\r\n\tFinally, I want to emphasize that, in this book, I expect to have excellent contributons on the subjects other than muscle systems, so that the book will be widely read by people interested in non-muscle motile systems as well as by muscle researchers.
",isbn:null,printIsbn:null,pdfIsbn:null,doi:null,price:0,priceEur:null,priceUsd:null,slug:null,numberOfPages:0,isOpenForSubmission:!1,hash:"862ba53997da17b644b918fe44e97d4a",bookSignature:"Emeritus Prof. Haruo Sugi",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/7021.jpg",keywords:"Musculo-skeletal system, Cardio-vascular system, Porter myosins, Cellular transport, Motile systems, cell division, Contractile ring formation, Mitotic apparatus, Ciliary Movement, Flagellar Movement, Amoeboid movement, Novel motile systems",numberOfDownloads:null,numberOfWosCitations:0,numberOfCrossrefCitations:0,numberOfDimensionsCitations:0,numberOfTotalCitations:0,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"August 13th 2018",dateEndSecondStepPublish:"September 3rd 2018",dateEndThirdStepPublish:"November 2nd 2018",dateEndFourthStepPublish:"January 21st 2019",dateEndFifthStepPublish:"March 22nd 2019",remainingDaysToSecondStep:"3 years",secondStepPassed:!0,currentStepOfPublishingProcess:5,editedByType:null,kuFlag:!1,biosketch:null,coeditorOneBiosketch:null,coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"140827",title:"Emeritus Prof.",name:"Haruo",middleName:null,surname:"Sugi",slug:"haruo-sugi",fullName:"Haruo Sugi",profilePictureURL:"https://mts.intechopen.com/storage/users/140827/images/system/140827.jpg",biography:"Haruo Sugi was appointed instructor in the Depertment of Physiology of the University of Tokyoin 1962, and worked at Columbia University and the National Instututes of Health, USA, from 1965 to 1967. 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Venkateswarlu",coverURL:"https://cdn.intechopen.com/books/images_new/371.jpg",editedByType:"Edited by",editors:[{id:"58592",title:"Dr.",name:"Arun",surname:"Shanker",slug:"arun-shanker",fullName:"Arun Shanker"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"72",title:"Ionic Liquids",subtitle:"Theory, Properties, New Approaches",isOpenForSubmission:!1,hash:"d94ffa3cfa10505e3b1d676d46fcd3f5",slug:"ionic-liquids-theory-properties-new-approaches",bookSignature:"Alexander Kokorin",coverURL:"https://cdn.intechopen.com/books/images_new/72.jpg",editedByType:"Edited by",editors:[{id:"19816",title:"Prof.",name:"Alexander",surname:"Kokorin",slug:"alexander-kokorin",fullName:"Alexander Kokorin"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"314",title:"Regenerative Medicine and Tissue Engineering",subtitle:"Cells and Biomaterials",isOpenForSubmission:!1,hash:"bb67e80e480c86bb8315458012d65686",slug:"regenerative-medicine-and-tissue-engineering-cells-and-biomaterials",bookSignature:"Daniel Eberli",coverURL:"https://cdn.intechopen.com/books/images_new/314.jpg",editedByType:"Edited by",editors:[{id:"6495",title:"Dr.",name:"Daniel",surname:"Eberli",slug:"daniel-eberli",fullName:"Daniel Eberli"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"57",title:"Physics and Applications of Graphene",subtitle:"Experiments",isOpenForSubmission:!1,hash:"0e6622a71cf4f02f45bfdd5691e1189a",slug:"physics-and-applications-of-graphene-experiments",bookSignature:"Sergey Mikhailov",coverURL:"https://cdn.intechopen.com/books/images_new/57.jpg",editedByType:"Edited by",editors:[{id:"16042",title:"Dr.",name:"Sergey",surname:"Mikhailov",slug:"sergey-mikhailov",fullName:"Sergey Mikhailov"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"1373",title:"Ionic Liquids",subtitle:"Applications and Perspectives",isOpenForSubmission:!1,hash:"5e9ae5ae9167cde4b344e499a792c41c",slug:"ionic-liquids-applications-and-perspectives",bookSignature:"Alexander Kokorin",coverURL:"https://cdn.intechopen.com/books/images_new/1373.jpg",editedByType:"Edited