By Henrik Bruus

Microfluidics is a tender and swiftly increasing clinical self-discipline, which offers with fluids and ideas in miniaturized structures, the so-called lab-on-a-chip platforms. It has purposes in chemical engineering, pharmaceutics, biotechnology and medication. because the lab-on-a-chip platforms develop in complexity, a formal theoretical figuring out turns into more and more important.The uncomplicated notion of the booklet is to supply a self-contained formula of the theoretical framework of microfluidics, and while supply actual motivation and instance from lab-on-a-chip know-how. After 3 chapters introducing microfluidics, the governing questions for mass, momentum and effort, and a few uncomplicated movement ideas, the next 14 chapters deal with hydraulic resistance/compliance, diffusion/dispersion, time-dependent stream, capillarity, electro-and magneto-hydydrodynamics, thermal delivery, two-phase stream, complicated circulate styles and acousto-fluidics, in addition to the recent fields of opto-and nano-fluidics. during the booklet basic versions with analytical ideas are awarded to supply the coed with an intensive actual figuring out of order of magnitudes and diverse chosen micorfluidic phenomena and devices.The ebook grew out of a collection of well-tested lecture notes. it truly is with its many pedagogical routines designed as a textbook for a complicated undergraduate or first-year graduate direction. it's also well matched for self-study.

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**Example text**

55) The initial condition is given by x(0) = and ∂t x(0) = 0. We study the solution of this problem using perturbation theory using the damping as the perturbation. The unperturbed oscillator has γ = 0 and the solution x0 (t) = cos(ω0 t) with ω0 ≡ k/m. (a) Introduce the following dimensionless variable x ˜ and t˜ by the deﬁnitions x ≡ x ˜ and t ≡ t˜/ω0 , and let α = γ/ω0 be the dimensionless perturbation parameter. Calculate the ˜1 . ﬁrst-order perturbation result x ˜=x ˜0 + α x (b) Find the exact solution using a trial solution of the complex form x ˜ = exp(iβ t˜), and compare a ﬁrst-order expansion in α of the result with the ﬁrst-order perturbation result.

58) is indeed a heat-transfer equation, it is customary to rewrite it in terms of the entropy s per unit mass times ρT . 8) and the equation of motion Eq. 27) have been used to rewrite ∂t ρ and ∂t vj , respectively. The last term ρ∂t ε can be rewritten by using the ﬁrst law of thermodynamics, Eq. 50), and thereby bringing the entropy s into play ρ∂t ε = ρT ∂t s + p p ∂t ρ = ρT ∂t s − ∂j (ρvj ). 61) Likewise, the third term containing vj ∂j p can also be rewritten by use of the ﬁrst law, d(ε + p/ρ) = [T ds − pd(1/ρ)] + [pd(1/ρ) + (1/ρ)dp] = T ds + (1/ρ)dp, from which follows −vj ∂j p = −ρvj ∂j ε + p + ρT vj ∂j s.

A ﬂuid where the density ρ may vary as function of space and time. Consider an arbitrarily shaped, but ﬁxed, region Ω in the ﬂuid as sketched in Fig. 1. The total mass M (Ω, t) inside Ω can be expressed as a volume integral over the density ρ, dr ρ(r, t). 2) Ω Since mass can neither appear nor disappear spontaneously in non-relativistic mechanics, M (Ω, t) can only vary due to a mass ﬂux through the surface ∂ Ω of the region Ω. 3) where v is the Eulerian velocity ﬁeld. Since the region Ω is ﬁxed the time derivative of the mass M (Ω, t) can be calculated either by diﬀerentiating the volume integral Eq.