear, and the Marangoni mode associated with the presence of the surfactant. In contrast, at finite Reynolds numbers, infinite number of normal modes are possible and by parameter continuation with respect to the Reynolds number the most dangerous Yih and Marangoni modes can be identified. In this article, the effect of an insoluble surfactant on the stability of two-layer couette channel flow is studied in detail for low to moderate values of the Reynolds number. To isolate the Marangoni effect, gravity was suppressed in this problem and this was done by considering equal density fluids. Linear stability analysis was carried out by formulating Orr–Sommerfeld boundary value problem, which was solved numerically using Chebyshev collocation method[1][2] for all wavenumbers. Both Marangoni mode and Shear mode are detected and utmost focus is given to Marangoni mode as Shear mode is always stable at moderate to long wavenumbers under the influence of inertia. The rest of the paper is organized as follows. In  § Model description, the governing equations for the system in question are laid out, Normal mode analysis of the physical system is carried out, Orr–Sommerfeld boundary value problem is formulated. General description of Chebyshev collocation method and detailed description of numerical simulation of Orr–Sommerfeld boundary value problem by Chebyshev collocation method and validation of numerical method with literature data is given in  § Numerical method. Detailed discussion of results done in  § Results. The concluding remarks and outlook for further-work in  § Conclusions. Finally acknowledgements in and  §Acknowledgemnts. Model description Consider two super-imposed immiscible liquid layers between two infinite parallel plates located at, as in Fig. 1. Let the basic flow be driven only by steady motion of plates. It is well known that the basic ‘‘Couette’’ velocity profiles are steady and vary only in the span-wise direction and in the basic state, the unperturbed interface between the liquids is flat and is located at. The gravity is suppressed in this problem by considering equal densities in order to investigate the effects of surfactant and inertia on the stability of the system under consideration. The subscripts 1 and 2 refer to the lower or upper fluid, respectively and channel walls move in the horizontal direction, x, with velocities and. The interface is occupied by an insoluble surfactant with surface concentration which is only convected and diffused over the interface, but not into the bulk of the fluids thus locally changing the surface tension . Governing equations The mass and momentum conservation equations governing the two-layer system are , (1) Where subscript represents lower and upper liquid layers respectively. Here , Figure 1: Schematic sketch of Couette-Poiseuille flow with surfactant laden interface. The perturbed interface is shown as sinusoidal curve. is the concentration of insoluble surfactant. The associated boundary conditions for the system are no slip and no penetration boundary conditions at the walls. ,at and ,at The associated interface conditions are continuity of velocity, tangential stress and normal stress. Continuity of velocity at the interface , at The tangential and normal stress conditions at the interface are given by (2) Where are stress tensors, is unit normal, is unit tangent and Kinematic interfacial condition is The surfactant transport equation[13] at the interface is given by (3) Where is surface molecular diffusivity of surfactant. is usually negligible and neglected in this case. We introduce dimension less variables as follows , , , , The dimensionless variables in base state for the couette flow with flat interface and uniform surfactant concentration are given by , ( ) and , () Where is shear of basic velocity at interface and is given by We consider the perturbed state with small deviation from the base state: ,,,, Now we represent disturbance velocity in disturbance stream-functions and such that ,,, Performing normal mode analysis by substituting Where is wave number of the disturbance, and are constants, and is the complex wave speed. Linearizing the kinematic boundary condition yields . Linerarizing the dimensionless x and y-components of Navier-Stokes equation (2) followed by subtraction from the corresponding base state equations and elimination of pressure terms, yields two 4th order Orr-Sommerfield ODEs in stream-functions, one for each fluid. (4a) (4b) Where is the Reynolds number and . (when,) Boundary conditions at wall in terms of stream-functions are (5a) (5b) Continuity of velocity at interface gives , (5c) Linearization of normal stress condition gives (5d) Linearization of surfactant transport equation gives Linearization of tangential stress balance condition gives Where is the Marangoni number. By substituting the value of from linearized surfactant transport equation in linearized tangential stress balance condition gives (5e) For each value of Eqs. (4),(5) forms a eigen value problem, which was numerically solved using chebyshev collocation method[1,2] and QZ algorithm for determining the complex phase velocity . Numerical method The two Orr-Sommerfield