by",editors:[{id:"19816",title:"Prof.",name:"Alexander",surname:"Kokorin",slug:"alexander-kokorin",fullName:"Alexander Kokorin"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},chapter:{item:{type:"chapter",id:"40414",title:"Breeding Dairy Cows to Reduce Greenhouse Gas Emissions",doi:"10.5772/50395",slug:"breeding-dairy-cows-to-reduce-greenhouse-gas-emissions",body:'The dairy industry has made large advances in efficiencies over the past 60 years as a result of changes in breeding, nutrition and management practices [1]. To meet the growing demand for dairy products, which is projected to continue out to the year 2050 [2], milk production per cow has increased over the last thirty years primarily by genetic selection and better nutrition. Genetic selection has tended to focus on mostly production traits (kilograms milk, kilograms fat and protein) rather than fitness (lameness, mastitis, fertility and lifespan) traits, although most countries now include fitness traits in addition to production traits in modern breeding goals. The Holstein Friesian is a popular breed due to its high genetic potential to produce milk; however it is characterised by having a lower body condition score, and reduced fertility and survival compared to other breeds [3]. Even with these negative attributes, in tandem with efficiencies in production in recent decades have come reductions in greenhouse gas (GHG) emissions and resource inputs per unit product [1, 4, 5], while emissions per unit area have increased.
The main GHGs attributed to livestock systems are methane and nitrous oxide emissions [6]. Due to the variability in lifespan of gases in the atmosphere and the ability of gases to reflect and trap radiant energy, the average potential of a GHG to warm the earth‘s near-surface air is expressed in carbon dioxide equivalents (CO2-eq.) emissions (its global warming potential). Methane and nitrous oxide are capable of trapping about 25 and 298 times more radiant energy respectively, over a 100-year time horizon, than one kilogram of carbon dioxide [6]. The dairy sector’s total CO2-eq. emissions are estimated to be 4% of total global GHG emissions, of which, about half are methane and a third nitrous oxide emissions [2].
The main benefits of selection to improve production efficiencies are by increased productivity and gross efficiency (i.e. the ratio of yield of milk to resource input) by firstly, diluting the maintenance cost of animals in the system and secondly, less animals are required to produce the same amount of product [1, 7]. Studies have found [8, 9] that more energy efficient animals produce less waste in the form of methane and nitrogen excretion per unit product. A study in the UK [4] calculated that the genetic improvement in dairy cows by economic and production efficiency in the last 20 years had reduced GHG emissions per unit product by 0.8% per year and would continue to reduce emissions at a rate of 0.5% per year over the next 15 years. A reduction of 0.6% per year in GHG emissions per unit product was found in the US [1] over a 63 year period. Emissions of methane and nitrous oxide per unit product were estimated to have shown large declines of about 1.3% and 1.5% per year respectively over the last 20 years in the UK, and will continue to decline over the next 15 years albeit at a slightly slower rate per year [4]. These rates of decline for methane are similar to those reported in other studies [10] for enteric methane emissions per unit product of 1.1% per year for cows selected on increased milk fat and protein production (Select line cows) and at 1.4% per year for cows selected to represent the UK average for milk fat and protein production over a similar time period.
In this review, we investigate the potential role of selective breeding in reducing GHG emissions.
