Riassunti Biomatematical modeling

Definition 0.2. Given

Mof X. For simplicity, we will also suppose that X is also a metric space, so that balls centered inx with radius r are well-defined and belong to . T is endowed with a total ordering compatibleMR,with the one of that is t < t if and only if t t > 0; in this way all intervals of T are1 2 2 1well defined. 2X will be called material manifold, T the time manifold, µ a motion and M a subbody.MSThe triple (T, X, ) will be called a continuous body and B = the material shape of theM M{P 2body. The set (x) = : P = µ(t, x) : t T (or an interval in T )} is called the trajectory of{P 2 }x, while the set µ(M ) = : P = µ(t, x) : x M is called (space) shape of M .E E3 3⇥ ⇥ !Finally, the map T T given by 1 3p = (t , t , P ) = µ(t , µ (t , P )) ( )1 2 2 1is called deformation.If t is held fixed and t = t is variable, we will also use the notation p = (t, P ) dropping t1 2 1and if also t is understood we could write p = (P ). E 32 F(t

operation of “di↵erence” such that t t is a1 2E E1 12real number. In this way does not have a preferred origin. If an instant t is fixed, then the set of all0E R.12numbers t t with t is isomorphic to02 E E E R3 1 3 32 2is the same as but if P, Q , then P Q . Here too, if a point O is fixed, then the setE R3 3{P 2 }O : P is isomorphic to .3 1 1·)] ·)µ (t , P ) means [µ(t , evaluated at P . Since µ(t, is injective, its inverse is well-defined.1 14 It is an useful exact form of an “infinitesimal of order greater than one”. Indeed,o(||P Q||)lim = lim !(P, Q) = 0||P Q||!Q !QP Pbut here there’s no need for denominators which reduce the domain of validity of the formula.12 Di↵erentiating (1) with respect to time and setting@V (t, P ) = (t, P )@tone has 5Ḟ(t,V (t, P ) V (t, Q) = Q)(P Q) + õ(||P Q||). ( )Definition 0.3. Let be a deformation. The vector field1v(t, p) = V (t, (t, p))E 32is called velocity at the point

It corresponds to the velocity V of the material particle passing through p.

Inverting (1), one has easily 1FP Q = (p q) + o(||p q||) so that 1ḞFv(t, p) v(t, q) = (p q) + o(||p q||) whence it is immediate to see that 1 1 1Ḟ(t,grad v(t, q) = (t, q))F (t, (t, q)) or simply, when the dependencies are understood, 1ḞF(2) grad v = .E 3⇥X

As a map between the manifolds T and , the map µ has a di↵erential dµ acting between the corresponding tangent spaces. We are interested in the part µ̇ of this di↵erential which in the T direction, that is, corresponding to the same material point. This operator generates the⇥rate of change of a quantity defined on T X with values on a metric space Y (which canR R 3be for scalar fields like the temperature, for vector fields, and so on). For example, for a1 02scalar quantity, at time t and for a fixed x X, one has (t, x) = (t, µ (t , P )) = (t, P ).

The function is called Lagrangian form of , and therefored ˙0 0(t,

P ) = (t, P )dt is the rate of change of “with respect to the reference coordinates”. On the other side, if one wants to compute derivatives at time t, the quantity to derive (the so-called Eulerian form of) is 1(t, p) = (t, µ (t, p)) 1 1 1·) ·))) and therefore the operator is di↵erent. It is not difficult to see (since µ (t, = µ (t , (t,1 that the corresponding derivative is @ ·+ v grad@t which is called material derivative and it is written d /dt when there is no danger of confusion, or D /Dt. If = v then its material derivative is the acceleration field a.

We state now two fundamental theorems (not really physical, but analytical).

F.Theorem 0.1 (Euler’s Formula). Let be a deformation with gradient Thend F(t, F(t,det Q) = det Q) div v(t, (t, Q)).dt5 Of course the time derivative of a “space infinitesimal quantity” is still a space infinitesimal quantity. 3The proof is based on the fact that by a result of linear

Algebrad 1F(t) F(t) Ḟ(t)F(t)det = det tr( )dtand the use of (2). !Let Y be a metric space ad : X Y be a integrable function. The structure of subbodiesM of X allows the definition of an integralZ dxM2 ·)such that for every t T there exists a density (t, such thatZ Z n·)dx = (t, dLM µ(t,M)Rn nwhere indicates the Lebesgue measure on .

L E 3M M)If is a subbody of X, then its shape M = µ(t, is a subset of . We will call also Ma subbody of the continuous body, even if it is referred to a given time. !Theorem 0.2 (Reynolds’ Theorem). Let Y be a metric space, : X Y be a integrablefunction and let a deformation. Then ✓ ◆Z Zd Dn ndL = + div v dL .dt Dt(t,M ) (t,M )The proof of this theorem follows easily by a change of variable p = (t, P ) whose jacobianF:is det ✓ ◆ ✓ ◆Z Z Z Z0d d dJ @ Dn n 0 n ndL = J( ) dL = + J dL = + J div v dLdt dt dt @t Dt(t,M ) M M (t,M )and observing that ✓ ◆D @( )= .Dt @t 1|P = (t,p)By the well-known formula

set-theoretic properties, that the applicatione7!k : M H(M, M ) is additive, i.e. ?[ \k(M N ) = k(M ) + k(N ) if M N =and therefore it is likely to happen that h will behave as an integral.

It is now fundamental to link the heat and power expended with the first law ofthermodynamics.

Axiom 0.1 (First Law of Thermodynamics). There exists an additive quantity E :R,! called internal energy, such that for all tM d e (i)E(M )(t) = H(M, M ) P (M, v(t))dtwhere v denotes the current real (not virtual) velocity.

Supposing that E is an integral and ⇢ be the mass density, we can write

Z Z Z Zd n n n @M n· ·T D⇢e dL = h(x, y) dL (x)dL (y) q n dLdt e⇥MM M @M Me(remember that M and M share the same boundary) and setting

Z n⇢(x)r(x) = h(x, y) dL (y)eM1 e( ) That is, union, intersection and complement M . This is not trivial since these operations must bee \internal: for example, the complement of an open set is not open, and therefore M must be int (B M ).

Another requirement is that

the boundary @M must admit almost everywhere an outer exterior normalMn and this complicates the situation. There are, however, suitable definitions of the three operationswhose analysis would lead us too far. 12where ⇢ is the mass density, one has easilyZ Z Zn n nh(x, y) dL (x)dL (y) = ⇢(x)r(x) dL (x)eM M Mand therefore Z Z Z Zd n n @M n· ·T D(0.1) ⇢e dL = ⇢r dL (x) q n + dL .dt M M @M MSetting now w = v into d’Alembert Principle, that we recallZ Zn n 1· ·T)(0.2) ( ṗ ⇢b r̃ v div w dL = (Tn t) w dHgM @Mwe get Z Zn n 1· ·T)( ṗ ⇢b r̃ v div v dL (Tn t) v dH = 0.gM @MBut clearly 2||v||d d2· ·ṗ w = (⇢v) v = ⇢v˙ + ⇢dt dt 2and then, using the formula T · ·C Cdiv(C w) = div w + grad wit is not difficult to getZ Z Z2||v||d d n 2 n· ||v|| · ·T D)(0.3) ⇢ dL = (⇢b v r dL + t v dSgdt dt 2M M @Mwhere r = r̃ + ⇢ div v.g gSumming equations (0.1) and (0.3) it is immediate