equations eqs. (4) along with eight boundary conditions eqs. (5) are solved numerically using Pseudo-spectral Chebyshev collocation method[1,2]. To implement the Chebyshev method, we transformed each of the two fluid domains into standard Chebyshev domain that is Fluid 1 domain is mapped to and Fluid 2 domain is mapped to by substituting and respectively. Next, we represent each stream function as truncated summation of orthogonal Chebyshev polynomials by setting. and(6) Where and are unknown Chebyshev coefficients and N is the number of Cheyshev collation points in each domain. Upon substituting eq. (6) in eq. (4) and projecting them on to arbitrary orthogonal functions and respectively by taking the Chebyshev inner product, . these two Chebyshev inner products forms N-3 equations each summing up to 2N-6 equations and N+1 coefficients in and N+1 coefficients . 2N-6 equations along with 8 boundary conditions obtained by substituting eq. (6) in eq. (5) and 2N+2 coefficients forms a linear system Where, and,are square matrices of size 2N+2. This generalized eigen value problem was solved by QZ algorithm to obtain and subsequently growth rate, .We used, above which the eigen values are independent of number of collocation points. The accuracy of the Numerical method is checked by comparing current results with published literature[10] for the Two layer couette flow with an insoluble surfactant in stokes flow limit. To make this comparison, growth rates are calculated by muting the inertial terms by settingin the our code and with same parameters as in Halpern’s[10] Fig 2a and Fig 2b, where growth rates are predicted by long-wave evolution equation. Fig xxx shows excellent agreement between two numerical procedures. Figure 2: Dispersion curves for the most (a)Unstable Figure 3: Dispersion curve for the (solid line), (dashed line), at, ,, RESULTS and discussions Blyth and Pozrikidis[14] observed that in the Stoke’s flow limit, there exists two modes that govern the stability of a two-layer couette flow system with surfactant: the Marangoni mode and the Yih mode associated with surfactant and the clean liquid-liquid interface respectively. But on the other hand, in flows with inertia, there exists more than two normal modes. From Fig. 3, the broken line corresponding to is above the solid line, which corresponds to , it is evident that the surfactant in the presence of inertia has significantly reduced the rate at which small perturbations decay. Earlier stability analysis for stoke flow in presence of surfactant opens up a range of unstable wave numbers extending from zero up to the critical wavenumber .The neutral stability curve Fig. 4 for values (,, and ) is in accordance with the earlier stokes flow stability analysis and in addition at , a second small window of stable wave numbers appears to form an island of stable modes, wit h the island tip located at . In Fig. 5 we plot the growth rate of the Marangoni mode against the Reynolds number, up to and beyond, for , corresponding to the stable island tip. At, linear stability for Stokes flow predicts the growth rate, for the Marangoni mode. The present results confirms that the Marangoni mode at marks the inauguration of the lower stable loop. In Fig. 6 for a fixed Reynolds number , we show the dependence of the growth rates of the Marangoni mode on the wave number. The close-up near , presented in Fig. 6(b), shows that the Marangoni mode has negative growth rate for small band of wave numbers ranging from and has positive growth rate thereafter up-to , beyond which the Marangoni mode is stable again. These results clearly demonstrate the crucial role of the surfactant, which either provokes instability or significantly lowers the rate of decay of infinitesimal perturbations. Figure 4: Neutral stability curves for ,, and Figure 5: Growth rate vs. Reynolds number for the Marangoni mode for, , , , , Figure 6: Dispersion curve for the Marangoni mode (solid line) for,, , ,, (b) Zoom-in of (a) around Figure 7: Neutral stability curves for , , and (a) (b) (c) (d) (e) (f) Further, we investigated the effect of Marangoni number on the stability of the system under consideration via Fig 7(a) and this shows that in the devoid of surfactant that is at there is very small band of wavenumbers where the system is unstable for any Reynolds number. Moreover around the band of unstable wavenumbers is slightly larger than at any arbitrary Re. In presence of surfactant, Fig. 7(b)-7(e) a second small window of stable wave numbers appears to form

Exemplification Essay