Level of feed intake and its composition are important factors influencing methane and nitrogen losses. As the feed intake of an animal increases, the percentage of dietary gross energy (GE) intake lost as methane decreases by an average of 1.6% per unit of intake [11]. A higher feed intake level increases its fractional passage rate through the rumen and reduces its retention time, rumen digestion (depending on the diet) and methane production [11, 12, 13]. As rumen retention time decreases with increased feed intake the rate of nitrogen excretion increases, increasing the potential for nitrous oxide emissions [9, 14]. However, if cows are to meet their genetic potential for milk production, they need to maximise their feed intake [15, 16]. At pasture the nutrient intake can vary and impair the milk production potential of the animal, particularly during the peak of lactation [17]. To meet the genetic potential for milk production forage based diets are supplemented with high energy dense feed in the form of concentrate. Supplementing the diet of high milk yielding dairy cows at pasture with concentrate was found [17] to result in a lower rate of pasture intake substitution and a higher response in improved milk yield compared to lower milk yielding cows.
Cows with a high body weight have been found to have a greater bite weight when eating and therefore are more efficient in their use of time spent feeding [16]. The larger North American Holstein genotype has been found to produce between 8 to 11% less methane as a percentage of GE intake, on both a total mixed ration and pasture-based diet, than a small New Zealand Holstein [18], presumably due to differences in level of feed intake. However, larger cows have greater maintenance requirements. For the same level of production, a smaller cow is obviously a more efficient converter of feed into milk. This is why selection programmes in both New Zealand and Australia, in particular have focused on increasing the rate of genetic gain in traits that contribute to profitability per unit of feed eaten [19].
Selecting dairy animals for efficient feed use could bring both higher production and reduced resource requirements. In comparison to other mitigation strategies, selective breeding offers a medium to long-term approach to GHG mitigation, which can be cost effective [20]. The response from selective breeding depends on the selection intensity, genetic variation, generation interval and the economic importance of the trait, with annual rates of response typically being between 1% and 3% [7] of the mean in the trait under selection. In intensive poultry and pig production, profitability on cereal-based diets has encouraged selection for feed efficiency (ranging from 1.7 to 2.4 kg cereal feed per kg animal weight) compared to ruminant systems [21]. The improvement in feed conversion efficiency in non-ruminants has been quite remarkable, for example, it has been reported [22] that the feed conversion efficiency (kg lean meat/t of feed) in pigs has nearly doubled from 85 kg/t in the 1960s to 12 170 kg/t in 2005. It is reasonable to assume that in non-ruminant animals where selection on feed use efficiency has been made that current selection goals will account for a moderate to high proportion of any genetic variation in methane output or nitrogen use efficiency.
Average daily dry matter intake and milk yield are moderately heritable in dairy cows at about 0.30 [15, 23] compared to about 0 to 0.15 for health and fertility traits [24]. Therefore, genetic improvement in dry matter intake and milk production traits is easier to achieve than for health and fertility. Increasing the genetic potential of a cow to produce milk increases total system GHG emissions, due a higher feed intake [25]. However, the milk produced per unit of feed eaten is likely to reduce due to improvements in gross efficiency. For example, the genetic correlation between intake and milk yield has been found [15] to account for just less than half the genetic improvement in milk production being covered by an increase in dry matter intake. This implies that the apparent improvement in gross efficiency is partly due to feed conversion efficiency. The remainder could be due to increased reliance of body tissue mobilisation.
The direct selection of animals on a trait such as methane production in ruminants may be of little importance given its relationship with feed intake [26, 27], which is a more easily measured trait. The additional benefit from directly measuring GHG emissions from animals would be if selection on a measure of feed use efficiency was not possible. Whether breeding goals are able to account for all the genetic variation in methane output or nitrogen efficiency is unlikely, and therefore there may be some benefit to directly selecting on these traits if possible. In ruminants, measurements of methane output, nitrogen efficiency and overall feed efficiency are difficult and costly to obtain, which has limited the direct selection of these traits in the past. A large part of the variation in methane emissions from dairy cows has been found to be genetic, with a heritability of 0.35 for methane output and 0.58 for methane output per unit product [28], presumably due to a prediction of methane being used and its close unity correlation with feed intake. In comparison, a lower heritability of 0.13 has been found [29] in sheep for methane output. Once individual measurements for total animal methane emissions become more affordable to carry out for a large number of animals, selecting animals on methane output will become possible.
Variation in feed use efficiency and enteric methane emissions between-animals, breeds and over time means there is potential to reduce GHG emissions through genetic selection [30, 31, 32]. With a positive genetic correlation between feed efficiency and methane output, with an estimated range from 0.18 to 0.84 [28], it can be inferred that selecting cows that are more efficient will reduce methane production, possibly in the order of 1.1% to 2.6% per year. From a range of production and fitness traits, breeding studies [4, 14] found feed efficiency to have a large impact on reducing the GHG emissions from dairy systems compared to other production or fitness traits. Feed efficiency can be assessed by feed intake required per unit product (gross efficiency) or by net or metabolic efficiency commonly calculated as residual feed intake [4]. Residual feed intake is the difference between the observed and predicted feed intake; where the predicted feed intake is often calculated as energy requirements. Studies [31, 33, 34] looking at selecting beef cattle based on a lower residual feed intake (difference between actual and expected feed intake) found that growth performance was not compromised and the lower expected feed intake resulted in less methane produced. Heritability estimates for feed efficiency tend to be moderate (0.16 to 0.46; [4]). Low correlations between residual feed intake and other production traits imply that little or no genetic improvement has previously been made in residual intake in beef cattle as a result of selection on production traits [4]. In dairy cows, calculating residual feed intake accurately is difficult as changes in body tissue composition need to be fully accounted for. This is because without accounting for body composition changes, residual feed intake is mathematically equivalent to energy balance [35]. Negative energy balance (often considered to be very similar to condition score loss) in early lactation has been the subject of intense phenotypic (and genetic) investigation [36]. Mobilisation of body tissue, or low body condition score is associated with reductions in fertility [37].
Taking direct feed intake measurements can be costly due to the equipment required, therefore an indirect measure of feed efficiency may be a more appropriate option for dairy animals, but further research is required to investigate measures that might be correlated with intake. Biologically inactive markers released in the rumen (such as
Recently, there has been interest in estimating genomic breeding values for traits in the feed conversion efficiency complex. The estimated accuracy of genomic prediction of RFI calculated in a population of 1000 New Zealand and 1000 Australian non-lactating heifers was around 0.4 [39]. The accuracy of the genomic prediction could be increased further still if countries were to pool together their phenotypes that are expensive to record and their genotype resources, as these data could be used to increase the accuracy of genomic selection further still. Collaborative efforts between research organizations in the Netherlands, the UK and Australia have already demonstrated that the accuracy of genomic predictions of dry matter intake can be increased by combining datasets (de Haas submitted, 2012). The ultimate aim of these collaborative research efforts is to develop genomic breeding values for dry matter intake or a feed conversion efficiency trait that could be used in breeding programs to improve efficiency and mitigate emissions.
Due to the profitability of Holstein cows, Holstein genes are present in a large proportion of dairy cows globally, particularly North American. Larger North American Holstein-Friesian cows have been found to show a better response in milk yield with a higher proportion of concentrate in their diet than smaller genotypes like the New Zealand Holstein-Friesian, which have been selected for higher milk yield performance from pasture [16]. Cows which were ~88% North American Holstein and selected on increased milk fat and protein production (Select line cows) were found to grow faster and had increased kg milk per kg dry matter intake during their productive life when on a high energy dense diet, compared to cows selected to represent the UK average for milk fat and protein production on the same diet [10]. Select genetic line animals have a high genetic potential for mobilising body energy reserves for production, which has been found to have deleterious effects on health and fertility [3, 24], particularly later in life [7]. However, it was found [14] that Select line cows responded to a diet containing a low proportion of forage, rather than a high forage diet, by having a significantly shorter calving interval. Select line animals on a low forage diet also produced lower CO2-eq. emissions per energy corrected milk compared to non-select and cows on a high forage diet over their lifetime (Figure 1).