Exemplification Essay I once heard a story about a restaurant manager named Jerry. Jerry was the type of guy that always was in a good mood no matter what. He was a natural motivator. When one of his employees would come in hating life he would be help them to look on the positive side of the situation. One of his other employees was curious, so one day he went up to Jerry and asked, â€Å"I don’t get it man, how can you be in a good positive mood all of the time. How do you do it? † Jerry replied, â€Å"Each morning I get up and I have a decision to make: to be in a good mood or to be in a bad mood. I choose to be in a good one. Every time something bad happens I can choose blame it on myself or I can choose to learn from the situation. † â€Å"It’s not that easy. † The employee protested. â€Å"Yeah it is. † Jerry said. â€Å"Life is all about the choices that you make and how you handle them, you choose to be in a good mood or to be in a bad mood. Bottom line: Attitude is everything. † The employee reflected on Jerry’s example. Later he left the restaurant business to start a business of his own. He lost touch with Jerry but would often use his teachings in his everyday choices in his life. A number of years later he heard that Jerry had left the back door open to his restaurant and he was robbed at gunpoint. While Jerry was opening the safe he was shaking and he slipped. One of the robbers, on edge, shot Jerry and they scurried away. Luckily, he wasn’t lying out for too long for an ambulance to come rush Jerry to the hospital; Jerry survived. Six months after the robbery the old employee met up with Jerry and had asked him about the incident. Jerry replied, â€Å"The Paramedics in the Ambulance were great. They kept on telling me that I was going to be fine until they handed me over to the doctors, that’s when I got worried. The doctors and the nurses eyes read that I was a dead man, not likely to survive. A nurse asked, â€Å"Do you have any allergies? †. I had a choice to make, to live or to die, to be in a good mood or a bad mood. â€Å"Yes† I replied to the nurse. The doctors stopped what they were doing waiting for my answer. â€Å"Bullets! † During their laughter they I told them â€Å"I’m not dead yet, make sure I don’t die. † Jerry lived by the outstanding skill of the doctors, but also due to his attitude. On July 19, 2011 my dad taught me the greatest lesson that I could ever be taught. Every year my paternal side of the family has a reunion in Capitol Reef National park, Southern Utah. My dad came up with the idea that we should leave a couple days earlier and ride our road bikes ahead of the rest of the family. Knowing me, he knew I was â€Å"in†. I just had had a friend move in with me because his family moved to Kentucky and he wanted to finish his senior year at our high school. He joined us on the ride. We started to do some training rides to get ready for the two hundred mile trek of which we were about to embark. July 19: woke up, ate a banana and oatmeal, got dressed in biking gear, took the first pedal and off we went. There were two different routes we could have taken: east of the lake or west of the lake. We chose to go west to avoid the traffic and the higher risk. When we reached the west side of the lake there was a head wind, so we started to draft off of each other. Every five minutes we would rotate who was in lead. It was my turn to lead, my dad following me, and my friend Kallen following him. Kallen’s headphones fell out of his ears and got wrapped up in his front wheels. While drafting he reached down to clear the remains of his headphones and his arm got sucked into the spokes which made him flip over the handle bars. I didn’t notice over the sound of my music that he had crashed but my dad did hear the accident and waved me down to go back. We frantically signaled a car down. A nice lady stepped out and offered to take Kallen to the hospital. My dad and I continued on our way. We reached the other side of the lake and I got a flat tire, which was my second for the day and I was already upset that we lost Kallen. We patched the flat and continued on our ride. Anger was just bottling up inside of me as we entered the next town. When we arrived we sat down to eat. It was about 3 o’clock and we still had seventy miles to go which added to my anger. After lunch we rode over to a park to use the public bathrooms. I take a short cut through a little dirt field. I used the restroom and hopped back on my bike to begin riding again and my tire was flat, again. â€Å"AHHH! †, I screamed. We went and sat on the grass to patch it. My attitude has not only has affected me, but has affected my dad too. â€Å"You better get in a better mood† he said†Ã¢â‚¬ ¦because you are acting like a little baby. You have a choice to make; be in a good mood or a bad one, because we are going to finish this ride whether you want to or not. † I was shocked. My dad had never talked to me like that before. I realized that attitude is everything. We rode into the dark that night till we reached one hundred and twenty miles and the next day we biked the remaining 80 and made it to Capitol Reef. July 19 is my birthday. It had to have been the worst but also the best birthday that I have ever had. Your attitude towards something can change your life. Whether it is as big as saving your life or as small as changing your mood such as finishing a biking trip with your dad. It will stick with you and make you a better person and a happy person. Bottom line: Attitude is Everything. Exemplification Essay Exemplification Essay I once heard a story about a restaurant manager named Jerry. Jerry was the type of guy that always was in a good mood no matter what. He was a natural motivator. When one of his employees would come in hating life he would be help them to look on the positive side of the situation. One of his