Systems emissions can be minimised by improvements in herd health and fertility (improving longevity and productivity), and by reducing the number of replacement animals retained on the farm to reduce wastage [13, 40, 41]. Improving herd fertility in the UK back to 1995 levels could amount to a 24% reduction in methane emissions per cow by improved efficiencies of reduced herd replacements and calving interval length [40]. Cows of predominantly North American Holstein genes may be better suited to a high energy dense feeding system, typically found in the US, rather than a diet containing a high proportion of forage. In contrast, the performance of animals of New Zealand origin had higher yields of milk solids and better fertility compared to animals of North American origin when compared on a range of New Zealand grazing systems [42]. Therefore, selecting animals for an environment is important. In a study in the US [1], good health and welfare in modern high input systems (cows of 90% Holstein genes) was reported, with better production efficiency and CO2-eq. emissions per unit product compared to the past. This may be explained by optimal nutrition being provided to these animals, which may not hold true for the same cows on a lower quality forage diet.
Carbon dioxide equivalent (CO2-eq.) emissions per kg energy corrected milk (ECM) for cows selected for increased milk fat and protein fed a low proportion of forage (●) and a high proportion (×) of forage in their diet and cows selected to represent the average for milk fat and protein production fed a low proportion of forage (■) and a high proportion (▲) of forage in their diet (from [
To bring about reductions in livestock GHG emissions, it has been suggested [21] that significant technological innovations will be required in the future, in addition to managing our consumption of meat and milk products. Technologies that can bring affordable efficiencies to production are being developed. Using genomic information, such as through genomic breeding values for feed related traits, described previously, and sexed semen [43] offer the potential for better selective breeding.
Reductions in GHG emissions by genetic selection of dairy cows in the past has been achieved largely by increased productivity and gross efficiency, whereby the maintenance cost of animals in the system has been reduced and less animals are required to produce the same amount of product. Based on current breeding goals, a similar rate of reduction in emissions intensity can be expected in the near future. Selecting dairy cows on feed efficiency, and possibly methane and nitrogen losses, will have a large impact on the environmental footprint of milk production, once implemented in breeding schemes. Further research and development of novel technologies to better understand the physiological and genetic differences between animals that lead to differences in energy and nitrogen efficiencies (or overall feed use efficiency) are still required.
Dairy farming is a highly managed system and has the potential to make reductions in GHG emissions intensity though increased efficiencies, such as optimum animal performance (less non-productive and ill animals) and reduced inputs as shown in Figure 2, whilst still maintaining productivity.
Production efficiencies using management (---) that can reduce GHG emissions beginning with selective breeding of a genotype for a particular system (from [
Genetic improvement is a relatively cost-effective mechanism by which to achieve reductions in emissions, as the effect is cumulative and permanent. Many of the strategies for minimising emissions bring economic benefits to dairy farming through a reduction in production costs, with the added benefit of also reducing GHG emissions at little extra cost. The success of selective breeding as a mitigation strategy however, is dependent on producers being committed to its implementation.
Until direct measurements on GHG traits become available on sufficiently large numbers of animals, selecting for improved efficiency, RFI, or a measure of gross efficiency (unit of output per unit of feed eaten) offer attractive ways of reducing GHG emissions. However, the direct impact on GHG emissions is currently approximate and more research is needed to accurately assess the impact. Furthermore, selection strategies in dairy cattle need to be considered in a multi-trait framework, so that unfavourable correlated responses to selection (such as reduced fertility and excessive mobilisation of body reserves) are avoided.
This work was supported by funding from Dairy Australia, Meat and Livestock Australia and the Australian Government Department of Agriculture, Fisheries and Forestry under its Australia’s Farming Future Climate Change Research Program. Research on genomic breeding values for feed conversion efficiency was funded by the Dairy Future’s CRC and Geoffrey Gardiner Foundation.
The descriptions of the behaviors of the real world phenomena and systems through the use of mathematical models often involve developments of nonlinear equations which are difficult to solve exactly and analytically. Consequently, recourse is always made to numerical methods as alternative methods in solving the nonlinear equations. However, the developments of analytical solutions are obviously still very important. Analytical solutions for specified problems are also essential and required to show the direct relationship between the models parameters. When analytical solutions are available, they provide good insights into the significance of various system parameters affecting the phenomena. Such solutions provide continuous physical insights than pure numerical or computation methods. Indisputably, analytical solutions are convenient for parametric studies, accounting for the physics of the problem and appear more appealing than the numerical solutions. Also, they help in reducing the computation and simulation costs as well as the task involved in the analysis of real-life problems.
Although, there is no general exact analytical method to solve all nonlinear problems, over the years, the nonlinear problems have been solved using different approximate analytical methods such as regular perturbation, singular perturbation method, homotopy perturbation method, homotopy analysis method, methods of weighted residual, variational iterative method, differential transformation method, variation parameter method, Adomian decomposition method, etc. The non-perturbative approximate analytic methods present explicit approximate analytical solutions which often involve complex mathematical analysis leading to analytic expressions involving large number terms. Furthermore, the methods are inherently with high computational cost and time accompanied with the requirement of high skills in mathematics. Moreover, in practice, analytical solutions with large number of terms and conditional statements for the solutions are not convenient for use by designers and engineers. Also, in these methods, there are always search for particular value(s) that will satisfy the end boundary condition(s). This always necessitates the use of software and such could result in additional computational cost in the generation of solution to the problem. Also, the quests involve applications of numerical schemes to determine the required value(s) that will satisfy the end boundary condition(s). This fact renders most of the approximate analytical methods to be taken as more of semi-analytical methods than total approximate analytical methods. Moreover, these methods have their own operational restrictions that severely narrow their functioning domain and when they are routinely implemented, they can sometimes lead to erroneous results. Specifically, the transformation of the nonlinear equations and the development of equivalent recurrence equations for the nonlinear equations using differential transformation method proved somehow difficult in some nonlinear system such as in rational Duffing oscillator, irrational nonlinear Duffing oscillator, finite extensibility nonlinear oscillator. There is difficulty in the determination of Adomian polynomials for the application of Adomian decomposition method for nonlinear problems. There are lack of rigorous theories or proper guidance for choosing initial approximation, auxiliary linear operators, auxiliary functions, and auxiliary parameters in the use of homotopy analysis method. Therefore, the need for comparatively simple, flexible, generic and high accurate total approximate analytical solutions is well established. One of the techniques that can be applied for such quest is the perturbation method. Perturbation method, although comparably old, as a pioneer method for finding approximate analytical solutions to nonlinear problems, it offers an alternative approach to solving certain types of nonlinear problems. In the limit of small parameter, perturbation method is widely used for solving many heat transfer, vibration, fluid mechanics and solid mechanics problems. It is capable of solving nonlinear, inhomogeneous and multidimensional problems with reasonable high level of accuracy. The most significant efforts and applications of the method were focused on celestial mechanics, fluid mechanics, and aerodynamics. Although, the solutions reported for other sophisticated methods to difference problems have good accuracy, they are more complicated for applications than perturbation method. Therefore, over the years, the relative simplicity and high accuracy especially in the limit of small parameter have made perturbation method an interesting tool among the most frequently used approximate analytical methods. Although, the perturbation method provides in general, better results for small perturbation parameters, besides having a handy mathematical formulation, it has been shown to have a good accuracy, even for relatively large values of the perturbation parameter [1, 2, 3, 4, 5].
Consider a convective-radiative fin of temperature-dependent thermal conductivity
The heat flow in the fin and its temperatures remain constant with time.
The temperature of the medium surrounding the fin is uniform.
The temperature of the base of the fin is uniform.