other employees was curious, so one day he went up to Jerry and asked, â€Å"I don’t get it man, how can you be in a good positive mood all of the time. How do you do it? † Jerry replied, â€Å"Each morning I get up and I have a decision to make: to be in a good mood or to be in a bad mood. I choose to be in a good one. Every time something bad happens I can choose blame it on myself or I can choose to learn from the situation. † â€Å"It’s not that easy. † The employee protested. â€Å"Yeah it is. † Jerry said. â€Å"Life is all about the choices that you make and how you handle them, you choose to be in a good mood or to be in a bad mood. Bottom line: Attitude is everything. † The employee reflected on Jerry’s example. Later he left the restaurant business to start a business of his own. He lost touch with Jerry but would often use his teachings in his everyday choices in his life. A number of years later he heard that Jerry had left the back door open to his restaurant and he was robbed at gunpoint. While Jerry was opening the safe he was shaking and he slipped. One of the robbers, on edge, shot Jerry and they scurried away. Luckily, he wasn’t lying out for too long for an ambulance to come rush Jerry to the hospital; Jerry survived. Six months after the robbery the old employee met up with Jerry and had asked him about the incident. Jerry replied, â€Å"The Paramedics in the Ambulance were great. They kept on telling me that I was going to be fine until they handed me over to the doctors, that’s when I got worried. The doctors and the nurses eyes read that I was a dead man, not likely to survive. A nurse asked, â€Å"Do you have any allergies? †. I had a choice to make, to live or to die, to be in a good mood or a bad mood. â€Å"Yes† I replied to the nurse. The doctors stopped what they were doing waiting for my answer. â€Å"Bullets! † During their laughter they I told them â€Å"I’m not dead yet, make sure I don’t die. † Jerry lived by the outstanding skill of the doctors, but also due to his attitude. On July 19, 2011 my dad taught me the greatest lesson that I could ever be taught. Every year my paternal side of the family has a reunion in Capitol Reef National park, Southern Utah. My dad came up with the idea that we should leave a couple days earlier and ride our road bikes ahead of the rest of the family. Knowing me, he knew I was â€Å"in†. I just had had a friend move in with me because his family moved to Kentucky and he wanted to finish his senior year at our high school. He joined us on the ride. We started to do some training rides to get ready for the two hundred mile trek of which we were about to embark. July 19: woke up, ate a banana and oatmeal, got dressed in biking gear, took the first pedal and off we went. There were two different routes we could have taken: east of the lake or west of the lake. We chose to go west to avoid the traffic and the higher risk. When we reached the west side of the lake there was a head wind, so we started to draft off of each other. Every five minutes we would rotate who was in lead. It was my turn to lead, my dad following me, and my friend Kallen following him. Kallen’s headphones fell out of his ears and got wrapped up in his front wheels. While drafting he reached down to clear the remains of his headphones and his arm got sucked into the spokes which made him flip over the handle bars. I didn’t notice over the sound of my music that he had crashed but my dad did hear the accident and waved me down to go back. We frantically signaled a car down. A nice lady stepped out and offered to take Kallen to the hospital. My dad and I continued on our way. We reached the other side of the lake and I got a flat tire, which was my second for the day and I was already upset that we lost Kallen. We patched the flat and continued on our ride. Anger was just bottling up inside of me as we entered the next town. When we arrived we sat down to eat. It was about 3 o’clock and we still had seventy miles to go which added to my anger. After lunch we rode over to a park to use the public bathrooms. I take a short cut through a little dirt field. I used the restroom and hopped back on my bike to begin riding again and my tire was flat, again. â€Å"AHHH! †, I screamed. We went and sat on the grass to patch it. My attitude has not only has affected me, but has affected my dad too. â€Å"You better get in a better mood† he said†Ã¢â‚¬ ¦because you are acting like a little baby. You have a choice to make; be in a good mood or a bad one, because we are going to finish this ride whether you want to or not. † I was shocked. My dad had never talked to me like that before. I realized that attitude is everything. We rode into the dark that night till we reached one hundred and twenty miles and the next day we biked the remaining 80 and made it to Capitol Reef. July 19 is my birthday. It had to have been the worst but also the best birthday that I have ever had. Your attitude towards something can change your life. Whether it is as big as saving your life or as small as changing your mood such as finishing a biking trip with your dad. It will stick with you and make you a better person and a happy person. Bottom line: Attitude is Everything.