The fin thickness is small compared with its width and length, so that temperature gradients across the fin thickness and heat transfer from the edges of the fin is negligible compared with the heat leaving its lateral surface.
(a) Schematic of the convective-radiative longitudinal straight fin with magnetic field. (b) Schematic of the longitudinal straight fin geometry showing thermal contact resistance and boundary conditions.
Applying thermal energy balance on the fin and using the above model assumptions, the following nonlinear thermal model is developed
The boundary conditions are
Considering a case when a small temperature difference exists within the material during the heat flow. This actually necessitated the use of temperature-invariant physical and thermal properties of the fin. Also, it has been established that under such scenario, the term
On substituting Eq. (4) into Eq. (1), one arrives arrived at
The boundary conditions
On introducing the following dimensionless parameters in Eq. (8) into Eq. (5),
The dimensionless form of the governing Eq. (5) is arrived at as
On expanding Eq. (9), one has
The boundary conditions are
It is very difficult to develop closed-form solution for the above non-linear Eq. (10). Therefore, in this work, recourse is made to apply a relatively simple and accurate method approximate analytical method, the perturbation method. Perturbation theory is based on the fact that the equation(s) describing the phenomena or process under investigation contain(s) a small parameter (or several small parameters), explicitly or implicitly. Therefore, the perturbation method is applicable to very small magnitudes of ε where the nonlinearity is slightly effective. Although, it has been shown to have a good accuracy, even for relatively large values of the perturbation parameter,
In solving Eq. (10), one needs to expand the dimensionless temperature as
Substituting Eq. (13) into Eq. (10), up to first order approximate, we have
Leading order and first order equations with the appropriate boundary conditions are given as:
Subject to:
Subject to:
The boundary conditions
It can be shown from Eq. (15), (18) and (21) with the corresponding boundary conditions of Eqs. (16), (19) and (22) that the:
Leading order solution for θo is
While the first order solution θ1 is
The second-order solution θ2 is too huge to be included in the manuscript.
On substituting Eqs. (24) and (25) into Eq. (13) up to the first order (i.e. neglecting the higher orders), one arrives at
Consider a Casson nanofluid flowing between two parallel plates placed at time-variant distance and under the influence of magnetic field as shown in the Figure 2. It is assumed that the flow of the nanofluid is laminar, stable, incompressible, isothermal, non-reacting chemically, the nanoparticles and base fluid are in thermal equilibrium and the physical properties are constant. The fluid conducts electrical energy as it flows unsteadily under magnetic force field. The fluid structure is everywhere in thermodynamic equilibrium and the plate is maintained at constant temperature.
Model diagram of MHD squeezing flow of nanofluid between two parallel plates embedded in a porous medium.
Following the assumptions, the governing equations for the flow are given as
where
and the magnetic field parameter is given as
The Casson fluid parameter,
The radiation term is given as
The appropriate boundary conditions are given as
On introducing the following dimensionless and similarity variables
One arrives at the dimensionless equations
with the boundary conditions as follows
where
The comparative advantages and the provision of acceptable analytical results with convenient convergence and stability coupled with total analytic procedures of homotopy perturbation method compel us to consider the method for solving the system of nonlinear differential equations in Eqs. (40) and (41) with the boundary conditions in Eq. (42).
In order to establish the basic idea behind homotopy perturbation method, consider a system of nonlinear differential equations given as
with the boundary conditions
where
The operator
By the homotopy technique, a homotopy
or
In the above Eqs. (47) and (48),
Also, from Eq. (47) and Eq. (48), one has
or
The changing process of
It should be pointed out that of all the values of
The basic idea expressed above is a combination of homotopy and perturbation method. Hence, the method is called homotopy perturbation method (HPM), which has eliminated the limitations of the traditional perturbation methods. On the other hand, this technique can have full advantages of the traditional perturbation techniques. The series Eq. (29) is convergent for most cases.
According to homotopy perturbation method (HPM), one can construct an homotopy for Eq. (36)–(39) as
Taking power series of velocity, temperature and concentration fields, gives
and
Substituting Eqs. (55) and (56) into Eq. (53) and (54) as well as the boundary conditions in Eq. (42), and grouping like terms based on the power of
the boundary conditions are
In a similar way, the higher orders problems are obtained.
On solving Eqs. (57), (61) and (64) with their corresponding boundary conditions, we arrived at
In the same manner, the energy equations are solved. Following the definition of the homotopy perturbation method as presented in Eq. (52), one could write the solution of the fluid flow equation as
Consider a nanobeam embedded in linear and nonlinear elastic media as shown in Figure 3. The nanobeam is subjected to stretching effects and resting on Winkler, Pasternak and nonlinear elastic media in a thermo-magnetic environment as depicted in the figure.
A nanobeam embedded in linear and nonlinear elastic media (note: Only the bottom side of the elastic media is shown).
Following the nonlocal theory and Euler-Bernoulli theorem, the governing equation of the structure is developed as
It is assumed that the midpoint of the nanobeam is subjected to the following initial conditions
The following boundary conditions for the multi-walled nanotubes for simply supported nanotube is given,
Using the following adimensional constants and variables
The adimensional form of the governing equation of motion for the nanobeam is given as
And the boundary conditions become
The method of solution for the governing equation includes Galerkin decomposition and homotopy perturbation methods. As the name implies the Galerkin decomposition method is used to decompose the governing partial differential equation of motion can be separated into spatial and temporal parts. The resulting temporal equations are solved using homotopy perturbation method.
The procedures for the analysis of the equations are given in the proceeding sections as follows:
With the application of Galerkin decomposition procedure, the governing partial differential equations of motion can be separated into spatial and temporal parts of the lateral displacement function as
Using one-parameter Galerkin decomposition procedure, one arrives at
where
where
Substituting Eqs. (75) into (74), then multiplying both sides of the resulting equation by
where
The initial conditions are given as
A is the maximum vibration amplitude of the structure.
From the initial conditions in Eq. (83), one can write the initial approximation,
Eq. (22) satisfies the initial conditions in Eq. (83).
The homotopy perturbation representation of Eq. (77) is
From the procedure of homotopy perturbation method, assuming that the solution of Eq. (77) takes the form of:
On substituting Eqs. (86) into the homotopy Eq. (85)
rearranging the coefficients of the terms with identical powers of
with the conditions
with corresponding initial conditions
with corresponding initial conditions
The solution of the zero-order is given by.
From Eq. (27), we have
On substituting Eq. (94) into Eq. (90) and using trigonometric identities, after the colllection of like terms, one arrives at
The solution of the above Eq. (95) provides
Based on the procedure of HPM, setting
On substituting Eqs. (94) and (96) into Eq. (97), the result is
In order to find the natural frequency,
After simplification of Eq. (99), we have
The sextic equation can be written as
Eq. (101) can be written as
where
The roots of the sextic equation are
In this chapter, the applications of regular and homotopy perturbation methods to thermal, fluid flow and dynamic behaviors of engineering systems have been presented. Regular perturbation was used in the first example to developed approximate analytical solutions for thermal behavior of convective-radiative fin with end cooling and thermal contact resistance. In the second example, homotopy perturbation method utilized to study squeezing flow and heat transfer of Casson nanofluid between two parallel plates embedded in a porous medium under the influences of slip, Lorentz force, viscous dissipation and thermal radiation. The same method was used in the third example to analyze the dynamic behavior of piezoelectric nanobeam embedded in linear and nonlinear elastic foundations operating in a thermal-magnetic environment. It is hoped that the vivid presentation and applications of these perturbation methods in this chapter will advance better understanding of methods especially for real world applications.
IntechOpen - where academia and industry create content with global impact
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\n\nSara Uhac, COO
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\n\nDr Alex Lazinica
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