Management Team Essay

Many scholars believe that â€Å"there is a strong connection between the growth potential of a venture and the quality of its management team†, (Timmons and Spinelli, 2009). Describe what is meant by â€Å"management team† philosophy and attitude of entrepreneurial ventures that will eventually contribute to business venture growth? From Collins dictionaries the management team is defined as â€Å"a team of managers in charge of direction a company, business, etc. Wikipedia define the management team as â€Å"senior management, executive management, or management team is generally a team of individuals at the highest level of organizational†. For general definition, management team is a set of peoples that come from various function or responsibility that responsible to manage the organization. â€Å"there is a strong connection between the growth potential of a venture and the quality of its management team†, (Timmons and Spinelli, 2009). To relate this statement, I had found some statement from (Krishnan et al. 1997, p. 363). Differences between the top management teams on important dimensions such as backgrounds of managers has more potential to create unique value because it makes the combined organization stronger by offsetting weaknesses in both firms, thereby creating or maintaining a competitive advantage I also refer to www. jeffcobizjournal. com that mentioned about â€Å"a bad manager will make bad decisions, will hurt the morale of the employees and your relationship with customers, you can’t afford to keep them around†. This situation can give impact the organization directly, so it is really importance to have good quality of management team in order to ensure growth potential to organization business. This idea clearly directed the right people for the right jobs its crucial during the selecting the management team in the organization. Cited from the Marriott management philosophy â€Å"’A business succeeds not because it is long established or because it is big, but because there are men and women in it who live it, sleep it, dream it, and build great future plans for it. † Robinson Finkelstein, Hambrick, and Cannella (2009: 3) wrote, â€Å"The small group of people at the top of an organization can dramatically affect organizational outcomes. Becker (1964), training and wages for experiences and skilled managers can be seen as a firm’s investment in human capital, expecting to benefit from higher productivity and added economic value. From this statement we can relate the bad impact of the small group of people to it is because of the individual factor also. Research shows that effective communication in a team is a critical factor determining team performance (Hitt, et al. 2006). Robinson et al. found that an entrepreneurial attitude orientation scale significantly differentiated between entrepreneurs and non-entrepreneurs. Therefore, it is mentioned that: attitude towards entrepreneurship is a function of the demographic and psychological characteristics and their interaction. Baum, Locke and Smith (2001) reported significant correlations between self-efficacy and venture growth, Douglas and Shepherd (2005) define entrepreneurial capital as the composite of the individual’s entrepreneurial attitudes and abilities. Entrepreneurial attitudes are those toward autonomy, risk, work, income and (other net) perquisites, while entrepreneurial abilities include opportunity recognition, viability screening, and creative problem solving skills. Hofer and Sandberg (Summer 1987), stated there are three factors have a substantial impact on a new venture’s performance. In order of importance, these factors affecting new venture success are (1) the structure of the industry entered, (2) the new venture’s business strategy, and (3) behavioral characteristics of the entrepreneur. My focus will be on the behavioral characteristics of entrepreneur. Sources from K. Axelton, â€Å"Fever Pitch,† Entrepreneur (December 2004), p. 74; N. L. Torres, â€Å"Think Outside the Box,† Entrepreneur (February 2004), pp. 108–111; A. Pennington, â€Å"Una Cassidy,† Entrepreneur (November 2003), p. 24 found four entrepreneurial characteristics are key to a new venture’s success. Successful entrepreneurs have: 1) the ability to identify potential venture opportunities better than most people. 2) a sense of urgency that makes them action oriented, 3) switch the niche, 4) borrow a business model.

The Between Muslims And Christians - 1141 Words

When a Christian tries to start a conversation with a Muslim, he or she needs to have in mind all of what implies. That includes, a cultural-historical context that has shaped the mindset and worldview of Muslim from the seventh century to the present day. It is necessary to carry on with a lot of sensitivity and compassion to the feelings and prejudices between Muslim and Christians and Christian towards Muslins. The historical relationship between Muslims and Christians through the centuries has not been the best, in certain ways shameful. Christians have not complied with the duty to love their neighbor as themselves, whatever their philosophy or faith. For their part, Muslims have not always had in mind the words of his prophet:†¦show more content†¦In these verses Jesus gave us the key to our attitude and behavior towards Muslims. The Christian model should be best described as Godly, one who is a replica of His love. It is necessary that an absence of any neither dimin ishing nor discourteous words be found when addressing them. In class one Thursday night, one of the students said something that I have not seen that way before. She is an American, and said; â€Å" how can we be patriotic and prideful, and them diminish those who have not been born in the United States. We did not choose to be born here; it was by the grace of God.† I found that to be very profound and true. This comportment will not help when it come to evangelize a non-American. In other words, patriotism does not attract nor convince of the love of God. Lets be fair-minded and consider what would Jesus do if he was in our place, He would love them the way they are. Christian Hybrids have always existed within the Christian Church. These kinds of Christians are those who call themselves free. Not to be confused with those who seek freedom from the bondage of sin, the flesh, the world and the law, which is the Christian freedom found in Christ. In that sense, every true believer is free, although he is a slave of God and servant of others. Paul speaks of this in Romans 6. Liberal Christians and Christians who are free seem to be in the same context; however, there is a difference. The